Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom 2020940418, 9780691202532, 9780691202525, 9780691204932

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Annals of Mathematics Studies Number 208

Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom

Vadim Kaloshin Ke Zhang

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2020

c 2020 Princeton University Press

Requests for permission to reproduce material from this work should be sent to Permissions, Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 6 Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu All Rights Reserved Library of Congress Control Number: 2020940418 ISBN: 9780691202532 ISBN (pbk.): 9780691202525 ISBN (e-book): 9780691204932 British Library Cataloging-in-Publication Data is available Editorial: Susannah Shoemaker and Kristen Hop Production Editorial: Nathan Carr Cover Design: Leslie Flis Production: Brigid Ackerman Publicity: Matthew Taylor and Katie Lewis Copyeditor: Theresa Kornak The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed This book has been composed in LaTeX Printed on acid-free paper. Printed in the United States of America 10

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Dedicated to the memory of John Mather, a great mathematician and a remarkable person

Contents

Preface

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Acknowledgments

xiii

I Introduction and the general scheme 1 Introduction 1.1 Statement of the result . . 1.2 Scheme of diffusion . . . . 1.3 Three regimes of diffusion 1.4 The outline of the proof . 1.5 Discussion . . . . . . . . .

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2 Forcing relation 2.1 Sufficient condition for Arnold diffusion . . . . . . 2.2 Diffusion mechanisms via forcing equivalence . . . 2.3 Invariance under the symplectic coordinate changes 2.4 Normal hyperbolicity and Aubry-Mather type . . .

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3 Normal forms and cohomology classes at single resonances 24 3.1 Resonant component and non-degeneracy conditions . . . . . . . 24 3.2 Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 The resonant component . . . . . . . . . . . . . . . . . . . . . . . 29 4 Double resonance: geometric description 4.1 The slow system . . . . . . . . . . . . . . . . . 4.2 Non-degeneracy conditions for the slow system 4.3 Normally hyperbolic cylinders . . . . . . . . . . 4.4 Local maps and global maps . . . . . . . . . . .

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5 Double resonance: forcing equivalence 5.1 Choice of cohomologies for the slow system . . . 5.2 Aubry-Mather type at a double resonance . . . . 5.3 Connecting to Γk1 ,k2 and ΓSR k1 . . . . . . . . . . . 5.3.1 Connecting to the double resonance point 5.3.2 Connecting single and double resonance .

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viii

CONTENTS

5.4 5.5

Jump from non-simple homology to simple homology . . . . . . . Forcing equivalence at the double resonance . . . . . . . . . . . .

II Forcing relation and Aubry-Mather type

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6 Weak KAM theory and forcing equivalence 6.1 Periodic Tonelli Hamiltonians . . . . . . . . . . . . 6.2 Weak KAM solution . . . . . . . . . . . . . . . . . 6.3 Pseudographs, Aubry, Ma˜ n´e, and Mather sets . . . 6.4 The dual setting, forward solutions . . . . . . . . . 6.5 Peierls barrier, static classes, elementary solutions 6.6 The forcing relation . . . . . . . . . . . . . . . . . 6.7 The Green bundles . . . . . . . . . . . . . . . . . .

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7 Perturbative weak KAM theory 7.1 Semi-continuity . . . . . . . . . . . . . . . . . . . 7.2 Continuity of the barrier function . . . . . . . . . 7.3 Lipschitz estimates for nearly integrable systems 7.4 Estimates for nearly autonomous systems . . . .

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8 Cohomology of Aubry-Mather type 8.1 Aubry-Mather type and diffusion mechanisms 8.2 Weak KAM solutions are unstable manifolds 8.3 Regularity of the barrier functions . . . . . . 8.4 Bifurcation type . . . . . . . . . . . . . . . .

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III Proving forcing equivalence 9 Aubry-Mather type at the single resonance 9.1 The single maximum case . . . . . . . . . . . 9.2 Aubry-Mather type at single resonance . . . . 9.3 Bifurcations in the double maxima case . . . 9.4 Hyperbolic coordinates . . . . . . . . . . . . . 9.5 Normally hyperbolic invariant cylinder . . . . 9.6 Localization of the Aubry and Ma˜ n´e sets . . . 9.7 Genericity of the single-resonance conditions .

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10 Normally hyperbolic cylinders at double resonance 10.1 Normal form near the hyperbolic fixed point . . . . . 10.2 Shil’nikov’s boundary value problem . . . . . . . . . 10.3 Properties of the local maps . . . . . . . . . . . . . . 10.4 Periodic orbits for the local and global maps . . . . 10.5 Normally hyperbolic invariant manifolds . . . . . . . 10.6 Cyclic concatenations of simple geodesics . . . . . .

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CONTENTS

11 Aubry-Mather type at the double resonance 11.1 High-energy case . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Simple non-critical case . . . . . . . . . . . . . . . . . . . . 11.3 Simple critical case . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Proof of Aubry-Mather type using local coordinates 11.3.2 Construction of the local coordinates . . . . . . . . .

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12 Forcing equivalence between kissing cylinders 12.1 Variational problem for the slow mechanical system . 12.2 Variational problem for original coordinates . . . . . 12.3 Scaling limit of the barrier function . . . . . . . . . . 12.4 The jump mechanism . . . . . . . . . . . . . . . . .

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IV Supplementary topics

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13 Generic properties of mechanical systems on the two-torus 13.1 Generic properties of periodic orbits . . . . . . . . . . . . . . . 13.2 Generic properties of minimal orbits . . . . . . . . . . . . . . . 13.3 Non-degeneracy at high-energy . . . . . . . . . . . . . . . . . . 13.4 Unique hyperbolic minimizer at very high energy . . . . . . . . 13.5 Generic properties at the critical energy . . . . . . . . . . . . .

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14 Derivation of the slow mechanical system 162 14.1 Normal forms near maximal resonances . . . . . . . . . . . . . . 162 14.2 Affine coordinate change, rescaling, and energy reduction . . . . 172 14.3 Variational properties of the coordinate changes . . . . . . . . . . 177 15 Variational aspects of the slow mechanical system 15.1 Relation between the minimal geodesics and the Aubry sets 15.2 Characterization of the channel and the Aubry sets . . . . . 15.3 The width of the channel . . . . . . . . . . . . . . . . . . . 15.4 The case E = 0 . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix: Notations

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References

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Preface The question of Arnold diffusion has been at the center stage of Hamiltonian dynamics ever since V. I. Arnold came up with the first example in 1964, after which his name was permanently attached to this topic. The coiner of the term, B. Chirikov, originally meant it as the stochastic diffusion. However, mathematical understanding of the stochastic side has been scarce. In the current literature, Arnold diffusion mostly means topological instability. In this book, we discuss Arnold diffusion for a smooth system with two and a half degrees of freedom. The main theorem is that Arnold diffusion occurs for a cusp-residual perturbation of an integrable system. This result was announced by J. Mather in 2003, whose proof was unfortunately unfinished before his passing. The many works on this topic in the past two decades, ours included, largely follow the blueprint laid out by him. His influence on this topic cannot be understated. With this said, we also try to incorporate the latest developments in the field and to take a different point of view. Our approach combines the geometrical and variational aspects of the theory and is heavily influenced by the weak KAM approach advanced by A. Fathi and P. Bernard. There are other works that address the same problem, most notably the works by C.-Q. Cheng, and the works by J.-P. Marco (some in collaboration with M. Gidea). We provide references and some discussions at the end of Chapter 1. Our main goal for this book is to present a relatively self-contained full proof of this theorem. While it is not possible to give textbook-level details, we have tried to include all the important statements and proofs. Given the complex nature of the proof, we present all of its main structures in the first five chapters. It is our hope that the reader will be able to grasp the main ideas after reading these chapters. This book has benefited from discussions with and suggestions from many people. We are grateful to the numerous discussions with J. Mather, and have benefited from the lectures he gave in the spring of 2009. The draft version has received helpful comments from A. Bounemoura, C.-Q. Cheng, and G. Forni. We deeply appreciate the efforts of the referees for their careful reading and numerous suggestions, which greatly improved the substance and presentation of the book. Vadim Kaloshin and Ke Zhang

Acknowledgments V. K. has been partially supported by National Science Fundation grant DMS1402164 and enjoyed the hospitality of the ETH Institute for Theoretical Studies. He has been partially supported by Dr. Max R¨ossler, the Walter Haefner Foundation, and the ETH Zurich Foundation. K. Z. is supported by a National Sciences and Engineering Research Council of Canada grant, reference number REGPIN-2019-07057, and thanks the ETH Institute for Theoretical Studies for hosting him in 2017. Both authors visited the Institute of Advanced Study in the spring of 2012, and acknowledge its hospitality and a highly stimulating research atmosphere.

Part I

Introduction and the general scheme

Chapter One Introduction The famous question called the ergodic hypothesis, formulated by Maxwell and Boltzmann, suggests that for a typical Hamiltonian on a typical energy surface all but a set of initial conditions of zero measure, have trajectories dense in this energy surface. However, Kolmogorov-Arnold-Moser (KAM) theory showed that for an open set of (nearly integrable) Hamiltonian systems there is a set of initial conditions of positive measure with almost periodic trajectories. This disproved the ergodic hypothesis and forced reconsideration of the problem. A quasi-ergodic hypothesis, proposed by Ehrenfest [36] and Birkhoff [17], asks if a typical Hamiltonian on a typical energy surface has a dense orbit. A definite answer to whether this statement is true or not is still far out of reach of modern dynamics. There was an attempt to prove this statement by E. Fermi [39], which failed (see [40] for a more detailed account). To simplify the problem, Arnold [4] asks: Does there exist a real instability in many-dimensional problems of perturbation theory when the invariant tori do not divide the phase space? For autonomous nearly integrable systems of two degrees or time-periodic systems of one and a half degrees of freedom, the KAM invariant tori divide the phase space. These invariant tori forbid large scale instability. When the degrees of freedoms are larger than two, large scale instability is indeed possible, as evidenced by the examples given by Arnold [5]. This book answers the question of the typicality of these instabilities in the two and a half degrees of freedom case.

1.1

STATEMENT OF THE RESULT

Let (θ, p) ∈ T2 × B 2 be the phase space of an integrable Hamiltonian system H0 (p) with T2 being 2-dimensional torus T2 = R2 /Z2 3 θ = (θ1 , θ2 ) and B 2 being the unit ball around 0 in R2 , p = (p1 , p2 ) ∈ B 2 . H0 is assumed to be strictly convex with the following uniform estimate: there exists D > 1 such that |H0 (0)|,

k∂p H0 (0)k ≤ D,

2 D−1 I ≤ ∂pp H0 (p) ≤ DI,

where I is the 2 × 2 identity matrix.

∀p ∈ B 2 ,

(1.1)

4

CHAPTER 1

Figure 1.1: Resonant net

Consider a smooth time periodic perturbation H (θ, p, t) = H0 (p) + H1 (θ, p, t),

t ∈ T = R/T.

We study Arnold diffusion for this system, namely, topological instability in the p variable. Arnold [5] proved existence of such orbits for an example and conjectured that they exist for a typical perturbation (see e.g. [4, 6, 7]). Denote Z3∗ = Z3 \ (0, 0, 1)Z, the integer equation k · (ω, ω, 1) = 0 for ω ∈ R2 is called a resonance relation. The submanifold Sk = {p ∈ R2 : k · (∂p H0 , 1) = 0}. is called the resonance submanifold for k. If curves Sk and Sk0 are given by two linearly independent resonances vectors k, k 0 , they either have no intersection or intersect at a single point in B 2 . Write k = (k 1 , k 0 ) ∈ Z2 × Z, k is called space irreducible if the greatest common divisor of components of k 1 is one. The union of resonances Sk with space irreducible k’s form a dense subset of R2 . Consider a finite collection of tuples:  K = (k, Γk ) : k ∈ Z3∗ , Γk ⊂ Sk ∩ B 2 , where k is space-irreducible, and Γk ⊂ Sk is a closed segment. We say K defines a diffusion path if [ P= Γk (k,Γk )∈K

is a connected set. We would like to construct diffusion orbits along the path P (see Figure 1.1). Remark 1.1. The requirement of space-irreducibility is only for technical convenience, and not essential to the construction.

5

INTRODUCTION

Figure 1.2: Description of cusp-generic perturbations

Theorem 1.2. Let 5 ≤ r < ∞ and suppose H0 ∈ C r (R2 ) satisfy condition (1.1). Let P ⊂ B 2 be a diffusion path, and U1 , . . . , UN be open sets such that Ui ∩ P 6= ∅, i = 1, . . . , N . Then there exist: • a C r open and dense set U = U(P) ⊂ S r = {kH1 kC r = 1} depending on the diffusion path P (but not on the open sets U1 , . . . , UN ), • a nonnegative lower semi-continuous function 0 = 0 (H1 ) with 0 |U > 0 , • a “cusp” set V := V(U, 0 ) := {H1 : H1 ∈ U, 0 <  < 0 (H1 )}, • a C r open and dense subset of H1 ∈ W ( V, such that for each H1 ∈ W there is an orbit (θ, p)(t) of H and times 0 < T1 < · · · < TN with the property p(Ti ) ∈ Ui ,

i = 1, . . . , N.

Remark 1.3. The condition that 0 is lower semi-continuous implies the set V is open. Remark 1.4. The set W may be understood as follows. The open and dense set U represents the set of “good directions” of perturbations, and the set of exceptional directions is nowhere dense. Around each exceptional direction we remove a cusp and call the complement V. For this set of perturbations we establish a connected collection of invariant manifolds. Then in the complement to some exceptional perturbations W in V we show that there are diffusing orbits “shadowing” these cylinders. This notion is due to Mather [62], who called it cusp residual. See Figure 1.2. Remark 1.5. The openness of the set W follows immediately from the continuous dependence of the solution on the Hamiltonian. The main challenge is to prove density.

6

CHAPTER 1

Given (k1 , Γk1 ) ∈ K and λ > 0, we define in Section 3.1 a quantitative non-degeneracy hypothesis relative to the resonant segment Γk1 : SR(k1 , Γk1 , λ) = [SR1λ ] − [SR3λ ]. λ Let us denote by USR (k1 , Γk1 ) the set of H1 that satisfies SR(k1 , Γk1 , λ). Suppose H1 satisfies the condition SR(k1 , Γk1 , λ). We define a finite subset of Z3∗ called the strong additional resonances. In Theorem 3.3 we define a large constant K = K(k1 , Γk1 , λ) and call k2 ∈ Z3∗ strong if:

• (k2 , Γk2 ) ∈ K and such that Γk1 ∩ Sk2 6= ∅ and |k2 | ≤ K(k1 , Γk1 , λ). We emphasize that strong additional resonances are taken from the set Z3∗ , not just the space-irreducible ones. Denote the set of strong additional resonances Kst (k1 , Γk1 , λ). If k2 is strong, then it defines a unique double resonance Γk1 ,k2 = Γk1 ∩ Sk2 . For each double resonance Γk1 ,k2 , we associate non-resonance conditions of two types: • high energy [DR1h ] – [DR3h ] (Section 4.2), • low energy [DR1c ] – [DR4c ] (Section 4.2). For each (k1 , Γk1 ) ∈ K and k2 ∈ Kst (k1 , Γk1 , λ) consider the set of H1 which satisfy the above conditions and denote it by UDR (k1 , Γk1 , k2 ). Remark 1.6. All our non-degenerate conditions at a double resonance are stated relative to a single resonance. In particular, the condition DR(k1 , Γk1 , k2 ) may differ from the condition DR(k2 , Γk2 , k1 ). Roughly speaking, condition DR(k1 , Γk1 , k2 ) is used for diffusion along Γk1 across the double resonance Γk1 ,k2 , while DR(k2 , Γk2 , k1 ) is for diffusion along Γk2 across Γk1 ,k2 . When both conditions are satisfied, our diffusion mechanism also ensures the existence of an orbit that switches from Γk1 to Γk2 at the intersection Γk1 ,k2 , using an equivalence relation that is sufficient for the existence of diffusion. This will be explained in Section 1.2. The following theorem (Theorem 1.7) is immediate given that S λ λ • (Proposition 3.2) Each USR (k1 , Γk1 ) is open and the union λ>0 USR (k1 , Γk1 ) is dense. • (Proposition 4.3) The set UDR (k1 , Γk1 , k2 ) is open and dense. Theorem 1.7. For each λ > 0, the set  [ \ λ USR U = U(P) := (k1 , Γk1 ) ∩ λ>0 (k1 ,Γk1 )∈K

is open and dense in S r .

 \ k2

∈Kst (k

1 ,Γk1 ,λ)

UDR (k1 , Γk1 , k2 )

7

INTRODUCTION

As a corollary of Theorem 1.2, we obtain Theorem 1.8 (Almost Density Theorem). For any ρ > 0 there are • • • •

an open dense set U = U(ρ) ⊂ S r , a non-negative lower semi-continuous function 0 : S r → R+ with 0 |U > 0, a cusp set V := V(U, 0 ) := {H1 : H1 ∈ U, 0 <  < 0 (H1 )}, an open dense subset W ( V,

such that for any H1 ∈ W there is a ρ-dense orbit on T2 × B 2 × T. Proof. Suppose p∗ ∈ B 2 and let ω = ∇H0 (p∗ ). We say ω is ρ-irrational if there exists T > 0 such that {t(ω, 1) : t ∈ [−T, T ]} ⊂ T3 is ρ-dense, and let T (ω) be the smallest such T . Using the fact that p˙ = O(), θ˙ = ∇H0 (p) + O(), there is 0 > 0 depending on ρ, T (ω) and kH0 kC 2 such that if 0 <  < 0 , ! [ t B2ρ φH (θ0 , p0 , t0 ) ⊃ T2 × Bρ (p∗ ) × T, (1.2) t∈R for all p0 ∈ Bρ (p∗ ), (θ0 , t0 ) ∈ T2 × T. A resonance k ∈ Z3∗ is called ρ-irrational if the sub-torus {x ∈ T3 : x · k = 0} is ρ-dense. Dirichlet’s Theorem implies that there are at most finitely many k ∈ Z3∗ which is not ρ-irrational (called ρ-rational). We claim that if k is ρirrational, then the ρ-irrational frequencies form a dense subset of Sk = {ω ∈ R2 : (ω, 1) · k = 0}. Indeed, if ω ∈ Sk is ρ-rational, then Tω = {t(ω, 1) : t ∈ R}, which is not ρ-dense and is a sub-torus of dimension 1 or 2. If dim Tω = 2, we must have Tω = Tk , which is impossible, since Tk is ρ-dense. Therefore dim Tω = 1 and (ω, 1) is doubly resonant. The set of such ω is at most countable. Since there are infinitely many space-irreducible resonances, there is a diffusion path P consisting only of ρ/2-irrational resonances. Moreover, we may choose the path P to be ρ/2-dense in B 2 , since space-irreducible resonances are dense. We now apply Theorem 1.2 to the path P, and pick pSi , i = 1, . . . , N ∈ P n such that (∇H0 (pi ), 1) is (ρ/2)-irrational, and such that i=1 Bρ/2 (pi ) ⊃ B 2 . According to our theorem, there is an orbit whose p component visits every Bρ (pi ). Then the orbit must be ρ-dense in view of (1.2).

1.2

SCHEME OF DIFFUSION

A Hamiltonian H : T2 × R2 × T → R is called Tonelli if it satisfies: 2 • (Strictly convex) For each p ∈ R2 , ∂pp H(θ, p, t) is strictly positive definite. • (Super linear) limkpk→∞ H(θ, p, t)/kpk = ∞. • (Complete) The Hamiltonian flow generated by H is complete.

8

CHAPTER 1

If H0 satisfies condition (1.1), then for all  < D−1 , H = H0 + H1 are Tonelli. Indeed, convexity and super-linearity are immediate, while the flow of H is complete, since the Hamiltonian vector field is uniformly Lipschitz. A Tonelli Hamiltonian can always be associated with a Lagrangian L(θ, v, t) = sup {p · v − H(θ, p, t)}, p∈R2

and one can study the dynamics in the Lagrangian setting. Using Lagrangian dynamics, Mather developed a theory ([37], [61]) for special invariant sets of Tonelli Hamiltonians. He showed that for each cohomology class c ∈ R2 ' H 1 (T2 , R), the Hamiltonian H admits families of invariant sets of the Hamiltonian flow on T2 × R2 × T, called the Mather, Aubry, and Ma˜ n´e sets. In our perturbative setting, one has the relation fH (c) ⊂ AeH (c) ⊂ N eH (c) ⊂ T2 × BC √ (c) × T, M    where C is a constant depending only on D (see Corollary 7.7). Their projections to the (θ, t) components are denoted MH (c),

AH (c),

NH (c).

f0 (c), Ae0 (c), and N e 0 (c) to denote their intersections with the We use M H H H time-0 section {t = 0}, which are invariant under the time-1 map. Throughout the book, we may switch between the two equivalent settings: consider either continuous invariant sets of the flow, or discrete invariant sets under the time-1 map. Our main strategy is to pick a subset Γ∗ ⊂ R2 of cohomologies very close to the diffusion path P, then find an orbit that shadows a sequence of Aubry sets AeH (ci ), i = 1, . . . , N , ci ∈ Γ∗ . We show that these Aubry sets admit nondegenerate heteroclinic connections between them, called a transition chain by Arnold ([5]). The existence of these connections from one of the four diffusion mechanisms. We give a general introduction to these mechanisms in the text that follows, and refer to Section 2.1 for precise definitions. Mather mechanism In the case that H is defined on T × R × T, the time-1 map of H is a monotone twist map on the infinite cylinder T × R. An essential invariant curve is an invariant curve that is homotopic to T × {0}. It is known since Birkhoff that a region free of essential invariant curves is unstable, namely there exist orbits that drift from one boundary of the region to another. Mather [61] gave a conceptual description of this phenomenon and generalized it to higher dimensions. We say that the pair (H , c) satisfies the Mather mechanism if 0 0 eH NH (c) = πθ N (c) ⊂ T2  

9

INTRODUCTION

is contractible, where πθ : T2 × R2 is the projection to the first component. (In e 0 (c) is not an essential invariant curve.) Mather the twist map case this means N proved that ([61]) in this case, Ae0H (c) admits a heteroclinic connecting orbit to Ae0H (c0 ) if c, c0 are close. Arnold mechanism In Arnold’s original paper [5], Arnold showed the existence of a family of invariant tori, whose own stable and unstable manifolds intersect transversally. In our setting, the tori are the Aubry sets AeH (c) contained in a 3-dimensional normally hyperbolic invariant cylinder (which means homotopic to T2 × R). Then Ae0H (c) is contained in a 2-dimensional cylinder. To consider homoclinic connections, we lift the system to a double covering map, then a homoclinic connection of AeH (c) becomes a heteroclinic connection between the two copies. We say that the pair (H , c) satisfy the Arnold mechanism if Ae0H (c) is an invariant curve and there exists a symplectic double covering map Ξ : T2 ×R2 → T2 × R2 , such that the set 0 eH N (Ξ∗ c) \ Ae0H ◦Ξ (Ξ∗ c)  ◦Ξ

is totally disconnected. If Ae0H (c) is a smooth invariant curve with transversal intersection of stable and unstable manifolds, then the above set is discrete. This mechanism is inspired by Arnold’s pioneering paper [5], hence its name. In our setting, Cheng-Yan [26, 27] and Bernard [11] showed that if (H , c) satisfy the Arnold mechanism, Ae0H (c) admits a heteroclinic connecting orbit to Ae0H (c0 ) if c, c0 are close. Bifurcation mechanism This is technically similar to the Arnold mechanism, but happens when the Aubry set AeH (c) is contained in two disjoint normally hyperbolic invariant cylinders. We say that the pair (H , c) satisfies the bifurcation mechanism if the set 0 eH N (c) \ Ae0H (c) 

is totally disconnected. In this case, the same conclusion as in the Arnold mechanism holds. Normally hyperbolic invariant cylinders and Aubry-Mather types The three mechanisms given do not apply to all cases, even after a generic perturbation. The main observation is that they do apply to cohomologies that satisfy:

10

CHAPTER 1

Figure 1.3: Diffusion along a cylinder.

1. (2D NHIC) The Aubry set Ae0H (c) is contained in a 2-dimensional normally hyperbolic invariant cylinder. 2. (1D Graph Theorem) The Aubry set Ae0H (c) is contained in a Lipschitz graph over the circle T. In this case, we say that the pair (H , c) is of the Aubry-Mather type. Under these assumptions, the Aubry set resembles the Aubry-Mather sets for twist maps, and in particular, generically we have the following dichotomy: e 0 (c) is contractible or Ae0 (c) is a Lipschitz invariant curve. In the either π N H H latter case, we can show that the Arnold mechanism applies after an additional perturbation. Since either the Mather or Arnold mechanism applies, we conclude e is connected to A(c e 0 ) for c, c0 close. Moreover, this argument can be that A(c) 0 continued if c is also of Aubry-Mather type. Dynamically, the orbit is either diffusing along the heteroclinic orbits of invariant curves or diffusing in a Birkhoff region of instability within the cylinder. See Figure 1.3. While a cohomology of Aubry-Mather type is robust, namely it can be extended along a continuous curve, in a one-parameter family one may encounter a bifurcation where the Aubry set jumps from one cylinder to another one. At the bifurcation, the Aubry set is contained in both cylinders. We say that the pair (H , c) is of bifurcation Aubry-Mather type if the Aubry set is possibly contained in two cylinders. For technical reasons, we have to involve a different bifurcation type, called asymmetric bifurcation type. This is very similar to the bifurcation AubryMather type, the main difference is, on one side of the bifurcation, the Aubry set is an Aubry-Mather type set contained in an invariant cylinder, while on the other side we have a hyperbolic periodic orbit. This happens when we cross double resonance. Forcing relation and jump mechanism The rigorous formulation of the three diffusion mechanisms will be given using the concept of forcing equivalence defined by Bernard in [11], which is a general-

11

INTRODUCTION

ization of an equivalence relation defined by Mather; see [61]. If c, c0 are forcing equivalent (denoted c a` c0 ), then there is a heteroclinic orbit connecting the associated Aubry sets. Moreover, there exists an orbit shadowing an arbitrary sequence of cohomologies, as long as they are all equivalent. See Section 2.1 for more details. The main theorem reduces to Theorem 2.2, which proves forcing equivalence of a net of cohomologies, called Γ∗ . The set Γ∗ consists of finitely many smooth curves. On each of the smooth curves, we prove the cohomologies are of AubryMather type, and therefore one of the three mechanisms applies, after a generic perturbation. We prove forcing equivalence of different connected components directly, using the definition of the forcing relation. We call this the jump mechanism.

1.3

THREE REGIMES OF DIFFUSION

Our plan is to choose a net Γ∗ of cohomology classes, and prove their forcing equivalence, by first proving they are of Aubry-Mather type or bifurcation Aubry-Mather type. This is done in three distinct regimes. Single resonance Let (k1 , Γk1 ) ∈ K be one of the single-resonance component, and Kst (k1 , Γk√ 1 , λ) be the collection of the strong additional resonances. Then for p in a O( ε)neighborhood of the set   [  ΓSR BM √ (Γk1 ,k2 ) , k1 (M, λ) := Γk1 \ k2 ∈Kst (k1 ,Γk1 ,λ)

where M is a large parameter, the system admits the normal form NSR = H0 + Z(θs , p) + O(δ),

(θs , θf , t) ∈ T3 ,

where how small δ is depends on how many double resonances we exclude. Under the non-degeneracy conditions SR(k1 , Γk1 , λ), the above system admits 3-dimensional (for the flow) normally hyperbolic invariant cylinders, and one can prove each c ∈ ΓSR k1 (M, λ) is of AM or of bifurcation AM type. Double resonance, high energy Let p0 = Γk1 ,k2 be a strong double resonance. On the set BM √ (p0 ),

p0 = Γk1 ∩ Γk2

12

CHAPTER 1

we perform a√normal form transformation and then rescale the p variable via I = (p − p0 )/ . One can show that the system is conjugate to √
1 K(I) − U (ϕ) + O( ) , β where K : R2 → R is a positive quadratic form and U : T2 → R, and β > 0 is a constant depending only on k1 , k2 . The system H s = K(I) − U (ϕ) is a two degrees of freedom mechanical system, and is called the slow mechanical system. In the text that follows we use the shifted energy E := H s (ϕ, I) − min H s as a parameter. By the Maupertuis principle, for each E > 0, the Hamiltonian flow in the energy surface H s − min H s = E is conjugate to a geodesic flow, whose metric is called the Jacobi metric gE . When the shifted energy E is not too close to 0, we are in the high-energy regime. By imposing the conditions [DR1h ]−[DR3h ], one shows the existence of 2-dimensional normally hyperbolic invariant cylinders associated to the shortest loops for the associated Jacobi metric. This cylinder persists under perturbation, and one can show that the associated cohomologies are of Aubry-Mather type. Double resonance, low energy As the energy decreases, the cylinder constructed in the high-energy regime may not persist. Under the non-degeneracy conditions DR(k1 , k2 ), we distinguish two separate cases: 1. Simple cylinder: In this case the cylinder extends across zero shifted energy to negative shifted energy. In this case one can still show the associated cohomologies are of Aubry-Mather type. 2. Non-simple cylinder: In this case the cylinder may be destroyed before the shifted energy becomes zero. However, we show the existence of two simple cylinders near the non-simple one, and one can “jump” from one cylinder to another one. This is the only case where the jump mechanism is used.

1.4

THE OUTLINE OF THE PROOF

We structured this book into four parts: Part I (Chapters 1 to 5) We formulate all the major definitions and theorems used in the main proof, and prove our main theorem (Theorem 1.2) assuming results proven in the later chapters. We hope the reader will be able to get a good idea of the whole proof

INTRODUCTION

13

by just reading Part I. Chapter 2 describes forcing relations, different diffusion mechanisms, and Aubry-Mather types. It also formulates Theorem 2.2 and shows that it implies our main theorem. Chapter 3 first describes the single resonance non-degeneracy conditions and normal forms. It then formulates Theorem 3.3, which covers the forcing equivalence in the single resonance regime. The double resonance is split into two chapters. Chapter 4 describes the geometrical structure for the system at double resonance. After the normal form near a double resonance is desribed, the system is reduced to the slow mechanical system with perturbation. We formulate the non-degeneracy conditions and theorems about their genericity. The normally hyperbolic invariant cylinders are described, and the proof using local and global maps is sketched. Chapter 5 describes the choice of cohomology and Aubry-Mather type at the double resonance. First, cohomology classes are chosen for the (unperturbed) slow mechanical system. We prove Aubry-Mather type for the perturbed slow mechanical system and revert to the original coordinates. As the system has been perturbed, we need to modify the choice of cohomology classes to connect the single and double resonances. At the end of this chapter, Theorem 2.2 is proved, proving our main theorem assuming statements proved in the rest of this book. Part II (Chapters 6 to 8) We introduce most of the variational theory used in this book. Chapter 6 is an overview of standard weak KAM theory, Chapter 7 concerns perturbation aspects of the weak KAM theory, and Chapter 8 defines cohomology of AubryMather type and explains why it implies one of the diffusion mechanisms, after a generic perturbation. Part III (Chapters 9 to 12) We prove that the Aubry-Mather type holds in various regimes, and also describes the jump mechanism. Chapter 9 proves Aubry-Mather type in the singleresonance regime, Chapter 10 proves the geometric picture of double-resonance described in Chapter 4, and Chapter 11 proves Aubry-Mather type for the double resonance regime. The jump mechanism is formulated and proved in Chapter 12. Part IV (Chapters 13 to 15) These chapters include topics that more or less stand alone. Chapter 13 proves genericity of the double resonance conditions. The flavor is similar to that of Kupka-Smale theorem. Chapter 14 proves various normal form results, and formulates the coordinate changes that are used to derive the slow system at

14

CHAPTER 1

the double resonance. Chapter 15 describes the variational property of the slow mechanical system. Much of this topic is covered in Mather’s works [64, 65], but we give a self-contained presentation.

1.5

DISCUSSION

Relation with Mather’s approach Theorem 1.2 was announced by Mather in [62], where he proposed a plan to prove it. Some parts of the proof are provided in [63]. Our work realizes Mather’s general plan using weak KAM theory and Hamiltonian point of view. In the following we summarize the new techniques and tools introduced. • We utilize Bernard’s forcing relation to simplify the construction of a diffusion orbit. This allows a more Hamiltonian treatment of the variational concepts and allows us to reduce the main theorem to local forcing equivalence of cohomology classes. • We use Hamiltonian normal forms to construct a collection of normally hyperbolic invariant cylinders along the chosen diffusion path. We obtain precise control of the normal forms (via an anisotropic C 2 norm) at both single and double resonances. Mather’s method uses mostly the Lagrangian point of view. • We introduce the concept of Aubry-Mather type, which generalizes the work done in Bernard-Kaloshin-Zhang [13] to a more abstract setting, applicable to both single and double resonances. Heuristically, this means the Aubry set has the same topological structures as Aubry-Mather sets in twist maps, in particular, the set has a proper ordering. Our approach can be seen as a generalization of the variational technique for a priori unstable systems from [11, 13, 27]. • One important obstacle is the problem of regularity of barrier functions (see Section 8.3), which outside of the realm of twist maps is difficult to overcome. Our definition of Aubry-Mather type allows proving this statement in a general setting. It is our understanding that Mather [58] handles this problem without proving the existence of invariant cylinders. • In a double resonance we also construct normally hyperbolic invariant cylinders. This leads to a fairly simple and explicit structure of minimal orbits near a double resonance. In particular, in order to switch from one resonance to another we need only one jump (see Section 5.5 for the formulation of the statement). • It is our understanding that Mather’s approach [58] requires an implicitly defined number of jumps. His approach resembles his proof of existence of diffusing orbits for twist maps inside a Birkhoff region of instability [59].

15

INTRODUCTION

Other results on a priori stable systems A different approach to prove a theorem of similar flavor using variational methods is given in the series of works [24, 23, 25]. In [54] (see also [44, 55]), the authors develop a geometric method to prove Arnold diffusion. An important aspect for studying a nearly integrable system is a partial averaging theory based on arithmetic properties of the frequency. We develop a version of this theory in [52]. Generic instability of resonant totally elliptic points In [50] stability of resonant totally elliptic fixed points of symplectic maps in dimension 4 is studied. It is shown that generically a convex, resonant, totally elliptic point of a symplectic map is Lyapunov unstable. Non-convex Hamiltonians In the case the Hamiltonian H0 is non-convex or non-strictly convex for all p ∈ B 2 , for example, H0 (p) = p21 + p32 , the problem of global Arnold diffusion is wide open. Some results for the Hamiltonian H0 (p) = p21 − p22 are in [16, 20]. To apply a variational approach one faces another deep open problem of extending Mather theory and weak KAM theory beyond convex Hamiltonians or developing a new technique to construct diffusing orbits. Other diffusion mechanisms Here we would like to give a short review of other diffusion mechanisms. In the case n = 2 Arnold proposed the following example: H(q, p, φ, I, t) =

I2 p2 + + (1 − cos q)(1 + µ(sin φ + sin t)). 2 2

This example is a perturbation of the product of a 1-dimensional pendulum and a 1-dimensional rotator. The main feature of this example is that it has a 3dimensional normally hyperbolic invariant cylinder. There is a rich literature on the Arnold example and we do not intend to give n extensive list of references; we mention [2, 9, 14, 15, 79] and references therein. This example gave rise to a family of examples of systems of n + 1/2 degrees of freedom of the form H (q, p, φ, I, t) = H0 (I) + K0 (p, q) + H1 (q, p, φ, I, t), where (q, p) ∈ Tn−1 ×Rn−1 , I ∈ R, φ, t ∈ T . Moreover, the Hamiltonian K0 (p, q) has a saddle fixed point at the origin and K0 (0, q) attains its strict maximum at q = 0. For small  a 3-dimensional NHIC C persists. Several geometric mechanisms of diffusion have evolved. In [31, 32, 33, 43] the authors carefully analyze two types of dynamics induced

16

CHAPTER 1

on the cylinder C. These two dynamics are given by so-called inner and outer maps. In [35, 34], these techniques are applied to a general perturbation of Arnold’s example. In [30, 75, 76, 77] a return (separatrix) map along invariant manifolds of C is constructed. A detailed analysis of this separatrix map gives diffusing orbits. In [22, 45, 51] for an open set of perturbations of Arnold’s example, one constructs a probability measure µ in the phase space such that the pushforward of µ projected onto the I component in the proper time scale weakly converges to the stochastic diffusion process. This, in particular, implies the existence of diffusing orbits. In [42], the authors treats the a priori chaotic setting, but prove diffusion in the real analytic category, which is much more difficult. A different mechanism related to the slow–fast system is given by the same authors in [41]. For a priori unstable systems the works [11, 26, 27] are inspired and influenced by Mather’s approach [59, 60, 61] and build diffusing orbits variationally. Recentlyan a priori unstable structure was established for the restricted planar three-body problem [38]. It turns out that for this problem there are no large gaps. A multidimensional diffusion mechanism of different nature, but also based on existence and persistence of a 3-dimensional NHIC C is proposed in [20]. There are many studies of Arnold diffusion using computer numerics: see for example [46] and the references therein.

Chapter Two Forcing relation

2.1

SUFFICIENT CONDITION FOR ARNOLD DIFFUSION

We will utilize the concept of forcing equivalence, denoted c a` c0 . The actual definition will not be important for the current discussions, instead, we state its main application to Arnold diffusion. Proposition 2.1 (Proposition 0.10 of [11]). Let {ci }N i=1 be a sequence of cohomology classes which are forcing equivalent with respect to H. For each i, let Ui f0 (ci ). Then there is a trajectory be neighborhoods of the discrete Mather sets M H of the Hamiltonian flow visiting all the sets Ui in the prescribed order. Let σ > 0 and Vσ (H) denote the σ neighborhood of H in the space C r (T2 × B × T) with respect to the natural C r topology. The following statement is a “local” version of our main theorem, where we state that given H1 ∈ U, we can 2

• Choose 0 to be locally constant on a neighborhood of H1 • Prove forcing equivalence on a residual subset of a neighborhood of H . Theorem 2.2. Let P be a diffusion path and U1 , . . . , UN be open sets intersecting P. Then there is an open and dense subset U(P) ⊂ S r , and for each H1 ∈ U(P), there are δ = δ(H0 , H1 ) > 0, 1 = 1 (H0 , H1 ) > 0 such that for each H10 ∈ Vδ (H1 ), 0 <  < 1 , there is a subset Γ∗ (, H0 , H10 ) ⊂ R2 satisfying Γ∗ = Γ∗ (, H0 , H10 ) ∩ Ui 6= ∅,

i = 1, . . . , N,

with the property that there is σ = σ(, H0 , H10 ) > 0 and a residual subset Rσ (H0 + H10 ) ⊂ Vσ (H0 + H10 ), such that for each H 0 ∈ R, with respect to the Hamiltonian H 0 , all the c ∈ Γ∗ (, H0 , H10 ) are forcing equivalent to each other. Proposition 2.1 and Theorem 2.2 imply our main theorem. Proof of Theorem 1.2. First of all, let us define the lower semi-continuous function 0 . For each H1 ∈ U, define 1 H 2 (·) = 1 (H0 , H1 )1Vδ (H1 ) (·),

18

CHAPTER 2

where 1V denote the indicator function of V. An indicator function of an open 1 set is lower semi-continuous by definition. For H1 ∈ S r \ U, let H ≡ 0. We 2 then define 1 0 (·) = sup H 2 (·), H1 ∈S r

which is lower semi-continuous, being an (uncountable) supremum of lower semi1 continuous functions. Note that 0 is positive on each H1 ∈ U since H 2 (H1 ) = 1 (H0 , H1 ) > 0. Consider H1 ∈ U and 0 <  < 0 (H1 ) as defined earlier. Let Γ∗ (0 , H0 , H1 ) be as in Theorem 2.2. Let ci ∈ Ui ∩ Γ∗ (, H0 , H1 ). For any kH − H0 kC r ≤ , f0 (c) ⊂ by Corollary 7.7, there is C > 0 depending only on D such that M H 2 √ T × BC  (c). As a result, reducing 0 if necessary (note that minimum of an lower semi-continuous function and a constant is still lower semi-continuous), f0 (ci ) ⊂ T2 × Ui . Since ci are all forcing equivalent by Theorem 2.2, we have M H Proposition 2.1 implies the existence of an orbit visiting each neighborhood T2 × Ui . Since the above discussion applies to all H ∈ Rσ (H0 + H1 ), where H1 ∈ U and 0 <  < 0 (H1 ), we conclude that for a dense subset of V(U, 0 ) (as defined in Theorem 1.2), there is an orbit visiting each T2 × Ui . Since this property is open due to the smoothness of the flow, it holds on an open and dense subset W of V. The set Γ∗ (, H0 , H1 ) will be chosen to be the union of finitely many smooth curves, √ and will coincide with P except on finitely many neighborhoods of size O( ) of strong double resonances.

2.2

DIFFUSION MECHANISMS VIA FORCING EQUIVALENCE

We reformulate the diffusion mechanisms introduced in Section 1.2 using forcing equivalence. We start with the Mather mechanism. Proposition 2.3 (Theorem 0.11 of [11]). Suppose 0 NH (c) is contractible

(Ma)

as a subset of T2 , then there is σ > 0 such that c is forcing equivalent to all c0 ∈ Bσ (c). To define the Arnold mechanism, we consider a finite covering of our space. Let ξ : Tn → Tn be a linear double covering map, for example: (θ1 , θ2 ) 7→ (2θ1 , θ2 ). Then ξ lifts

19

FORCING RELATION

to a symplectic map Ξ : Tn × Rn → Tn × Rn ,

Ξ(θ, p) = (ξθ, ξ ∗ p),

where ξ ∗ (p) is defined by the relation ξ ∗ (p) · v = p · dξ(v) for all v ∈ R2 . For example, if n = 2 and ξ(θ1 , θ2 ) = (2θ1 , θ2 ) we have ξ ∗ (p1 , p2 ) = (p1 /2, p2 ). This allows us to consider the lifted Hamiltonian H ◦ Ξ. Lemma 2.4. (Section 7 in [11]) We have Ae0H◦Ξ (ξ ∗ c) = Ξ−1 Ae0H (c),

0 0 eH◦Ξ eH N (ξ ∗ c) ⊃ Ξ−1 N (c).

Moreover, ξ ∗ c ` ξ ∗ c0 relative to H ◦ Ξ implies c ` c0 relative to H. The Aubry set can be decomposed into disjoint invariant sets called static classes, which gives important insight into the structure of the Aubry set. In e 0 (c). In the case particular, when there is only one static class, then Ae0H (c) = N H 0 0 0 0 e e e e AH (c) 6= NH (c), the difference NH (c)\ AH (c) consists of heteroclinic orbits from e 0 (c), it may one static class to another [11]. Using Lemma 2.4, when Ae0H (c) = N H e 0 (ξ ∗ c), and the difference provides additional happen that Ae0H◦Ξ (ξ ∗ c) ( N H◦Ξ heteroclinic orbits to the Aubry set that are not contained in the Ma˜ n´e set before the lifting. This can be exploited to create diffusion orbits. Proposition 2.5 (Proposition 7.3 and Theorem 9.2 of [11]). Suppose, either: e 0 (c) \ Ae0 (c) is totally disconnected, Ae0H (c) has two static classes, and N H H (Bif ) or 0 eH◦Ξ Ae0H (c) has one static class, and N (ξ ∗ c) \ Ae0H◦Ξ (ξ ∗ c) is totally disconnected. (Ar) Then there is σ > 0 such that c is forcing equivalent to all c0 ∈ Bσ (c).

As implied by the labeling, the first item is called the bifurcation mechanism, and the second the Arnold mechanism. We obtain the following immediate corollary: Corollary 2.6 (Mather-Arnold mechanism). Suppose Γ ⊂ B 2 is a connected set, and for each c ∈ Γ one of the diffusion mechanisms (Ma), (Bif ), or (Ar) holds. Then all c ∈ Γ are forcing equivalent. Recall that Γ∗ (, H0 , H1 ) can be chosen as a union of finitely many smooth curves. • We will later show that for c ∈ Γ∗ (, H0 , H1 ) in Theorem 2.2, one of the two applies: Proposition 2.3 or Proposition 2.5. As a result, each connected component of Γ∗ (, H0 , H1 ) consists of equivalent

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CHAPTER 2

c’s. • We prove the forcing equivalence between different connected components using directly the definition of forcing relation. We call this the “jump” mechanism.

2.3

INVARIANCE UNDER THE SYMPLECTIC COORDINATE CHANGES

n A diffeomorphism Ψ = Ψ(θ, p) : Tn × Rn → TP × Rn is called exact symplectic n ∗ if Ψ λ − λ is an exact one-form, where λ = i=1 pi dθi is the canonical form. n n n n We say Φ : T × R × T → T × R × T is exact symplectic if there exists a smooth map Φ1 : Tn × Rn × T → Tn × Rn such that

Φ(θ, p, t) = (Φ1 (θ, p, t), t), e = E(θ, e p, t), such that and there is E e p, t)) Ψ(θ, p, t, E) = (Φ(θ, p, t), E + E(θ,

(2.1)

(called the autonomous extension of Φ) is exact symplectic. The new term e p, t) is defined up to adding a function f 0 (t), where f (t) is periodic in t. E(θ, e 0, t) ≡ 0; therefore the choice of E e is unique. Let us assume E(0, Let H = H(θ, p, t), and Φ is exact symplectic with extension Ψ. Then for G(θ, p, t, E) = H(θ, p, t) + E, we define e p, t). Φ∗ H = Ψ∗ H = G ◦ Ψ(θ, p, t, E) − E = H ◦ Φ + E(θ,

(2.2)

The Aubry, Mather and Ma˜ n´e sets are invariant under exact symplectic coordinate change in the following sense. Proposition 2.7 ([10], [67]). Suppose H and Φ∗ H are Tonelli, and let Ψ be the extension of Φ. Let (c, α) ∈ Rn × R ' H 1 (Tn × T, R), and let Ψ∗ (c, α) = (c∗ , α∗ ) be the push forward of the cohomology class via the identification H 1 (Tn × T, R) ' H 1 (Tn × Rn × T × R, R). Then α = αH (c)

⇐⇒

α∗ = αΦ∗ H (c∗ ).

Let us denote (Φ∗H c, α∗ ) = Ψ∗ (c, αH (c)). Then

    fH (c) = Φ M fΦ∗ H (Φ∗H c) , AeH (c) = Φ AeΦ∗ H (Φ∗H c) , M   eH (c) = Φ N eΦ∗ H (Φ∗H c) . N

21

FORCING RELATION

Note in the particular case when Φ is homotopic to identity, Φ∗H c = c and αΦ∗ H (c) = αH (c). Lemma 2.8. Let Φ be an exact symplectic coordinate change on Tn × Rn × T. Then tuple (H, c) satisfies (Bif ) or (Ar) if and only if (Φ∗ H, Φ∗H c) satisfies e 0 (c). Then (H, c) the same conditions. Suppose in addition that Ae0H (c) = N H ∗ ∗ satisfies (Ma) if and only if (Φ H, ΦH c) satisfies the same conditions. Proof. Since our symplectic coordinate changes are always identity in the t components, the invariance of Aubry and Ma˜ n´e sets implied the invariance of their zero section under the map Φ(·, ·, 0). The invariance of (Bif ) follows. For the invariance of (Ma), note that due to the graph property, Ae0H (c) is contractible in Tn × Rn if and only if A0H (c) is contractible in Tn . Therefore the contractibility of Aubry set is invariant, and since the Aubry set coincides with the Ma˜ n´e set by assumption, (Ma) is invariant. For (Ar), let Ψ be the extension of Φ as in (2.1), and let us extend Ξ trivially to Tn × Rn × T or Tn × Rn × R without changing its name. Let Φ1 be an exact symplectic change homotopic to Φ, with extension Ψ1 , such that Φ ◦ Ξ = Ξ ◦ Φ1 ,

Ψ ◦ Ξ = Ξ ◦ Ψ1 .

Note that Ξ∗ c, defined as the push forward of H 1 (Tn × Rn × T × R, R) under the identification with H 1 (Tn × T, R), is identical to ξ ∗ c. We have Ψ∗1 (Ξ∗ c, αH◦Ξ (Ξ∗ c)) = Ψ∗1 Ξ∗ (c, αH (c)) = Ξ∗ Ψ∗ (c, αH (c)). Since Ξ is independent of t, Ξ∗ is identity in the last component. We conclude that (Φ1 )∗H◦Ξ Ξ∗ c = Ξ∗ (Φ∗H c). Moreover, (Φ ◦ Ξ)∗ H = (Φ∗ H) ◦ Ξ,

(Ξ ◦ Φ1 )∗ H = Φ∗1 (H ◦ Ξ).

Then     e(Φ◦Ξ)∗ H ((Φ ◦ Ξ)∗H c) = Φ1 N eΦ∗ (H◦Ξ) ((Φ1 )∗H◦Ξ Ξ∗ c) = N eH◦Ξ (Ξ∗ c) Φ1 N 1 and       eΦ∗ H (Φ∗H c) = Ξ−1 Φ Ξ−1 N eΦ∗ H (Φ∗H c) = Ξ−1 N eH (c) , Φ1 Ξ−1 N therefore   e(Φ∗ H)◦Ξ (Ξ∗ (Φ∗H c)) \ Ξ−1 N eΦ∗ H (Φ∗H c) = N eH◦Ξ (Ξ∗ c) \ Ξ−1 N eH (c). Φ1 N This implies invariance of (Ar) after considering the time-zero section of the above equality. Our definition of exact symplectic coordinate change for a time-periodic

22

CHAPTER 2

system is somewhat restrictive, and in particular, it does not apply directly to the linear coordinate change performed at the double resonance. In that setting, we will prove invariance of the Mather, Aubry, and Ma˜ n´e sets directly.

2.4

NORMAL HYPERBOLICITY AND AUBRY-MATHER TYPE

A 2-dymensional normally hyperbolic invariant cylinder is called symplectic if the restriction of the canonical form dθ ∧ dp is non-degenerate on its tangent space. Loosely speaking, a pair (H∗ , c∗ ) is called of Aubry-Mather type (AM type for short, refer to Definition 8.1 for details) if: 1. The discrete Aubry set Ae0H∗ (c∗ ) is contained in a 2-dimensional normally hyperbolic invariant cylinder on which the restriction of the symplectic form is non-degenerate. 2. There is σ > 0 such that the following holds for c ∈ Bσ (c∗ ) and H ∈ Vσ (H∗ ): a) The discrete Aubry set satisfies the graph property under the local coordinates of the cylinder. b) When the Aubry set is an invariant graph, then locally the unstable manifold of the Aubry set is a graph over the configuration space T2 . This definition gives an abstract version of the setting seen in the a priori unstable systems. Theorem 2.9 (See Theorem 8.6). Suppose H∗ ∈ C r , r ≥ 2, (H∗ , c∗ ) is of Aubry-Mather type, and Γ ⊂ R2 is a smooth curve containing c∗ in the relative interior. Then there is σ > 0 such that for all c ∈ Bσ (c∗ ) ∩ Γ, the following dichotomy holds for a C r -residual subset of H ∈ Vσ (H∗ ): 0 1. Either the projected Ma˜ n´e set NH (c) is contractible as a subset of T2 (Mather mechanism (Ma)); 2. Or there is a finite covering map Ξ such that the set

e 0 (ξ ∗ c) \ Ξ−1 N e 0 (c) N H◦Ξ H is totally disconnected (Arnold mechanism (Ar)). We say (H∗ , c∗ ) is of bifurcation Aubry-Mather type if there exist two normally hyperbolic invariant cylinders, such that the local Aubry set restricted to each cylinder satisfies the conditions of Aubry-Mather type. The precise definition is given in Definition 8.3. We will also consider a particular (and simpler) bifurcation. We say (H∗ , c∗ ) is of asymmetric bifurcation type if there exists one normally hyperbolic invariant cylinder and a hyperbolic periodic orbit, such that the Aubry set is either

FORCING RELATION

23

contained in the cylinder (and of Aubry-Mather type) or the periodic orbit, see Definition 8.5. We state the consequence of these definitions in terms of diffusion. Theorem 2.10 (See Theorem 8.7). Suppose H∗ ∈ C r , r ≥ 2, (H∗ , c∗ ) is of bifurcation Aubry-Mather type or asymmetric bifurcation type, and Γ ⊂ R2 is a smooth curve containing c∗ in the relative interior. Then there is σ > 0 and an open and dense subset R ⊂ Vσ (H∗ ) such that for each H ∈ R and each c ∈ Γ ∩ Bσ (c∗ ), (Bif ) holds on at most finitely many c’s, and for all other c’s either (Ma) or (Ar) holds. The following Proposition is a direct consequence of Theorem 2.9 and Theorem 2.10. Proposition 2.11. Suppose Γ ⊂ B 2 is a piecewise smooth curve of cohomologies such that for each c ∈ Γ, the pair (H∗ , c) is of Aubry-Mather type, bifurcation Aubry-Mather type, or asymmetric bifurcation type. Then there is σ > 0 and a residual subset Rσ (H∗ ) ⊂ Vσ (H∗ ), such that either (Ma), (Bif ) or (Ar) holds for each H ∈ Rσ (H∗ ) and each c ∈ Γ. Sm Proof. For a piecewise smooth Γ = i=1 Γi , we can extend each Γi to Γ0i smoothly, such that Γi is contained in the relative interior of Γ0i . We then apply Theorem 2.9 and Theorem 2.10 to each c ∈ Γi relative to the smooth curve Γ0i , to get the conclusion of our Proposition for c0 ∈ Bσ(c) (c), and H ∈ Rσ(c) (H∗ ) ⊂ Vσ(c) (H∗ ). The proposition then follows by considering a finite covering of Γ by Bσ(cj ) (cj ), and taking finite intersection of residual subsets Rσ(cj ) (H∗ ). We now describe the selection of cohomologies and prove AM type in each of the two regimes. Single resonance is covered in Chapter 3. Double resonance is split into two chapters: Chapter 4 covers the geometrical part, while Chapter 5 covers the variational part.

Chapter Three Normal forms and cohomology classes at single resonances

3.1

RESONANT COMPONENT AND NON-DEGENERACY CONDITIONS

Let (k1 , Γk1 ) ∈ K be a resonant segment in the diffusion path. Define the resonant component of H1 relative to the single resonance k1 as follows: X [H1 ]k1 (θ, p, t) = hk (p)e2πik·(θ,t) , k∈k1 Z

where hk are the Fourier coefficients of H1 (θ, p, t). Since [H1 ]k1 only depends on the variables k1 · (θ, t) and p, we define a function Zk1 = Zk1 (θs , p) : T × R2 → R by the relation Zk1 (k1 · (θ, t), p) = [H1 ]k1 (θ, p, t). For p0 ∈ Γk , define the following conditions: [SR1λ ] For all p ∈ Bλ (p0 ), the function Zk1 (·, p) achieves a global maximum at θ∗s (p) ∈ T, and for all θs ∈ T, Zk1 (θs , p) − Zk1 (θ∗s (p), p) < λd(θs , θ∗s (p))2 ,

∀θs ∈ T.

[SR2λ ] For all p ∈ Bλ (p0 ), there exist two local maxima θ1s (p) and θ2s (p) of the function Zk1 (., p) in T satisfying ∂θ2s Zk1 (θ1s (p), p) < λI,

∂θ2s Zk1 (θ2s (p), p) < λI

and for all θs ∈ T, Zk1 (θs , p)
2 < max{Zk1 (θ1s (p), p), Zk1 (θ2s (p), p)} − λ min{d(θs − θ1s ), d(θs − θ2s )} . Definition 3.1. For a given λ > 0, we say that H1 satisfies the condition SR(k1 , Γk1 , λ) if for each p0 ∈ Γk1 , at least one of [SR1λ ] and [SR2λ ] holds for the function Zk1 (θs , p).

NORMAL FORMS AND COHOMOLOGY CLASSES AT SINGLE RESONANCES

25

Figure 3.1: Single resonance after removing punctures

Proposition 3.2 (Proof is in Section 9.7). The set of H1 ∈ S r such that SR(k1 , Γk1 , λ) holds for some λ > 0 is open and dense. Let K be a large parameter, recall the strong additional resonances are defined by Kst (k1 , Γk1 , K) = {k2 ∈ Z3∗ : |k2 | ≤ K, Γk1 ∩ Sk2 6= ∅}. We√show generic forcing equivalence on each connected components of Γk1 minus O( )-neighborhoods of the strong double resonances, called punctures. The following theorem is the main result of this section, and the proof is given in Section 3.3 assuming propositions proved in the later sections. For M, K > 0 denote   [  ΓSR B2M √ (Γk1 ,k2 ) . (3.1) k1 (M, K) := Γk1 \ k2 ∈Kst (k1 ,Γk1 ,K)

Theorem 3.3. Suppose H1 satisfies the condition SR(k1 , Γk1 , λ) on Γk1 . Then there exist • K, M, 1 > 0 depending on (D, k1 , λ), σ = σ(k1 , H0 , , H1 ) > 0, • for every 0 <  < 1 , a residual subset R ⊂ Vσ (H0 + H1 ), such that the following holds for all H ∈ R: for each c ∈ ΓSR k1 (M, K) the associated Aubry or Ma˜ n´e sets satisfy (Ma), (Bif ), or (Ar). As a result, each connected component of ΓSR k1 (M, K) is contained in one forcing equivalent class. Refer to Figure 3.1 for an illustration.

26 3.2

CHAPTER 3

NORMAL FORM

The classical partial averaging theory indicates that after a coordinate change, the system has the normal form H0 + Zk1 + h.o.t. away from punctures. In order to state the normal form, we need an anisotropic norm adapted to the perturbative nature of the system. Define |β| α kH1 (θ, p, t)kCIr = sup  2 sup ∂(θ,t) ∂pβ H1 (θ, p, t) , (3.2) |α|+|β|≤r

where α ∈ (Z+ )3 , β ∈ (Z+ )2 are multi-indices and | · | denotes the sum of r the indices. The rescaled norm is similar √ to the C norm, but replace the p derivatives by the derivatives in I = p/ , hence the name. Theorem 3.4 (See end of this section). Let κ ∈ (0, 1) there is C = C(D, k1 , κ) > 1 such that the following hold. Suppose 0 < δ < C −1 κ; set K = Cδ −1 κ−1 and −1 M = δ −2 . Then for any c ∈ ΓSR ) there exists pc ∈ Γk1 such that k1 (M, δ c ∈ BκK √ (pc ),

(3.3)

and an C ∞ exact symplectic coordinate change homotopic to the identity Φ : T2 × BK √ (pc ) × T → T2 × BK √/2 (pc ) × T such that: 1. (Φ )∗ H = H0 + [H1 ]k1 + R, 2.

kRkCI2 ≤ Cδ.

(3.4)

√ kΠθ (Φ − Id)kCI2 ≤ Cδ 2 and kΠp (Φ − Id)kCI2 ≤ Cδ 2 .

Here the CI2 norm is evaluated on the set T2 × BK √/2 (pc ) × T. Remark 3.5. • The C ∞ coordinate change is obtained by approximating a coordinate change that is only C r−1 , see Section 14.1. The reason this can be done is that we only need C 2 estimates of the coordinate change. • When applying the theorem, κ will be chosen depending only on k1 , and in particular, independent of the “true small parameters” δ and . We use the idea of Lochak (see for example [53]) to cover the action space with double resonances. A double resonance p0 = Sk1 ∩ Sk2 corresponds to a periodic orbit of the unperturbed system H0 . More precisely, we have ω0 = Ω0 (p0 ) := ∇H0 (p0 ) which satisfies R(ω, 1) ∩ Z3 6= ∅. Denote by Tω0 = min{t > 0 : t(ω0 , 1) ∈ Z3 } the minimal period. The resonant lattice for p0 is Λ = SpanR {k1 , k2 } ∩ Z3 , and the resonant

27

NORMAL FORMS AND COHOMOLOGY CLASSES AT SINGLE RESONANCES

component is [H1 ]k1 ,k2 =

X

hk (p)e2πik·(θ,t) .

k∈Λ

Proposition 3.6 (See Section 14.1). Let p0 = Γk1 ,k2 , T = Tω0 (p0 ) . Then for a parameter C1 > 1, there exists C = C(r, C1 ) > 1, 0 = 0 (r) > 0 such that if K1 > C satisfies C1 T < 2√ , K1  then for each 0 <  < 0 there exists a C ∞ exact symplectic map Φ : T2 × BK1 √/2 × T → T2 × BK1 √ × T such that (Φ)∗ H = H0 + [H1 ]k1 ,k2 + R1 , where kR1 kCI2 ≤ CK1−1 , and kΠθ (Φ − Id)kC 2 ≤ CK1−2 , I

√ kΠp (Φ − Id)kC 2 ≤ CK1−2 . I

3

Denote Λ1 = SpanR {k1 } ∩ Z and Λ2 = SpanR {k1 , k2 } ∩ Z3 . Lemma 3.7. There is an absolute constant C > 0 such that if min{|k| : k ∈ Λ2 \ Λ1 } ≥ K > 0, we have

1

k[H1 ]k1 ,k2 − [H1 ]k1 kC 2 ≤ CK − 2 . Proof. Note that there is an absolute constant C > 0 such that for each two dimensional lattice Λ ⊂ Z3 , we have X 1 |k|−2− 2 < C. k∈Λ\{0}

Indeed, we can bound the sum above using the integral Z 1 |z|−2− 2 dz. |z|≥1, z∈SpanR Λ

Using the fact that khk kC 2 ≤ |k|2−r kH1 kC r , we have X X 1 k[H1 ]k1 ,k2 − [H1 ]kC 2 ≤ |k|2−r ≤ K − 2 k∈Λ2 \Λ1

k∈Λ2 \{0}

1

1

|k|2+ 2 −r < CK − 2 .

28

CHAPTER 3

The following lemma is an easy consequence of the Dirichlet theorem (see [53]). Lemma 3.8. There is C = C(D, k1 ) > 0 such that for each Q1 > 1 and each c ∈ Sk1 , there is a double resonance p0 with Tω0 < CQ1 , and kc − p0 k < C(Tω0 Q1 )−1 , where ω0 = ∇H0 (p0 ). √ Proof of Theorem 3.4. Denote τ = δ −2 ε and let C1 = C1 (k1 ) be the larger of the two constants in Lemmas 3.7 and 3.8. Apply Lemma 3.8 using √ the parameter τ 2 Q1 = C1 τ −1 , then each c ∈ Sk1 is contained in the T (p ≤ K  neighborhood c) of a double resonance pc , whose period T (ωc ) is at most C1 Q1 , where ωc = ∇H0 (pc ). Since Tω0 ≥ 1, we get a first estimate √ −2 kpc − ck < C1 Q−1 . 1 =τ =δ In particular, this implies if c is in the set (3.1) with M = δ −2 , we have pc ∈ / Kst (k1 , Γk1 , δ −1 ). Suppose k2 ∈ Z3∗ is such that pc = Γk1 ,k2 , and since Λ2 \ Λ1 (see Lemma 3.7) contains only vectors larger than δ −1 , we have Tωc ≥ C1−1 δ −1 . We get the improved estimate √ √ √ δ −2  −1 kpc − ck ≤ C1 (Tωc Q1 ) = C1 ≤ C12 δ −1  = κK , Tωc where we’ve set K = C12 κ−1 δ −1 . Moreover, from Lemma 3.7, k[H1 ]k1 ,k2 − [H1 ]k1 kC 2 < C1 δ = C1 κ−1 K −1 . C 3 κ−2

C1√ C√ 1 2 √ = Set C2 = C12 κ−2 , then Tωc ≤ δ−2 = C 2 κ−2 , and therefore  δ −2  K2  1 Proposition 3.6 applies with the parameters C2 and K. We obtain, for a constant C3 = C3 (C2 , κ), kR1 kCI2 ≤ C3 K −1 .

Therefore H ◦ Φ = H0 + [H1 ]k1 + R, where

kRkCI2 = k([H1 ]k1 ,k2 − [H1 ]k1 ) + R1 kCI2 1

1

≤ C3 C1 κ−1 K −1 + C1 K − 2 ≤ 2C1 K − 2 as long as K > C3 κ−2 , which is ensured if δ < C3−1 κ. Moreover, kΠθ (Φ − Id)kCI2 ≤ C3 K −2 ≤ C3 δ 2 , √ √ kΠp (Φ − Id)kCI2 ≤ C3 K −2  ≤ C3 δ 2 .

29

NORMAL FORMS AND COHOMOLOGY CLASSES AT SINGLE RESONANCES

The theorem follows by setting C = C3 .

3.3

THE RESONANT COMPONENT

Using the fact that k1 = (k11 , k10 ) ∈ Z2 × Z is space irreducible, there is k2 = (k21 , k20 ) such that B0T := k11 k21 ∈ SL(2, Z). (This is the only pace where the space irreducibility condition is used.) Define   (k1 )T B =  (k2 )T  ∈ SL(3, Z), 0 0 1 and    θs θf  = B θ , t t  s

f

s

f

ΦL (θ, p, t) = (θ , θ , p , p , t),

 ps = (B0T )−1 p. pf



Let us note that kBk, kB −1 k depends only on k1 . One verifies that ΦL is a linear exact symplectic coordinate change. Note that θs = k1 · (θ, t) and [H1 ]k1 ◦ ΦL depend only on θs , ps , pf . Let us write N = Φ∗L (Φ∗ H ) = H0 (ps , pf ) + Z(θs , ps , pf ) + R(θs , θf , ps , pf , t),

(3.5)

where we abused notation by keeping the name of H0 and R after the coordinate change. Let us also abuse notation by writing θ = (θs , θf ) and p = (ps , pf ). Note that N is defined on the set T2 × BK1 √ (p1 ) × T,

where

K1 =

K , 2kB −1 k

p1 = (B0T )−1 p0

and the resonant segment Γk1 is represented by Γs = {p : ∂ps H0 = 0} in the new coordinates. For , δ > 0 and p1 ∈ R2 , define: R(, δ, p1 , K1 ) n = N = H0 + Z(θs , p) + R(θ, p, t) :

kRkCI2 (T2 ×BK

1

√

 (p1 )×T)

o 0 depending on D, λ such that if K1 > C, 0 <  < 0 , and 0 < δ < δ0 , for each N ∈ R(, δ, p1 ) and each c ∈ BK1 √/2 (p1 ) ∩ Γs the pair (N, c) is of Aubry-Mather type. Proposition 3.10 (See Theorem 9.4). Consider N as in Proposition 3.9, and assume that Z(θs , p) satisfies condition [SR2λ ] at p2 ∈ Γs . Then there is δ0 , 0 , C > 0 depending on D, λ such that if K1 > C, 0 <  < 0 , and 0 < δ < δ0 , there is an open and dense subset R1 ⊂ R(, δ, p1 , K1 ), such that for each c ∈ BK1 √/2 (p1 ) ∩ Γs the pair (N, c) is of bifurcation Aubry-Mather type. Proof of Theorem 3.3. Denote Dr (p) = T2 × Br (p) × T for short. Let Γ be a connected component of (3.1) and c0 ∈ Γ. Setting κ = 18 (max{kBk, kB −1 k})−2 which depends only on k1 , we apply Theorem 3.4 with parameters κ, δ > 0 to obtain C0 = C0 (D, k1 ) > 1, p0 ∈ Γk such that c0 ∈ BκK0 √ (p0 ), and the normal form transformation Φ : DK0 √/2 (p0 ) → DK0 √ (p0 ). Let K1 = K/(2kB −1 k), Φ = ΦL ◦ Φ : DK1 √ (p1 ) → DK0 √ (p0 ) and N = Φ∗ H . The mapping Φ∗ : C r (DK0 √ (p0 )) → C r (DK1 √ (p1 )) is a continuous linear mapping between Banach spaces, and we have N ∈ R(, C0 δ, p1 , K1 ). Under Φ, the correspondence between cohomolgies is given by the linear map c 7→ (B0T )−1 c. Suppose √ c, p0 are mapped √ to c1 , p1 ; then we have kc1 − p1 k < kB −1 kkc − p0 k < K0 /(8kB −1 k) = K1 /4, and hence c1 ∈ BK1 √/4 (p1 ) ∩ Γs . For 1 small enough, we choose δ sufficiently small such that Proposition 3.9 or 3.10 applies, depending on whether c0 satisfies [SR1λ ] or [SR2λ ]. It follows that for each c ∈ BK1 √/2 (p1 ) ∩ Γs , (Ma), (Bif ) or (Ar) holds on a C r -residual subset Rσ (N ) of N ∈ Vσ (N ), in the space C r (DK1 √ (p1 )). By continuity of Φ∗ , (Φ∗ )−1 Vσ (N ) is an open set of C r (DK0 √ (p0 )) containing H , and therefore contains a neighborhood Vσ0 (H ) of H . The subset Rσ0 (H ) = (Φ∗ )−1 Rσ (N ) ∩ Vσ0 (H ) is a residual subset of Vσ0 (H ). Applying Lemma 2.8 (invariance of diffusion mechanisms under symplectic coordinate changes), for each H ∈ Rσ0 (H ) and c ∈ (B0T )BK1 √/2 (p1 ) ∩ Γs ⊃ Bσ (c0 ) ∩ Γ, one of (Ma), (Bif ) or (Ar) hold. We now apply the above argument to each c ∈ Γ, and establish (Ma), (Bif ) or (Ar) for a neighborhood Bσ(c) (c) of c, on a C r residual subset Rc of Vσ(c) (H). By compactness, Γ can be covered by finitely many Bσ(ci ) (ci )’s, then by taking intersections over Rci , we conclude that our conditions hold on all c ∈ Γ, over a residual subset of Vσ0 (H ), where σ0 = min σ(ci ). The theorem follows.

Chapter Four Double resonance: geometric description In this chapter we describe the non-degeneracy condition and hyperbolic cylinders in the double-resonance regime. In the next chapter, we will return to variational setting, define the cohomology classes, and prove their forcing equivalence.

4.1

THE SLOW SYSTEM

Let k ∈ Kst (k1 , Γ, K), and denote p0 = Γk1 ,k , and ω0 = ∇H0 (p0 ). Define Λ = SpanR {k1 , k} ∩ Z3 , and choose k2 ∈ Z3∗ such that Λ = SpanZ {k1 , k2 }. It is always possible to choose |k2 | ≤ |k1 | + K. P Given H1 = k∈Z3 hk (p)e2πik·(θ,t) , let [H1 ]Λ =

X

hk (p)e2πik·(θ,t) .

k∈Λ

After a symplectic coordinate change defined on the set T2 × BK √ (p0 ) × T (see Theorem 14.1), the system has the normal form: 3

N = Φ∗ H = H0 + [H1 ]Λ + O( 2 ). The system N is conjugate to a perturbation of a two degrees of freedom mechanical system after a coordinate change and an energy reduction. The details are given in Chapter 14, here we give a brief description. Let k3 ∈ Z3 be such that   B T = k1 k2 k3 ∈ SL(3, Z). To define a symplectic coordinate change, we consider the corresponding autonomous system N (θ, p, t) + E, and consider the coordinate change (θ, p, t, E) =     θ ϕ√ −1 =B , t τ/ 

ΦL (ϕ, I, τ, F ),   √  p − p0 I T =B . E + H0 (p0 ) F

(4.1)

32

CHAPTER 4

One checks that T2 × R2 × T × R,


dθ ∧ dp + dt ∧ dE  ΦL → T2 × R2 × T × R,

1 √ (dϕ ∧ dI + dτ ∧ dF ) 



is an exact symplectic coordinate change. The transformed Hamiltonian (N + E) ◦ ΦL is no longer Tonelli in the standard sense, however by using a standard energy reduction on the energy level 0, with τ as the new time takes the system to
√ √ 1 K(I) − U0 (ϕ) + P (ϕ, I, τ ) , ϕ ∈ T2 , I ∈ R2 , τ ∈ T, β where β = k3 · (ω0 , 1),


1 2 K(I) = B0 ∂pp H0 (p0 )B0T , 2

 T k B0 = 1T , k2

and U (k1 · (θ, t), k2 · (θ, t)) = −[H1 ]Λ (θ, p0 , t). The system H s (ϕ, I) = K(I) − U (ϕ) = K − U

(4.2)

is called the slow mechanical system (or slow system for short) and also denote Hs (ϕ, I, τ ) =


√ 1 K(I) − U0 (ϕ) + P (ϕ, I, τ ) . β

(4.3)

The non-degeneracy conditions at the double resonance p0 are stated in terms of H s .

4.2

NON-DEGENERACY CONDITIONS FOR THE SLOW SYSTEM

We consider the (shifted) energy as a parameter. For each E > 0, by the Maupertuis principle, the Hamiltonian dynamics on the energy surface SE = {(ϕ, I) : H s (ϕ, I) + min U (ϕ) = E} is a reparametrization of the geodesic flow for the Jacobi metric gE (ϕ)(v) = 2(E + U (ϕ)) K −1 (v),
−1 2 where K −1 (v) = 12 ∂II K v · v is the Lagrangian associated to the Hamiltonian K(I). We will be interested in a special homology class h = (0, 1) ∈ Z2 ' H1 (T2 , Z).

DOUBLE RESONANCE: GEOMETRIC DESCRIPTION

33

˙ 1) ' 0, i.e. orbits It represents orbits of the original system satisfying k1 · (θ, that travel close to the resonance Γk1 . We assume the following non-degeneracy conditions: [DR1h ] For each E ∈ (0, ∞), each shortest closed geodesic (called a loop) of gE in the homology class h is a hyperbolic orbit of the geodesic flow. [DR2h ] At all but finitely many bifurcation values, there is only one gE -shortest loop. At each bifurcation value E, there are exactly two shortest gE loop denoted γhE and γ¯hE . h [DR3 ] At bifurcation value E∗ , d(`E (γ E d(`E (γhE )) h )) |E=E ∗ 6= |E=E ∗ , dE dE where lE denote the gE length of a loop. We now discuss the conditions at the critical shifted energy E = 0. The Jacobi metric g0 becomes degenerate at one point ϕ∗ = argminϕ U (ϕ). By performing a translation, we may assume ϕ∗ = 0. While g0 is no longer a Riemannian metric, it still makes sense to talk about shortest geodesics in the class h. Let γh0 be one such shortest loops. Consider the following cases: 1. 0 ∈ γh0 and γh0 is not self-intersecting. Call such homology class h simple critical and the corresponding geodesic γh0 simple loop. 2. 0 ∈ γh0 and γh0 is self-intersecting. Call such homology class h non-simple and the corresponding geodesic γh0 non-simple. 3. 0 6∈ γh0 , then γh0 is a regular geodesic. Call such homology class h simple non-critical. Mather [65] proved that generically only these three cases occur (see the text that follows for the precise claim). Lemma 4.1. Let h be a non-simple homology class. Then for a generic potential U the curve γh0 is the concatenation of two simple loops, possibly with multiplicities. More precisely, given h ∈ H1 (T2 , Z) generically there are simple homology classes h1 , h2 ∈ H1 (T2 , Z) and integers n1 , n2 ∈ Z+ such that the corresponding minimal geodesics γh01 and γh02 are simple and h = n1 h1 + n2 h2 . We call γh0 extensible if there exists a family of shortest curves γhE converging to it in the Hausdorff topology. Consider the lift of γhE to the universal cover R2 , then as E → 0 it converges to a periodic curve in R2 that consists of concatenation of γh01 and γh02 . Suppose the order that the curves are concatenated is given by γh0σ , . . . , γh0σn . 1

Lemma 4.2 (See Section 10.6). Assume that γh0 is extensible, then the sequence (σn , . . . , σn ), as described, is uniquely determined up to cyclic permutation. We

34

CHAPTER 4

write γhE → γh0σ1 ∗ · · · ∗ γh0σn ,

as

E → 0.

We note that (0, 0) is a fixed point of the Hamiltonian flow H s (ϕ, I) which 2 is hyperbolic if ∂θθ U (0) > 0. Any simple g0 -shortest loop corresponds to a homoclinic orbit of the fixed point (0, 0). We impose the following non-degeneracy conditions: [DR1c ] (0, 0) is a hyperbolic fixed point with distinct eigenvalues −λ2 < −λ1 < 0 < λ1 < λ2 . Let v1± , v2± be the eigendirections for ±λ1 , ±λ2 . c [DR2 ] There is a unique g0 -shortest loop in the homology h. If it is non-simple, then it is the concatenation of two simple loops γh01 and γh02 . c [DR3 ] If γh0 is simple critical, then it is not tangent to the hv2+ , v2− i plane. If γh0 is non-simple, then each of γh01 and γh02 is not tangent to the hv2+ , v2− i plane. c [DR4 ] – If γh0 is simple non-critical, then γh0 is hyperbolic. – If γh0 is simple critical, then γh0 is non-degenerate in the sense that it is the transversal intersection of the stable and unstable manifolds of (0, 0). – If γh0 is non-simple, then each of γh01 and γh02 is non-degenerate. The genericity of these conditions is summarized in the following statement. Proposition 4.3. The conditions [DR1h − DR3h ] and [DR1c − DR4c ] hold on an open and dense set of potentials U ∈ C r (T2 ), for r ≥ 4. This proposition is the consequence of several statements given in the next section, namely Theorem 4.5, Proposition 4.6, and Proposition 4.7.

4.3

NORMALLY HYPERBOLIC CYLINDERS

Conditions [DR1h ] − [DR3h ] ensures that for each E0 > 0, the set [ ηhE E∈(E0 −δ,E0 +δ)

is a normally hyperbolic invariant cylinder. This cylinder does not necessarily extend to the shifted energy E = 0. The following statement ensures existence of cylinders near critical energy, using the conditions [DR1c ] – [DR4c ]. Recall that γh0 is a shortest curve of homology h in the critical energy. The corresponding Hamiltonian orbit is called ηh0 . Due to the symmetry of the 0 system, we also have the shortest curve γ−h which coincides with γh0 but has a different orientation. Theorem 4.4 (See Chapter 10). Suppose that H s satisfies condition [DR1c ] − [DR4c ].

35

DOUBLE RESONANCE: GEOMETRIC DESCRIPTION

Figure 4.1: Extension of homoclinics to periodic orbits, simple and non-simple.

1. If γh0 is simple, then there is e > 0 depending on H s such that: E a) For each 0 < E < e, there exist periodic orbits ηhE and η−h , such that the E 0 E 0 projections γh → γh and γ−h → γ−h in the Hausdorff topology. b) For each −e < E < 0, there exists a periodic orbit ηcE which shadows the 0 concatenation of ηh0 and η−h .

The union [ 0 0, let c¯h : (0, E] ¯ → H 1 (T2 , R) be a C 1 function such that c¯h (E) is For E in the relative interior of LF βH (λE h h). The following hold. 1. If E is not a bifurcation energy, then AH s (¯ ch (E)) = γhE . 2. If E is a bifurcation energy, AH s (¯ ch (E)) = γhE ∪ γ¯hE . 3. If h is simple, then the limit limE→0 LF βH s (λE h h) contains a segment of nonzero width. We assume, in addition, that c¯h (0) is in the relative interior of this segment. a) If h is simple critical, then AH s (¯ ch (0)) = γh0 ; for each 0 ≤ λ < 1, we have AH s (λ¯ ch (0)) = {ϕ = 0}. b) If h is simple non-critical, then AH s (¯ ch (0)) = γh0 ∪ {ϕ = 0}; for each 0 ≤ λ < 1, AH s (λ¯ ch (0)) = {ϕ = 0}. 4. If h is non-simple, then the limit limE→0 LF βH s (λE h h) is a single point. Note that due to symmetry of the system, LF βH (−λh) = −LF βH (λh). Denote c¯−h (E) = −¯ ch (E). We now choose the cohomology classes for H s as

41

DOUBLE RESONANCE: FORCING EQUIVALENCE

Figure 5.1: Choice of cohomology for H s . Left: simple case; right: non-simple case.

follows: if h is simple (either critical or non-critical), we choose     [ [ ¯h = Γ ¯ h (E) ¯ = Γ c¯h (E) ∪  λ¯ ch (0) , ¯ 0≤E≤E

0≤λ≤1

(5.2)

¯ DR ¯ DR ¯ ¯ ¯ ¯ ¯ Γ =Γ h h (E) = Γh ∪ Γ−h = Γh ∪ −Γh .


¯ DR is a continuous curve connecting c¯h (E), ¯ 0, and c¯−h (E). ¯ The curve Γ h If h is non-simple, then h = n1 h1 + n2 h2 is a combination of simple homologies h1 and h2 . Let 0 < µ < e be parameters. Define [ ¯ eh = Γ c¯h (E). (5.3) ¯ e≤E≤E

Since h1 is simple critical, we choose a continuous curve c¯µh1 (E) such that k¯ cµh1 (0) − c¯h (0)k < µ. We then define  ¯ e,µ =  Γ h1

 [

0≤E≤e+µ



c¯µh1 (E) ∪ 

 [

λ¯ ch1 (0) .

(5.4)

0≤λ≤1

¯ e,µ is a continuous curve connecting c¯µ (e + µ) with 0 (see Fig. 5.1, right). We Γ h1 h1 then define ¯h = Γ ¯ eh ∪ Γ ¯ e,µ , Γ ¯ DR ¯h ∪ Γ ¯ −h . Γ =Γ (5.5) h h1 ¯ DR consists of three connected components, Γ ¯e , Γ ¯e , Let us note that the set Γ h h −h e,µ e,µ ¯ ∪Γ ¯ and Γ h1 −h1 .

42 5.2

CHAPTER 5

AUBRY-MATHER TYPE AT A DOUBLE RESONANCE

Proposition 5.2. Suppose H s = K(I) − U (ϕ) satisfies the non-degeneracy assumptions. Given C > 0, there is 0 , δ > 0 depending on H s and C such that for each 0 <  < 0 , if U 0 ∈ Vδ (U ) and kP kC 2 < C, then for Hs =


√ 1 K(I) − U 0 (ϕ) + P (ϕ, I, τ ) , β

¯ DR is of one of four types: Aubry-Mather type, bifurcation Aubryeach c¯ ∈ Γ h Mather type, asymmetric bifurcation type, or AeHs is a hyperbolic periodic orbit. Note that this applies to all types of homologies: simple critical, simple noncritical, and non-simple. Proof. We prove our proposition by referring to technical statements proved in later chapters. (1) Simple, critical homology. Denote c0 = c¯h (0). Theorem 11.6 states that there exist 0 , , δ > 0 such that for all 0 <  < 0 , U 0 ∈ Vδ (U ), and c ∈ Be (c0 ), the pair Hs , c is of Aubry-Mather type. The same theorem also states Hs , λc0 for 0 ≤ λ ≤ 1 is of Aubry-Mather type. This covers the cohomologies [ [ c∈ c¯h (E) ∪ λ¯ ch (0). 0≤E≤e

0≤λ≤1

S

The cohomologies e≤E≤E¯ c¯h (E), are covered in Theorem 11.1, which states that for each e > 0, there is , δ as in our proposition, such that with respect to Hs , the cohomology c¯h (E), E ≥ e is of Aubry-Mather type if γhE is the unique shortest curve, and of bifurcation Aubry-Mather type if there are two shortest ¯ h are of Aubry-Mather or bifurcation curves. As a result, all cohomologies in Γ ¯ −h . This proves our Aubry-Mather type, and by symmetry, the same holds for Γ proposition in the simple homology case, see (5.2). ¯h (2) Non-simple homology. In the non-simple case, the cohomology curve Γ is the disjoint union of two parts, namely     [ [ [ ¯ eh = ¯ e,µ =  Γ c¯h (E), Γ c¯µh1 (E) ∪  λ¯ ch1 (0) . h1 ¯ e≤E≤E

0≤E≤e+µ

0≤λ≤1

¯ e,µ is of AubryWe note that h1 is a simple homology and therefore each c ∈ Γ h1 Mather type in the same way as in case (1). On the other hand, each homology ¯ e is of Aubry-Mather or bifurcation Aubry-Mather type since Theorem 11.1 in Γ h applies the same way to simple and non-simple homology. We conclude that our Proposition holds in the non-simple case. (3) Simple, non-critical homology. The high-energy case follows from Theorem 11.1 in the same way as before. The critical energy follows from Theorem 11.5, the main difference is that the critical energy is an asymmetric bifur-

43

DOUBLE RESONANCE: FORCING EQUIVALENCE

cation (see Definition 8.5). For c = λ¯ ch (0) where 0 ≤ λ < 1, the Aubry set is a single periodic orbit as a perturbation of the hyperbolic fixed point (0, 0). Corollary 5.3. Suppose Hs is from Proposition 5.2; then there is σ > 0 and a ¯ DR satisfies residual subset R of Vσ (Hs ), such that if H 0 ∈ R, then each c¯ ∈ Γ h one of the diffusion mechanisms (Ma), (Bif ), or (Ar). Proof. Note that Proposition 2.11 applies when c is of Aubry-Mather type, bifurcation Aubry-Mather type, or asymmetric bifurcation type. Then using Proposition 5.2, we only need to show the same conclusion holds if we add a fourth case, when the Aubry set is a single hyperbolic periodic orbit. However, e 0 = Ae0 is discrete, so (Ma) applies. in this case, N We now revert to the original coordinate system. Denote p0 = Γk1 ,k2 the double resonance point. For a given cohomology class c¯ ∈ R2 , we consider the pair c, α defined by the equality    √  c − p0 c¯  = BT . (5.6) −α + H0 (p0 ) αHs (¯ c)  Then α = αN (c) (compare to (4.1)). If (5.6) is satisfied, we denote (c, −α) = Φ∗L (¯ c, αHs (¯ c)),

c = Φ∗L,Hs (¯ c).

Moreover, let us consider the autonomous version of the Aubry set: AeHs +F (¯ c) = {(ϕ, I, τ, −Hs (ϕ, I, τ )) :

(ϕ, I, τ ) ∈ AeHs (¯ c)},

f and N e . It is proven in Proposition 14.9 that and similarly define M     eN +E (c) = ΦL N eH s +F (¯ AeN +E (c) = ΦL AeHs +F (¯ c) , N c ) .  

(5.7)

Proposition 5.4. Suppose c, c¯ are related by (5.6). Then with respect to N , c satisfies one of the diffusion mechanisms (Ma), (Bif ), or (Ar) if and only if c¯ does the same with respect to Hs . Proof. Suppose N , c and Hs , c¯ satisfy our assumption. Consider the following subsets: √ S = {(ϕ, I, τ, F ) : Hs + F = 0} ⊂ T2 × R2 × T × R, Σ0 = S ∩ {τ = 0}, ˆ 0 = Sˆ ∩ {t = 0}. Sˆ = {(θ, I, t, E) : N + E = 0} ⊂ T2 × R2 × T × R, Σ √ Note that S is a graph over T2 × R2 × T and Σ0 is a graph over T2 × R2 . We also have S is invariant under Hs , AeHs +F (¯ c) ⊂ S, and Σ0 is a global section of eH s (¯ the flow on S. Suppose (Hs , c¯) satisfies AeHs (¯ c) = N c), and (Ma) applies. 

44

CHAPTER 5

This implies AeHs +F (¯ c) ∩ Σ0 is a contractible subset of Σ0 . By (5.7),   AeN + (c) = ΦL AeHs +F , we get AeN +E (c) ∩ ΦL (Σ0 ) is contractible in ΦL (Σ0 ). Since ΦL is a conjugacy between Hamiltonian flows, ΦL (Σ0 ) is a global section for the flow of N + E ˆ Since Σ ˆ 0 is also a global section, the Poincar´e map on the energy surface S. ˆ P : ΦL (Σ0 ) → Σ0 is a diffeomorphism. As a result,   AeH  +E (c) ∩ Σ0 = P AeN +E (c) ∩ ΦL (Σ0 ) ˆ 0 . This proves (Ma) holds for (N , c). The proofs for the is contractible in Σ converse and the other mechanisms are similar.

CONNECTING TO ΓK1 ,K2 AND ΓSR K1

5.3

At this point it is natural to consider the cohomology class

¯ DR Φ∗L,Hs Γ := πc {Φ∗L (¯ c, αHs (¯ c)) : c¯ ∈ ΓDR h h }

(5.8)

(see (5.6)) where πc denotes the projection (c, −α) 7→ c ∈ R2 . We note that α is automatically determined by c via the relation α = α
N (c) = αH (c). ¯ DR for the original system, We would like to choose the cohomology Φ∗L,Hs Γ h but due to the  dependence of the map, the new set does not necessarily contain the double resonance point Γk1 ,k2 , nor does it connect to the cohomology ΓSR k1 already chosen at single resonance. To solve this problem, we will add three pieces of “connectors” to the set: Γcon is used to connect to Γk1 ,k2 , and Γcon ± to 0 SR connect to Γk1 . Then
∗ ¯ DR ∪ Γcon ∪ Γcon ∪ Γcon . ΓDR k1 ,k2 (, H1 ) = ΦL,Hs Γh 0 − +

(5.9)

See Figure 5.2. 5.3.1

Connecting to the double resonance point

We define c0 = Γk1 ,k2 ,

c0 = Φ∗L,Hs (0, αHs (0)),

Γcon = 0

[

{sc0 + (1 − s)c0 }

s∈[0,1]

and define h to be the homology class corresponding to k1 . Γcon is simply a line 0 segment connecting Γk1 ,k2 and c .

DOUBLE RESONANCE: FORCING EQUIVALENCE

45

Figure 5.2: Cohomology curve at double resonance, with connectors.

Proposition 5.5. Suppose H1 satisfies the conditions [DR1h − DR3h ] and [DR1c − DR4c ] at the double resonance Γk1 ,k2 relative to Γk1 . Then there is 0 , δ > 0 depending only on H s such that for 0 <  < 0 , H10 ∈ Vδ (H1 ), 0 ≤ s ≤ 1, and H0 = H0 + H10 : 0  0  0 0  NH 0 (sc0 + (1 − s)c0 ) = AH 0 (sc0 + (1 − s)c0 ) = AH 0 (c0 ) = AH 0 (c0 )    

is contractible in T2 . As a result the cohomologies in Γcon are forcing equivalent. 0 Proof. Consider c¯0 ∈ R2 , α ¯ 0 ∈ R defined by the formula  √    c¯0  0 −T = M . α ¯ 0  −αH (c0 ) + H0 (c0 ) Then according to (5.6), Φ∗L,Hs (¯ c0 , α ¯ 0 ) = c0 . According to Lemma 7.9, we have kαH√ (c ) − H (c )k ≤ C for some C depending only on H0 ; therefore 0 0 0  k¯ c0 k ≤ C . Recall that O = (0, 0) is a hyperbolic fixed point of the system H s , and AeH s (0) = O. By standard perturbation theory of hyperbolic sets, there is a √ neighborhood V 3 O, such that the system Hs = K −U 0 + P with U 0 ∈ Vδ (U ) and 0 <  < 0 , Hs admits a unique hyperbolic periodic orbit O contained in V . Moreover, using the upper semi-continuity of the Aubry set Corollary 7.2, by possibly choosing δ and 0 smaller, we ensure for all 0 ≤ λ ≤ 1, AeHs (λ¯ c0 ) ⊂ V ,  and therefore AeHs (λ¯ c0 ) = AeHs (0) = O . In this case the Aubry set has a unique static class; hence the Aubry set coincides with the Ma˜ n´e set, and also the discrete Aubry set is finite and therefore contractible. We now apply symplectic invariance (5.7) to get the same for the original system. 5.3.2

Connecting single and double resonance

The single-resonance cohomology ΓSR k1 (M, K) is defined in (3.1). For each double resonance Γk1 ,k2 , let ΓSR be the two connected components of (3.1) adjak1 ,k2 ,±

46

CHAPTER 5

cent to Γk1 ,k2 . We define connectors Γcon ± which connect the double-resonance
¯ DR to ΓSR cohomology curve Φ∗L Γ h k1 ,k2 ,± , respectively. 2 1 2 Let c ∈ R ' H (T , R) be a cohomology, and let ρH (c) denote the convex hull of rotation vectors of all c-minimal measures. This coincides with the Legendre-Fenchel transform relative to the alpha function ρH (c) = LF αH (c) ⊂ R2 . Note that ρH (c) is a set valued function taking values in convex sets. We have the following observations: • Let c ∈ ΓSR k1 . We showed in Section 3.3 that after a normal form and linear coordinate change and turning H to the normal form N , the Aubry set AeN (Φ∗L c) is a graph over the φf component. This implies the rotation vector ρN (Φ∗L c) is parallel to (0, 1) (in (θs , θf ) component). After reverting the coordinate change, this relation becomes k1 · (ρH (c), 1) = 0. • If c ∈ ΓDR ¯ be the associated cohomology for Hs (via (5.6)). Since the h , let c cohomologies we chose are in the channel associated to h = (0, 1), ρHs (¯ c) is parallel to h. By the same reasoning as in the single-resonance case, after reverting the coordinate change, we have k1 · (ρH (c), 1) = 0. We conclude that in both cases, the rotation vectors of the chosen cohomologies stay on the rational line Ωk1 = {ω :

k1 · (ω, 1) = 0}.

Let us also denote Ωk1 ,k2 = Ωk1 ∩ Ωk2 , c0 = Γk1 ,k2 , and ω0 = Ωk1 ,k2 . The main observation is that the rotation vectors of the curves ΓSR k1 (M, K) DR and Γk1 ,k2 are both contained in Ωk1 , and their intersection is non-empty; see Figure 5.3. To prove √ this statement, we show that the rotation vector of c ∈ ΓSR close to ω0 at its nearest point, while the rotation vector k1 (M, K) is O( ) √ −1 of c ∈ ΓDR is C E  away from ω0 when E is sufficiently large. Since both k1 ,k2 sets are connected and lie on the same line, they have to intersect. Proposition 5.6. Let ΓSR k1 ,k2 ,± and Γk1 ,k2 be as before. Suppose H1 satisfies the conditions [DR1h − DR3h ] and [DR1c − DR4c ] relative to Γk1 . Then there is 0 , δ > 0 depending only on H s such that such that if 0 <  < 0 , and H10 ∈ Vδ (H1 ), for H0 = H0 + H10 : 1. For each c ∈ ΓSR k1 (M, K), ρH0 (c) is a single point contained in Ωk1 . The

47

DOUBLE RESONANCE: FORCING EQUIVALENCE

Figure 5.3: Left: cohomology curves; right: rotation vectors.

function ρH (c) is continuous on ΓSR , and there is C > 0 such that kρH (c) − ω0 k ≤ Ckc − c0 k,

∀ c ∈ ΓSR k1 (M, K).

2. Let ch (E) = Φ∗L (¯ ch (E)) ∈ ΓDR k1 ,k2 . Then each ρH0 (ch (E)) is a single point contained in Ωk1 , and there are C, E0 > 0 such that √ kρH0 (ch (E)) − ω0 k ≥ C −1 E , E ≥ E0 . ¯ > 0 such that for E > E, ¯ the rotation vecAs a result, there exists E SR tors ρH0 (ch (E)) coincide with one of ρH0 (Γk1 ,k2 ,± ), see Figure 5.3. Similarly, for E sufficiently large, the rotation vectors ρH0 (c−h (E)) coincide with one of ρH0 (ΓSR k1 ,k2 ,± ) (which are not covered in the first case). Proof. In the single-resonance regime, after a linear coordinate change (Section 3.3) the system is converted to H0 + Z(θs , p) + R(θs , θf , ps , pf , t), where the invariant cylinder is given by (θs , ps ) = (Θs , P s )(θf , pf , t), and the Aubry set for any c ∈ Γs = {(∂p H0 (0), 1) · (1, 0, 0) = 0} is a graph over (θf , t) (see Theorem 9.1 and Theorem 9.2) This implies that any c admits a unique rotation vector, as different rotation vectors will result in intersecting minimizing orbits, which violates the graph theorem. By reverting the coordinate change, we also obtain ρH (c) ∈ Ωk1 . We then apply Corollary 7.8, to get (in general) √ kρH (c) − ∇H0 (c)k ≤ C , while k∇H0 (c) − ω0 k = k∇H0 (c) − ∇H0 (c0 )k ≤ Ckc − c0 k √ and recall that kc − c0 k ≥ K  for any c ∈ ΓSR k1 , item (1) follows. In the double-resonance case, the cohomology curves c¯h (E) at high-energy are of Aubry-Mather type corresponding to the homology class h = (1, 0). The same arguments as in the single-resonance case imply ρHs (c) is unique, and contained in the line ω1 = 0. Again by reverting the coordinate change, we

48

CHAPTER 5

obtain the desired property for ρH− (c). For the inequality, note that if any point (ϕ, I, t) in the Aubry set AeHs (c) satisfies H s (ϕ, I) ≥ E0 , we have K(I) ≥ H s (ϕ, I) − kU kC 0 ≥ 12 H s (ϕ, I) if E0 ≥ 2kU kC 0 . As a result min

kIk ≥ C −1

eH s (¯ (ϕ,I,τ )∈A ch (E)) 

p

min

√ K(I) ≥ C −1 E

eH s (¯ (ϕ,I,τ )∈A ch (E)) 

for a suitable C > 1. Reverting the coordinate change ΦL implies √ √ √ min kp − p0 k ≥ C −1 kIk  ≥ C −1 E . eH (ch (E)) (θ,p,t)∈A 

Finally, we apply Corollary 7.8 again to get our inequality. We have the following lemma: Lemma 5.7 (Proposition 6 of [56]). Let c1 , c2 ∈ H 1 (T2 , R) and ρ ∈ H1 (T2 , R) satisfy ρH (c1 ) = ρH (c2 ) = ρ, and both c1 , c2 lie in the relative interior of LF βH (ρ). Then AeH (c1 ) = AeH (c2 ). Proposition 5.8. Suppose H1 satisfies the conditions [DR1h − DR3h ] and [DR1c − DR4c ] relative to Γk1 . Then there is 0 , δ > 0 depending only on H s SR such that for 0 <  < 0 and H10 ∈ Rδ (H1 ), there is c± 1 ∈ Γk1 (M, K) and ± DR 0 0 c2 ∈ Γk1 ,k2 , such that for all 0 ≤ λ ≤ 1 and H = H0 + H1 ± ± ± ± ± ± 0 0 0 0 NH 0 (λc1 + (1 − λ)c2 ) = AH 0 (λc1 + (1 − λ)c2 ) = AH 0 (c1 ) = AH 0 (c2 )    

is contractible in T2 . As a result both cohomology curves [ [ + − Γcon {sc+ Γcon {sc− + = − = 1 + (1 − s)c2 }, 1 + (1 − s)c2 } s∈[0,1]

s∈[0,1]

are contained in a single forcing equivalent class. DR Proof. Proposition 5.6 implies the curves ρH (ΓSR k1 ) and ρH (Γk1 ,k2 ) overlap on 0 an interval contained in Ωk1 , for all H1 ∈ Vδ (H1 ). In particular, there must be DR c1 ∈ ΓSR k1 and c2 ∈ Γk1 ,k2 where they both have rational rotation vectors. We 0 now assume that H = H0 + H10 satisfies the residual condition that all Aubry sets with rational rotation vector are supported on a hyperbolic periodic orbit, in this setting A = N , c1 , c2 are contained in the relative interior of LF β (ρ). Lemma 5.7 now implies all the Aubry sets AH (λc1 + (1 − λ)c2 ), 0 ≤ λ ≤ 1 coincide, which implies (Ma) applies to the whole segment. The proposition follows.

DOUBLE RESONANCE: FORCING EQUIVALENCE

5.4

49

JUMP FROM NON-SIMPLE HOMOLOGY TO SIMPLE HOMOLOGY

As described, when h is not a simple homology, the cohomology class ΓDR k1 ,k2 as chosen is not connected. More precisely, ΓDR k1 ,k2 consists of three connected components (see also (5.3) and (5.9)):
¯ eh ∪ Γcon ¯ e−h ∪ Γcon ¯ e,µ ∪ Γ ¯ e,µ ∪ Γcon Φ∗L Γ Φ∗L Γ Φ∗L Γ + , − , 0 . h1 −h1 We will show forcing equivalence of the components by the following: Theorem 5.9 (Chapter 12). Suppose the slow system H s satisfies the condition [DR1c ] − [DR4c ], and that the associated homology h = n1 h1 + n2 h2 is nonsimple. Then there exist e, µ, 0 , δ > 0 depending on H s , such that there is a residual subset Rδ (H1 ) of Vδ (H1 ), and for each H10 ∈ Rδ (H1 ), there is E1 , E1 ∈ (e, e + µ) such that
Φ∗L (¯ ch (E1 )) , Φ∗L c¯hµ1 (E2 ) are forcing equivalent, with respect to H0 + H10 . The same conclusions apply when h, h1 are replaced with −h, −h1 . See the dashed line in Figure 5.2.

5.5

FORCING EQUIVALENCE AT THE DOUBLE RESONANCE

We summarize all of our constructions in the following theorem. Theorem 5.10. Suppose H1 satisfies all non-degeneracy conditions of Γk1 and Γk1 ,k2 . Then there is 1 = 1 (H0 , H1 ) > 0, δ = δ(H0 , H1 ) > 0, such that the following hold for all H10 ∈ Vδ (H1 ), and 0 <  < 1 . There is a subset DR 0 ΓDR k1 ,k2 = Γk1 ,k2 (H0 , H1 , ) satisfying • Γk1 ,k2 ⊂ ΓDR k1 ,k2 , SR • ΓDR intersects both ΓSR k1 ,k2 k1 ,k2 ,+ and Γk1 ,k2 ,− , σ = σ(H0 , H1 , ) > 0, and a residual subset Rσ (H0 + H10 ) of Vσ (H0 + H10 ), such that for all H ∈ R(H0 + H10 ), all of ΓDR k1 ,k2 are forcing equivalent. Proof. The set ΓDR k1 ,k2 is defined in (5.9), and its two properties hold by construction. During the proof, we say a property holds “after a residual perturbation” if it holds on a residual subset of a neighborhood of the corresponding Hamiltonian. If finitely many properties each hold after a residual perturbation, then they hold simultaneously after a residual perturbation. Moreover, if a property holds after a residual perturbation, and is invariant under coordinate changes, then it also holds in the new coordinate after a residual perturbation, provided

50

CHAPTER 5

the coordinate change is smooth enough. This is the case for our system since all coordinate changes are C ∞ . Let δ be the smallest parameter depending on H s such that all of Corollary 5.3, Proposition 5.5, and Proposition 5.8 hold. Let δ1 = δ1 (H0 , H1 ) be such that for any H10 ∈ Vδ1 (H1 ), the associated slow system is in Vδ (H s ). Then for 0 <  0 and S all k1 ∈ K, and that for all the strong double resonances k2 ∈ k1 ∈K Kst (k1 , Γk1 , λ), H1 satisfies the non-degeneracy conditions DR(k1 , Γk1 , k2 ). For each k1 , Theorem 3.3 applies. Therefore, there exists k11 (H0 , λ) > 0 such that the theorem applies for each H0 +H1 with 0 <  < 1 . Since SR(k1 , Γk1 , λ) is an open condition, there exists δ k1 = δ k1 (H0 , H1 ) such that the conclusion of the theorem holds for all H10 ∈ Vδk1 (H1 ). For each k1 , k2 , the conclusion of Theorem 5.10 holds for all H10 ∈ Vδk1 ,k2 (H1 ) and 0 <  < k11 ,k2 (H0 , H1 ). Define   k1 ,k2 k1 1 (H0 , H1 ) = min min 1 , min 1 , k1 ,k2

 δ(H0 , H1 ) = min

min δ

k1 ,k2

k1

k1 ,k2

, min δ

k1

k1

 ,

S S where the minimum is taken over all k1 ∈ K and k2 ∈ k1 ∈K Kst (k1 , Γk1 , λ). 1 and δ are positive since all minima taken are over a finite set of positive numbers. The conclusions of Theorems 3.3 and 5.10 apply to H10 ∈ Vδ (H1 ) and 0 <  < 1 . Define   [ [ ΓSR . Γ∗ (H0 , H10 , ) = ΓDR k1 ∪ k1 ,k2 k1 ∈K

k2 ∈Kst (k1 ,λ)

51

DOUBLE RESONANCE: FORCING EQUIVALENCE

For each single resonance k1 , the union SR ΓSR ∗ (k1 ) = Γk1 ∪

[

ΓDR k1 ,k2

k2 ∈Kst (k1 ,λ)

is contained in a single equivalent class, since each ΓDR k1 ,k2 is forcing equivalent, and they connect all the disconnected pieces from ΓSR k1 . If two single resonances SR 0 0 0 Γk1 and Γk1 intersect at Γk1 ,k1 , then Γ∗ (k1 ) and ΓSR ∗ (k1 ) also intersect at DR DR Γk1 ,k10 , since Γk1 ,k10 is contained in both Γk1 ,k0 and Γk0 ,k1 . As a result, the entire 1 1 Γ∗ (H0 , H10 , ) is contained in a single forcing equivalent class. Finally, if U1 , . . . , UN are S open sets which intersect P, by setting 0 small enough, they also intersect k1 ∈K ΓSR union is obtained from P k1 , since the said √ by removing finitely many neighborhoods of size O( ). Therefore, U1 , . . . , UN also intersect Γ∗ (H0 , H10 , ).

Part II

Forcing relation and Aubry-Mather type

Chapter Six Weak KAM theory and forcing equivalence In this chapter we give an introduction to weak Kolmogorov-Arnold-Moser (KAM) theory and forcing relation. Most of the presentations follow [11] and [37]. One change from the standard presentation is that we need to modify the definition of Tonelli Hamiltonians to allow different periods in the t component.

6.1

PERIODIC TONELLI HAMILTONIANS

A C 2 Hamiltonian H : Tn × Rn × R → R is called (time-periodic) Tonelli if it satisfies: 1. 2. 3. 4.

(Periodicity) There is 0 < $ = $H such that H(θ, p, t + $) = H(θ, p, t). 2 (Convexity) ∂pp H(x, p, t) is strictly positive definite as a quadratic form. (Superlinearity) limkpk→∞ H(x, p, t)/kpk = ∞. (Completeness) The Hamiltonian vector field generates a complete flow on Tn × Rn . We denote by φs,t H the flow from time s to time t, and by φH the flow φ0,$ . H

The Lagrangian associated to H is L = LH (θ, v, t) = sup{p · v − H(θ, p, t)}, and the Hamiltonian flow is conjugate to the Euler-Lagrange flow via the Legendre transform L : (θ, p, t) 7→ (θ, ∂p H(θ, p, t), t). For the most part, we will restrict to Hamiltonians with $ = 1, namely, defined n on Tn × R √ × T, but near double resonances we need to consider Hamiltonians that are -periodic. It is helpful to consider a family of Hamiltonians which satisfy these properties uniformly. For D > 0, consider n H(D) = H ∈ C 2 (Tn × Rn × R) : 2 D−1 I ≤ ∂pp H(θ, p, t) ≤ DI o kH(·, 0, ·)kC 0 , k∂p H(·, 0, ·)kC 0 ≤ D .

$H ≤ 1,

for all p ∈ Rn ,

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CHAPTER 6

We then check that each H ∈ H(D) is Tonelli, and it satisfies a list of uniform estimates called uniform family in [11]. In particular, if H0 ∈ H(D), then H ∈ H(2D) for all  ≤ 0 = 0 (D). Given C > 0, and let Ω ⊂ Rn be an open convex set. We say a function u : Ω → R is C semi-concave if for every x ∈ Ω, there is a linear function lx : Rn → R, such that u(y) − u(x) ≤ lx (y − x) + Cky − xk2 ,

y ∈ Ω.

The linear form lx is called a super-differential at x. The set of all superdifferentials at x is denoted ∂ + u(x). It is easy to see that if u is differentiable at x, then ∂ + u(x) = {du(x)}. A function u : Tn → R is semi-concave if it’s semi-concave as a function on Rn . √ Lemma 6.1. ([11]) If u : Tn → R is C semi-concave, then it is C n–Lipschitz. √ The super-differential set ∂ + u(x) ⊂ {kpk ≤ C n}. Given s < t ∈ R, x, y ∈ Tn , we define the Lagrangian action Z t AH (x, s, y, t) = AL (x, s, y, t) = inf LH (γ(τ ), γ(τ ˙ ), τ )dτ, γ(s)=x, γ(t)=y

(6.1)

s

where the infimum is taken over all absolutely continuous γ. We outline a series of useful results: Proposition 6.2. Let H ∈ H(D), then: 1. (Tonelli Theorem) (Appendix 2 of [60]) The infimum in (6.1) is always reached. It is then C 2 , which solves the Euler-Lagrange equation. Such a γ is called a minimizer. 2. (A priori compactness) (Section B.2 of [11]) There is a constant Cδ > 0 depending only on δ and D, such that for t − s > δ, any minimizer of AH (x, s, y, t) satisfies kγk ˙ ≤ Cδ . 3. (Uniform semi-concavity in space variable) (Theorem B.7 of [11]) For t − s > δ, the function AH (x, s, y, t) is Cδ semi-concave in x and y. Moreover, if γ : [s, t] → Tn is a minimizer, then p(s) ∈ −∂x+ AH (x, s, y, t),

p(t) ∈ ∂y+ AH (x, s, y, t),

(6.2)

where p(τ ) = ∂v LH (γ(τ ), γ(τ ˙ ), τ ). 4. (Uniform semi-concavity in the time variable, see Corollary 7.15) For t − s > 2, AH (x, s, y, ·) is C-semi-concave on (t − 1, t + 1), and AH (x, ·, y, t) is Csemi-concave on (s − 1, s + 1). If γ : [s, t] → Tn is a minimizer, −H(γ(s), p(s), s) ∈ −∂s+ AH (x, s, y, t), −H(γ(t), p(t), t) ∈ ∂t+ AH (x, s, y, t). (6.3)

WEAK KAM THEORY AND FORCING EQUIVALENCE

57

In particular, by Lemma 6.1, we have AH (·, ·, y, t) and AH (x, s, ·, ·) are uniformly Lipschitz. Remark 6.3. Item 4 seems to be known but we have not been able to locate a reference. We give a proof in Corollary 7.15 along with some more refined results about semi-concavity of action functions. For each c ∈ Rn , define LH,c (θ, v, t) = LH (θ, v, t) − c · v, and denote AH,c = ALH,c . If t − s > 2, we have + (p(t) − c, −H(γ(t), p(t), t)) ∈ ∂(y,t) AH,c (x, s, y, t)

and similarly for s.

6.2

WEAK KAM SOLUTION

We now define the (continuous) Lax-Oleinik semi-group Tcs,t : C(Tn ) → C(Tn ) via the formula Tcs,t u(x) = minn {u(z) + AH,c (z, s; x, t)} . z∈T

The semi-group property Tct1 ,t2 Tct0 ,t1 u = Tct0 ,t2 u is clear from definition. For a $-periodic Hamiltonian, the associated discrete semi-group is generated by the operator: Tc u = Tc0,$ u. Lemma 6.4. (See (A.3) and (A.10) of [11]) Let {uζ }ζ∈Z be a (possibly uncountable) family of C-semi-concave functions Tn → R, and v = inf uζ is finite for at least one point, then inf ζ∈Z uζ is also C semi-concave. Moreover, suppose for x0 ∈ Z, the infimum v(x0 ) is reached at uζ0 (x0 ), then ∂ + uζ0 (x0 ) = ∂ + v(x0 ). Using Proposition 6.2, the functions Tηn u for all n > 1/$ are C semi-concave with the constant depending only on the uniform family H(D). Proposition 6.5 (Ma˜ n´e ’s critical value, see Proposition 3.1 of [11]). There is a unique α ∈ R such that Tcn u(x) + nα is uniformly bounded over all n ∈ N. This value coincides with Mather’s alpha function: Z  αH (c) = − inf LH,c (θ, v, t)dµ(θ, v, t) (6.4) µ

where the infimum is taken over all invariant probability measures of the EulerLagrange flow on Tn × Rn × T$ .

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CHAPTER 6

ˆ p, c) = H(θ, p + c, t) − αH (c), then Let H(θ, LHˆ = LH,c + αH (c),

(6.5)

and αHˆ (0) = 0. This reduction is often used in proofs to reduce a statement about LH,c to the case c = 0 and αH (0) = 0. Let us also point out an alternative definition of the alpha function, namely, we can replace the class of minimal measures with the class of closed measures, which satisfies Z df (θ, t) · (v, 1)dµ(θ, v, t) = 0 for every C 1 function f : Tn × R → R satisfying f (θ, t + $) = f (θ, t). Proposition 6.6 (See [74]). Z αH (c) = − inf µ

 LH,c (θ, v, t)dµ(θ, v, t)

with µ taken over all closed probability measures. A function w : Tn × T$ → R is called a weak KAM solution if Tcs,t w(·, s) + αH (c)(t − s) = w(·, t),

s < t ∈ R,

i.e. the family w(·, t) is invariant under the semi-group up to a linear drift. The function u(θ) = w(θ, 0) is then a fixed point of the operator Tc + $αH (c). Proposition 6.7 (Existence of weak KAM solution, Proposition 3.2 of [11]). For c ∈ Rn and u0 ∈ C(Tn ), the function w(θ, t) = lim inf (Tct−N $,t u0 + N $α(c)), N →−∞

N ∈N

is a $-periodic weak KAM solution. Lemma 6.8. Weak KAM solutions are uniformly semi-concave over all H ∈ H(D). Proof. Let w(x, t) be a weak KAM solution; we view it as a function on Tn × R. The semi-concavity in x is clear by Lemma 6.4 and Proposition 6.2, item 3. We now prove the semi-concavity in t. By definition, for t − s > 2, w(x, t) = minn {w(z, s) + AH,c (z, s, x, t) + αH (c)(t − s)}, z∈T

which by Proposition 6.2, item 4, is C-semi-concave on any interval (t0 −1, t0 +1). Since w(x, ·) has period $ ∈ (0, 1], this implies it is also C-semi-concave on R.

59

WEAK KAM THEORY AND FORCING EQUIVALENCE

A function w : Tn × R → R is called dominated by (LH,c , α) if w(y, t) − w(x, s) ≤ AH,c (x, s, y, t) + (t − s)α,

x, y ∈ Tn ,

s < t.

γ : I → Tn , where I is an interval in R, is called calibrated by w if w(γ(t), t) − w(γ(s), s) = AH,c (x, s, y, t) + (t − s)α. Proposition 6.9 (Proposition 4.1.8 and Theorem 4.2.6 of [37]). w : Tn ×R → R is a weak KAM solution if and only if it is dominated by (LH,c , αH (c)) and for every (y, t) there is a calibrated curve γ : (−∞, t] → Tn such that γ(t) = y. Moreover, for each s ∈ (−∞, t), p(s) = ∂v LH (γ(s), γ(s), ˙ s), u(·, s) is differentiable at γ(s) and ∂x u(γ(s), s) = p(s) − c. The calibrated curve γ is unique if ∂x u(γ(t), t) exists.

6.3

˜ E, ´ AND MATHER SETS PSEUDOGRAPHS, AUBRY, MAN

Let u : Tn → R be semi-concave. By the Rademacher theorem, u is differentiable almost everywhere. For c ∈ Rn , we define the (overlapping) pseudograph Gc,u = Gc,H,u = {(x, c + ∇u(x)),

∇u(x) exists} .

In the 1-dimensional case, at every discontinuity of the function du(x), the left limit is larger than the right limit. In the time-dependent setting if w : Tn ×R → R is semi-concave, we write Gc,w = {(x, c + ∂x w(x, t), t) :

∂x w exists}.

The evolution by the Lax-Oleinik semi-group generates an evolution operator on the pseudograph. The following statement outlines its relation with the Hamiltonian dynamics. Proposition 6.10 ([11]). For each s < t, we have Gc,Tcs,t u ⊂ φs,t H (Gc,u ) , here φs,t H denotes the Hamiltonian flow. Corollary 6.11. Suppose w(θ, t) is a (time-periodic) weak KAM solution of LH,c , then
−1 φs,t Gc,w(·,t) ⊂ Gc,w(·,s) . H 0,$ In particular, (φ−1 is the associH ) Gc,u ⊂ Gc,u , where u = w(·, 0) and φH = φH ated discrete dynamics.

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CHAPTER 6

Let w = w(θ, t) be a continuous weak KAM solution for LH,c . Then Corollary 6.11 implies the set  Gc,w = (θ, p) : (θ, p) ∈ Gc,w(·,t) is backward invariant under the flow φs,t H . We then define ( ) \ s,t
−1 e w) = (θ, p, t) : (θ, p) ∈ I(c, φ Gc,w(·,t) , H

s τ ). w+ (x, t) is called a forward weak KAM solution if Tˇcs,t w(·, t) = w(·, s) − α(t − s) for some α ∈ R and all s < t. 2. The discrete-time version is Tˇc = Tˇc0,$ . 3. α = αH (c) is the unique number such that a weak KAM solution may exist. 4. For a semi-concave function u, we define Gˇc,u = {(θ, c + ∇x u(θ))}, and call it

WEAK KAM THEORY AND FORCING EQUIVALENCE

61

an anti-overlapping pseudograph. 5. Analogs of the previous section apply with appropriate changes. Let w(θ, t) be a weak KAM solution for LH,c , and w+ a forward weak KAM solution. We say that w, w+ are conjugate if they coincide on the set MH (c). Denote e w, w+ ) := {(x, c + ∂x w(x, t), t) : I(c,

(x, t) ∈ argmin(w − w+ )}.

Proposition 6.12 (Theorem 5.1.2 of [37]). For each weak KAM solution w, there exists a forward solution w+ conjugate to w satisfying w+ ≤ w, and e w) = I(c, e w+ ) = I(c, e w, w+ ). I(c, e w) is a Lipschitz One important consequence of Proposition 6.12 is that I(c, graph over its x component. This is immediate given the next lemma. Lemma 6.13 (Proposition 4.5.3 of [37]). If w, −w+ are C-semi-concave, then ∂x w is uniformly Lipschitz over the set argmin(x,t) {w(x, t) − w+ (x, t)} where the Lipschitz constant depends only on C. In fact, this approach gives a more general result which we will need later. Proposition 6.14. Let w : Tn ×T$ → R be a discrete-time weak KAM solution for H at cohomology c. For t > s, set wk+ (x, s) = Tˇcs,t w. Then e w, w+ ). φs−t Gc,w ⊂ I(c, k As a consequence φs−t Gc,w are uniformly Lipschitz graphs over x component. Proof. Suppose (xs , ps , s) ∈ φs−t Gc,w , and (xt , pt , t) = φt−s (xs , ps , s) ∈ Gc,w , then w(xt , t) = w(xs , s) + AH,c (xs , s, xt , t) since the backward orbit of (xt , pt ) is calibrated by Proposition 6.9. Then (Tˇcs,t w)(xs ) ≥ w(xs , s). On the other hand, since w a weak KAM solution, for arbitrary x, y, w(y, t) ≤ w(x, s) + AH,c (y, s, x, t). This implies Tˇcs,t w(·, t) ≤ w(·, s). It follows that (xs , s) ∈ argmin(w− Tˇs,t w) and

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e w, w+ ). Since I(c, e w, w+ ) is a compact set (since its a Lipschitz φs−t Gc,w ⊂ I(c, k k graph over a compact set argmin(w − Tˇs,t w)), the inclusion also holds for the closure.

6.5

PEIERLS BARRIER, STATIC CLASSES, ELEMENTARY SOLUTIONS

We define hH,c (x, s, y, t) = lim inf AH,c (x, s, y, t + N $) + N $αH (c) N →∞

called the Peierls barrier by Mather [61]. Then hH,c projects to a function on Tn × T$ × Td × T$ . The functions hH,c (·, ·, y, t) and hH,c (x, s, ·, ·) are uniformly Lipschitz, since they are limits of uniformly Lipschitz functions. We also have the triangle inequality: hH,c (x, s, y, t) + hH,c (y, t, z, τ ) ≥ hH,c (x, s, z, τ ). For any (x, s) ∈ Tn × T$ , we claim hH,c (x, s, x, s) ≥ 0. Suppose not, then we can always find a sequence of curves γk connecting (x, s) to itself such that the action goes to −∞, contradicting Proposition 6.5. Then the projected Aubry set AH (c) has the following alternative characterization: AH (c) = {(x, s) ∈ Tn × T$ : hH,c (x, s, x, s) = 0}. We will also consider the discrete-time barrier: hH,c (x, y) = hH,c (x, 0, y, 0),

(6.6)

then A0H (c) = {x : hH,c (x, x) = 0}. Proposition 6.15. For each (x, s) ∈ Tn × T$ , hH,c (x, s, ·, ·) is a weak KAM solution for LH,c . Define the Mather semi-distance: ¯ s, y, t) = hH,c (x, s, y, t) + hH,c (y, t, x, s), d(x, ¯ s, y, t) = 0 defines an equivalence relation (x, s) ∼ (y, t) then the condition d(x, on AH (c). The equivalence classes of this relation are called the static classes. Let S ⊂ AH (c) be a static class, it corresponds uniquely to an invariant set in the phase space: −1 Se = π(x,t) S. AH (c)

WEAK KAM THEORY AND FORCING EQUIVALENCE

63

For (ζ, τ ) ∈ S, the function hH,c (ζ, τ, ·, ·) is independent of the choice of (ζ, τ ) ∈ S, up to an additive constant. These functions are called elementary solutions, due to the representation formula below. Proposition 6.16. 1. (Representation formula, see Theorem 8.6.1 of [37] and Theorem 7 of [29].) Let w(x, t) be an LH,c weak KAM solution. Then w(x, t) =

min

{w(ζ, τ ) + hH,c (ζ, τ, x, t)}.

(ζ,τ )∈AH (c)

eH (c) is a hete2. (See Proposition 4.3 of [11].) Every orbit in the Ma˜ n´e set N e e e eH (c) if roclinic orbit between two static classes S1 , S2 . We have AH (c) = N e and only if AH (c) has only one static class. Suppose the Aubry set AH (c) is the union of two non-empty compact components. The following lemma allows us to isolate each component by adding a bump function to the Hamiltonian. Lemma 6.17. Suppose A1 ⊂ AH (c) and A2 = AH (c) \ A1 are both non-empty and compact, and V ⊂ A1 separates A1 and A2 . Let f ∈ C ∞ (Tn × T$ ) be such that f |V = 0, min f |A2 > 0. Then for the Hamiltonian H f (x, p, t) = H(x, p, t) − f (θ, t), we have αH (c) = αH f (c),

AH f (c) = A1 .

Proof. First we use (6.5) to reduce to the case c = 0 and αH (0) = 0. Given (θ0 , t0 ) ∈ A1 , since hH (θ0 , t0 , θ0 , t0 ) = 0, there exists a sequence of curves γk : Ik := [t0 , t0 + nk $] → Tn , such that nk → ∞, γ(t0 ) = θ0 = γ(t0 + nk $), and lim AH (γ(t0 ), t0 , γ(t0 + nk ), t0 + nk ) = 0.

k→∞

We claim that, for all k sufficiently large, γk (Ik ) ⊂ V . Suppose not, then there must exist kj → ∞ and tj ∈ Tkj such that (γk (tj ), tj mod $Z) ∈ ∂V . By passing to a subsequence, we may assume (γkj (tj ), tj ) → (y, s) ∈ ∂V . Using the representation formula (Proposition 6.16), hH (θ0 , t0 , y, s) ≤ hH (θ0 , t0 , θ0 , t0 ) + AH (θ0 , t0 , γkj (tj + nkj ), tj ). Taking lim inf, we get hH (θ0 , t0 , y, s) ≤ 0 and similarly hH (y, s, θ0 , t0 ) ≤ 0. It

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follows that 0 ≤ hH (θ0 , t0 , y, s) + hH (y, s, θ0 , t0 ) ≤ 0, implying (y, s) ∈ AH (0), a contradiction. Using our claim, we now prove the lemma. Let Lf , L denote the Lagrangians for H f and F , then Lf = L + f . First of all, since Lf ≥ L, it follows from Proposition 6.6 that αH f (0) ≤ αH (0) = 0. We now have 0 ≤ hH f (θ0 , t0 , θ0 , t0 ) ≤ lim inf ALf +αH f (0) (θ0 , t0 , γk (t0 + nk ), t0 + nk ) k→∞

≤ lim inf ALf (θ0 , t0 , γk (t0 + nk ), t0 + nk )

(αH f (0) ≤ 0)

= lim inf AL (θ0 , t0 , γk (t0 + nk ), t0 + nk )

(since γk (Tk ) ⊂ V eventually)

k→∞ k→∞

≤ 0, implying (θ0 , t0 ) ∈ AH f (0). The fact that hH f (θ0 , t0 , θ0 , t0 ) is bounded implies αH f (0) = αH (0) = 0, concluding the proof.

6.6

THE FORCING RELATION

Definition 6.18. Let u : Tn → R be semi-concave, and N ∈ N. We say that Gc,u `N c0 , if there exists a semi-concave function v : Tn → R such that Gc0 ,v ⊂

N [

φkH (Gc,u ) .

k=0

We say the c ` c0 if there exists N ∈ N such that Gc,u `N c0 for every pseudograph Gc,u . We say c a` c0 if c ` c0 and c0 ` c. In view of Proposition 6.10, we always have c ` c. The relation ` is transitive by definition (but not symmetric). The relation a` is an equivalence relation. We summarize the property of the forcing relation in the text that follows. • (Proposition 2.1) Let {ci }N i=1 be a sequence of cohomology classes which are forcing equivalent. f0 (ci ); For each i, let Ui be neighborhoods of the discrete-time Mather sets M H t then there is a trajectory of the Hamiltonian flow φ of H visiting all the sets Ui . • (Mather mechanism, Proposition 2.3)

WEAK KAM THEORY AND FORCING EQUIVALENCE

65

0 Suppose NH (c) is contractible as a subset of T2 , then there is σ > 0 such that c is forcing equivalent to all c0 ∈ Bσ (c). • (Arnold and bifurcation mechanism, Proposition 2.5) e 0 (c) \ Ae0 (c) Suppose that either Ae0H (c) has only two static classes and N H H 0 e is totally disconnected, or AH (c) has only one static class, and there is a e 0 (ξ ∗ c) \ Ξ−1 N e 0 is totally symplectic double covering map ξ such that N H◦Ξ H disconnected, then there is σ > 0 such that c is forcing equivalent to all c0 ∈ Bσ (c).

6.7

THE GREEN BUNDLES

In many parts of the proof, we study the hyperbolic property of a minimizing orbit, for which the concept of Green bundles is very useful. In this setting, we restrict to autonomous Tonelli Hamiltonians H on Tn × Rn . Consider V = {0} × Rn ⊂ Rn × Rn , viewed as a sub-bundle of T (Tn × Rn ), called the vertical bundle. An orbit (θ, p)(t) of the Hamiltonian flow φs,t H is called disconjugate if D(φs,t H )V t V,

∀s 6= t ∈ R.

Proposition 6.19 (See for example [3]). Let (θ, p)(t), t ∈ R be a minimizing orbit for LH . Then (θ, p)(t) is disconjugate, and the following hold: 1. For every t, the limits G− (θ(t), p(t)) = lim D(φs,t H )(θ(s), p(s))V s→−∞

G+ (θ(t), p(t)) = lim D(φs,t H )(θ(s), p(s))V s→∞

exist. The bundles G± , called the Green bundles, are given by graphs of symmetric matrices G± with uniformly bounded norms depending only on H. 2. If (θ, p)(t) is a hyperbolic orbit, then G− coincide with the span of the unstable bundle and the flow direction, and G+ coincide with the span of the stable bundle and the flow direction.

Chapter Seven Perturbative weak KAM theory By perturbative weak KAM theory, we mean two things: • How do the weak KAM solutions and the Mather, Aubry and Ma˜ n´e sets respond to limits of the Hamiltonian? • How do the weak KAM solutions change when we perturb a system, in particular, what happens when we perturb (1) completely integrable systems, and (2) autonomous systems by a time-periodic perturbation? In this chapter, we state and prove results in both aspects, as a technical tool for proving forcing equivalence.

7.1

SEMI-CONTINUITY

Let H(D) be the uniform family defined before, note that they are periodic of period 0 < $H ≤ 1, but not necessarily of the same period. It is known that in the case that all periods are fixed at 1, the weak KAM solutions are upper semi-continuous (see precise statements in the text that follows) under C 2 convergence over compact sets ([12]). The results generalize to the case when the periods are not the same, as we now show. Let us remark that if Hn ∈ H(D) is a family of Hamiltonians, and Hn → H uniformly over compact sets on Tn × Rn × R, then H is necessarily periodic of some period, and therefore H ∈ H(D). Let Ak be a sequence of sets, define lim supk→∞ Ak to be the set of all accumulation points of all sequences xk ∈ Ak . Lemma 7.1 (Lemma 7 of [12]). Suppose Hamiltonians Hk ∈ H(D), k ∈ N are a family of periodic Tonelli Hamiltonians. Suppose Hn → H in C 2 over compact sets, ck → c ∈ Rn , then αHk (ck ) → αH (c). If wk : Tn × R → R is a sequence of weak KAM solutions of Hk , ck which converges uniformly to w : Rn × R → R, then w is a weak KAM solutions of H, c. Moreover, we have lim sup Gck ,Hk ,wk ⊂ Gc,H,w , k→∞

lim sup IeHk (ck , wk ) ⊂ IeH (c, w), k→∞

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PERTURBATIVE WEAK KAM THEORY

˜H (ck ) ⊂ N ˜H (c). lim sup N k k→∞

Proof. The proof is an elaboration of Lemma 7 of [12]. Since all wk are defined up to an additive constant, we assume in addition that wk (0, 0) = w(0, 0) = 0. Note that if Hk ∈ H and ck uniformly bounded, then αHk (ck ) is uniformly bounded (6.4). By restricting to a subsequence, we may assume αHk (ck ) → α ∈ R. Then by taking the limit in Z t wk (γ(t), t) − wk (γ(s), s) ≤ LHk (γ(τ ), γ(τ ˙ ), τ ) − ck · γ(τ ˙ ) + αHk (ck ) dτ, s

we obtain Z w(γ(t), t) − w(γ(s), s) ≤

t

LH (γ(τ ), γ(τ ˙ ), τ ) − c · γ(τ ˙ ) + α dτ. s

Using boundedness of w, we obtain Z t 1 − lim LH (γ(τ ), γ(τ ˙ ), τ ) − c · γ(τ ˙ ) dτ ≤ α t−s→∞ t − s s

(7.1)

for any γ defined on R. If µ is any ergodic invariant measure for the EulerLagrange flow, for µ almost every orbit γ, the left-hand side of (7.1) converges R to − (L − c)dµ. We get α ≥ αH (c) in view of (6.4). Now given x ∈ Tn , t ∈ R, let γk : (−∞, t] → Td be a LHk ,ck , wk calibrated curve, then Proposition 6.2 implies γk are uniformly Lipschitz. Then γk has a 1 subsequence that converges in Cloc to a limit γ(t), which is LH,c +α, u calibrated. This implies both αH (c) = α and that u is a H, c weak KAM solution. We now prove the “moreover” part. Denote Gk = Gck ,Hk ,wk ,

G = Gc,H,w

Iek = IeH (ck wk ),

I = IH (c, w).

then if (xk , pk , tk ) ∈ Gk , there exist LHk ,ck , wk calibrated curves γk : (−∞, tk ] → Td with γk (tk ) = xk and ∂p Hk (xk , pk , tk ) = γ˙ n . Then by the same argument, 1 after restricting to a subsequence, γn converges in Cloc to γ : (−∞, t] → Td , and γ is a LH,c , w calibrated curve. This implies (t, x, p) ∈ G. For the set I, let us prove that for each fixed T , we have −T lim sup φ−T Hk Gk ⊂ φH G. k→∞

Indeed, for any (xk , pk , tk ) ∈ φ−T Hk Gck ,Hk ,wk , there exists LHk ,ck , wk calibrated d curves γ : (−∞, tk + T ] → T such that γ(tk ) = xk , ∂p Hk (xk , pk , tk ) = γ˙ k (tk ). Then exactly the same argument as before implies γn accumulates to a LH,c , u calibrated curve γ : (−∞, t + T ) → Td , and implying (t, x, p) ∈ Φ−T H G. We

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obtain e lim sup Iek ⊂ I. k→∞

Finally, we have ˜H (ck ) = N k

[

IeH (ck , w),

w

where the union is over all LHk ,ck weak KAM solutions w. For (xk , pk , tk ) ∈ ˜H (ck ), there exist uk such that (xk , pk , tk ) ∈ Gc ,H ,u . All wk are semiN k k k k concave with respect to a uniform constant, and hence they are equi-Lipschitz. Since all are normalized to wk (0, 0) = 0, periodicity implies that they are equibounded. Ascoli’s theorem implies there exists a subsequence that converges to u uniformly on the interval Tn × [0, K]. Since all wk ’s are periodic with period bounded by 1, this implies wk → w on Tn × R as well. We then apply the ˜. semi-continuity of G sets to get semi-continuity of N The theory built in [12] allows one to pass from semi-continuity of pseudographs to semi-continuity of the Aubry set under a condition called the coincidence hypothesis. A sufficient condition for this hypothesis is when the Aubry set has finitely many static classes. Corollary 7.2. Suppose A˜H (c) has at most finitely many static classes. Then if Hk ∈ H(D) C 2 converges to H over compact sets, and ck → c, we have lim sup A˜Hk (ck ) ⊂ A˜H (c) n→∞

as subsets of Tn × Rn × R. Proof. The proof follows in the same way as the proof of Theorem 1 in [12]. We also refer to Section 6.2 of [52] where this is carried out in detail.

7.2

CONTINUITY OF THE BARRIER FUNCTION

In general, the barrier function hH,c may be discontinuous with respect to H and c. However, the continuity properties hold in the particular case when the limiting Aubry set contains only one static class. Proposition 7.3. Assume that a sequence Hk ∈ H(D) converges to H in C 2 over compact sets, and ck → c ∈ Rn . Assume that the projected Aubry set AH (c) contains a unique static class. Let (xk , sk ) ∈ AHk (ck ) with (xk , sk ) → (x, s), then the barrier functions hHk ,ck (xk , sk ; ·, ·) converges to hH,c (x, s; ·, ·) uniformly. Similarly, for (yk , tk ) ∈ AHk (ck ) and (yk , tk ) → (y, t), the barrier functions hHk ,ck (·, ·; yk , tk ) converge to hH,c (·, ·; y, t) uniformly.

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PERTURBATIVE WEAK KAM THEORY

Proposition 7.4. Assume that a sequence Hn converges to H in C 2 over compact sets, ck → c ∈ Rn and the Aubry set AH (c) contains a unique static class. 1. For any (x, s) ∈ AH (c) we have lim

sup

k→∞ (y,t)∈M ×T

|hHk ,ck (xk , sk ; y, t) − hH,c (x, s; y, t)| = 0

uniformly over (xk , sk ) ∈ AHk (ck ) and (y, t) ∈ M × R. 2. For any lk ∈ ∂y+ hHk ,ck (xk , sk ; y, t) and lk → l, we have l ∈ ∂y+ hH,c (x, s; y, t). Moreover, the convergence is uniform in the sense that lim

k→∞ lk ∈∂y+ hH

inf (xk ,sk ;y,t) k ,ck

d(lk , ∂y+ hH,c (x, s; y, t)) = 0

uniformly in (x, s) ∈ AH (c), (xk , sk ) ∈ AHk (ck ). The following statement follows easily from the representation formula (see Proposition 6.16). Lemma 7.5. Assume that AH (c) has a unique static class. Let (x, t) ∈ AH (c), then any weak KAM solution differs from hH,c (x, t; ·, ·) by a constant. Proof of Proposition 7.3. We prove the second statement. By Proposition 6.2, all functions hHk ,ck (xk , sk ; ·, ·) are uniformly semi-concave and equi-Lipschitz. Since (xk , sk ) ∈ AH (ck ) implies hHk ,ck (xk , sk ; xk , sk ) = 0, the functions are equibounded. By Ascoli’s Theorem, the sequence hHk ,ck (xk , sk ; ·, ·) is pre-compact in uniform topology, and hH,c (x, s, ·, ·) is the unique accumulation point of the sequence. Indeed, due to Lemmas 7.1 and 7.5 any accumulation point must be hH,c (x, s; ·, ·) + C for some C ∈ R. But hH,c (xk , sk ; x, s) → hH,c (x, s; x, s) = 0, so C = 0. Statement 1 follows from the definition of the projected Aubry set AH (c) = {(x, s) ∈ M × T : hH,c (x, s; x, s) = 0} and statement 2. Proof of Proposition 7.4. Part 1. We argue by contradiction. Assume that there exist δ > 0, and by restricting to a subsequence, sup |hHk ,ck (xk , sk , y, t) − hH,c (x, s, y, t)| > δ. (y,t)

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By compactness, and by possibly restricting to a subsequence again, we may assume that (xk , sk ) → (x∗ , s∗ ), (yk , tk ) → (y, t). Using Proposition 7.3, take the limit as n → ∞, we have sup |hH,c (x∗ , s∗ , y, t) − hH,c (x, s, y, t)| > δ. (y,t)

By Lemma 7.5, the left-hand side is 0, which is a contradiction. Part 2. hHk ,ck (xk , sk , ·, t) converges to hH,c (x, s, ·, t) uniformly. Convergence of super-differentials follows directly from Proposition 6.2. It suffices to prove uniformity. Assume, by contradiction, that by restricting to a subsequence, we have (xk , sk ) → (x, s) ∈ AH (c), lk ∈ ∂y+ hHk ,ck (xk , sk , y, t) and (x, s) ∈ AH (c) such that lim lk ∈ / ∂y+ hH,c (x, s, y, t). k→∞

By Proposition 6.2, ln → l ∈ ∂y+ hH,c (x∗ , s∗ , y, t). Moreover, ∂y+ hH,c (x, s, y, t) = ∂y+ hH,c (x∗ , s∗ , y, t). Indeed, both functions hH,c (x, s, ·, ·) and hH,c (x∗ , s∗ , ·, ·) are weak KAM solutions, Lemma 7.5 implies they differ by a constant. This is a contradiction.

7.3

LIPSCHITZ ESTIMATES FOR NEARLY INTEGRABLE SYSTEMS

In this section we record uniform estimates for the nearly integrable system H = H0 (p) + H1 (θ, p, t),

(θ, p, t) ∈ Tn × Rn × T,

with the assumptions H0 ∈ H(D), kH1 kC 2 ≤ 1. n Proposition 7.6 (Proposition 4.3, [13]).√ For H as given, for any √ c ∈ R , any LH ,c weak KAM solution u(x, t) is 6D -semi-concave and 6D n-Lipschitz in x.

Corollary 7.7. For any LH ,c weak KAM solution u : Tn × T → R, let γ : (−∞, t0 ] → Tn be a calibrated curve. Then for any t ∈ (−∞, t0 ), p(t) := ∂v LH (γ(t), γ(t), ˙ t) √ satisfies kp(t) − ck ≤ 6D n. eH (c) satisfies the same estimate kp − ck ≤ Moreover, any (x, p, t) ∈ N  √ 6D n. Proof. Proposition 6.9 implies p(t)−c = ∂x u(γ(t), t), the conclusion follows from √ the fact that u(·, t) is 6D n-Lipschitz. The same holds for p(t0 ) by continuity.

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71

e follows from the fact that any (x, p, t) ∈ G ⊂ N is the end The statement for N point of a calibrated curve. Corollary 7.8. Let µ be any c-minimal measure of H , then there is C > 0 depending only on H0 such that √ kρ(µ) − ∇H0 (c)k ≤ C . Proof. Recall that µ is a measure on Tn × Rn × R, invariant under the EulerLagrange flow. By considering the Legendre transform L, we obtain Z Z ρ(µ) = vdµ(θ, v, t) = ∂p H (θ, p, t)dL∗ µ(θ, p, t). √ By Corollary 7.7, we have ∂p H (θ, p, t) = ∇H0 (c) + O( ) for all (θ, p, t) ∈ supp L∗ µ ⊂ AeH ( c), √ therefore, ρ(µ) = ∇H0 (c) + O( ).

7.4

ESTIMATES FOR NEARLY AUTONOMOUS SYSTEMS

The goal of this section is to derive a special Lipshitz estimate of weak KAM solutions for perturbations of autonomous systems. More precisely, consider H (x, p, t) = H1 (x, p) + H2 (x, p, t). Assume that H1 ∈ H(D/2) and kH2 kC 2 = 1. Then for  small enough all H ∈ H(D). Let L denote the associated Lagrangian. We first state an estimate for the alpha function. Lemma 7.9. There is C > 0 depending only on kH1 kC 2 and D, such that kαH (c) − αH1 (c)k < C. Proof. Let C always denote a generic constant depending on kH1 kC 2 . Let L1 be the Lagrangian for H1 ; then kL − L1 kC 0 ≤ C. As a result, the functionals Z Z (L − c · v)dµ, (L − c · v)dµ, defined on the space of closed probability measures, differ by at most C. The lemma follows immediately using the closed measure version for the definition of the alpha function, see Section 6.2.

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n Theorem 7.10. Let √ w : T × T$ → R be a weak KAM solution of L − c · v and suppose t > 1/ . Then there is C depending only on D and kck such that √ |H (x1 , p1 , t1 )−H (x2 , p2 , t2 )| ≤ C  k(x2 −x1 , t2 −t1 )k, (xi , pi , ti ) ∈ φ−t Gc,w .

In particular, the above estimate holds on the Aubry sets AeL (c). The proof relies on semi-concavity of weak KAM solution, following Fathi ([37]). However to get an improved estimate we need a notion of semi-concavity that is “stronger” in the t direction. Let Ω ⊂ Rn be an open convex set. Let A be a symmetric n × n matrix. We say that f : Ω → R is A-semi-concave if for each x ∈ Rn , there is lx ∈ Rn such that 1 f (y) − f (x) − lx · (y − x) ≤ A(y − x)2 , x, y ∈ Ω 2 where Ax2 denotes Ax·x. This definition generalizes the standard semi-concavity, as A-semi-concave functions are 21 kAk-semi-concave. We say f is A-semi-convex if −f is A-semi-concave. The following lemma follows from a direct computation. Lemma 7.11. f is A-semi-concave if and only if fA (x) = f (x) − 12 Ax2 is concave. The following lemma is proved in [80]. Lemma 7.12 (See Lemma 3.2 of [80]). Suppose f : Rn → R is B-semi-concave and g : Rn → R is (−A)-semi-convex, and S = B − A is positive definite. Suppose f (x) ≥ g(x) and M is the set on which f − g reaches its minimum. Then for all x1 , x2 ∈ M , we have 1 1 kdf (x2 ) − df (x1 ) − (A + B)(x2 − x1 )kS −1 ≤ kx2 − x1 kS , 2 2 √ where kxkS = Sx2 . Recall that the action function AL (x, s, y, t) is the minimal Lagrangian action of curves with end points γ(s) = x, γ(t) = y. Proposition 7.13. For the Hamiltonian H and c ∈ Rn , √ there is a constant C > 0 depending only on D and kck such that if t1 − t0 > 1/ , for all x0 , y0 ∈ Tn , n the action function AH ,c (x0 , t0 , ·, ·) (treated √ as a function √ on R ×R) is S -semin concave on the set (y, t) ∈ R ×(t1 −1/(2 ), t1 +1/(2 )), and AH ,c (·, ·, y0 , t1 ) √ √ on the set (x, s) ∈ Rn × (t0 − 1/(2 ), t0 + 1/(2 ), where   Idn×n √0 S = C . 0  We need the following lemma.

PERTURBATIVE WEAK KAM THEORY

73

Figure 7.1: Defining the curve γhλ .

Lemma 7.14. Suppose H ∈ H(D), and let A be the action function defined using the Lagrangian L associated to H. Then there exists a constant C depending on D such that if γ : [t0 , t1 ] → Tn is an extremal curve with T = t1 −t0 > 1, then 2 for h ∈ Rn , ∆T ∈ [−T /2, T /2], p(t) = ∂v L(γ(t), γ(t), ˙ t), κ = k∂(x,v,t)t LkC 0 , A(γ(t0 ), t0 , γ(t1 ) + h, t1 + ∆T ) − A(γ(t0 ), t0 , γ(t1 ), t1 )   1 2 ≤ p(t1 ) · h − H(γ(t1 ), p(t1 ), t1 ) · (∆T ) + Ckhk + C + κT (∆T )2 . T Proof. By considering the Lagrangian L(·, ·, · + t), it suffices to consider t = 0, s = T. Let γ : [0, T ] → Tn be an absolutely continuous curve, write λ = ∆T /T and define γhλ : [0, T + ∆T ] → Tn ,    t γ , t ∈ [0, (T − 1)(1 + λ)];  1+λ    γhλ (t) = t t γ 1+λ + 1+λ − T + 1 h, t ∈ [(T − 1)(1 + λ), T (1 + λ)]. The curve is obtained by adding a linear drift in x on the time interval [T −1, T ], then reparametrize time to the interval [0, T + ∆T ], see Figure 7.1. Let C denote an unspecified constant. Note that kγk ˙ ≤ C. AL (γhλ ) =

Z

(1+λ)T

L(γhλ , γ˙ hλ , t)dt  Z T −1  1 = (1 + λ) L γ(s), γ(s), ˙ (1 + λ)s ds 1+λ 0   Z T 1 + (1 + λ) L γ(s) + (s − T + 1)h, (γ(s) ˙ + h) , (1 + λ)s ds; 1+λ T −1 0

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then AL (γhλ ) Z T −1 L(γ, γ, ˙ s)ds + (−λ∂v L · γ˙ + (1 + λ)∂t L · λs) ds 0 0  Z T  1 + (1 + λ) ∂x L · (s − T + 1)h + ∂v L · (−λγ˙ + h) + ∂t L · λs ds 1+λ T −1
2 2 2 2 2 + CT k∂vv Lk + k∂vt LkT + k∂tt LkT λ
2 2 2 + Ck∂(x,v) 2 Lk khk + |λ|khk + λ
+ C k∂xt Lk|λ|T khk + k∂vt Lk(|λ| + khk)λT + k∂tt Lkλ2 T 2 . Z

T

≤ (1 + λ)

2 Using k∂ 2 Lk ≤ C, k∂(x,v,t)t Lk ≤ κ, and plug in ∆T = λT , we have

AL (γhλ ) − AL (γ) Z T Z 1 ≤λ (L − ∂v L · γ˙ + ∂t L) ds + (∂x L · sh + ∂v L · h) ds 0 0     1 1 1 +C + κT (∆T )2 + C khk2 + |∆T |khk + 2 (∆T )2 T T T
2 + Cκ |∆T |khk + (∆T ) Using the Euler-Lagrange equation, we have for p(t) = ∂v L(γ, γ, ˙ t), d d (−tH) = (t(L − ∂v L · γ)) ˙ = (L − ∂v L · γ) ˙ + t∂t L, dt dt d d (p · th) = (∂v L · th) = ∂v L · h + ∂x L · th, dt dt we get T 1 AL (γhλ ) − AL (γ) ≤ λ(−tH) + (p(T − 1 + t) · ht) 0    0 1 1 + Ckh2 k + C +  |∆T |khk + C + κT (∆T )2 T T   1 2 ≤ −H(γ(T ), p(T ), T )(∆T ) + p(T ) · h + 2Ckhk + 2C + κT (∆T )2 . T

Before returning to the proof of Proposition 7.13, we state a corollary of Lemma 7.14, which gives a proof of item 4 of Proposition 6.2. Corollary 7.15. Let H ∈ H(D), and A be its associated action function. Then there exists a constant C depending on D such that A(x, t0 , y, ·) is C-semi-

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concave on (t1 −1, t1 +1), and A(x, ·, y, t1 ) is C−semi-concave on (t0 −1, t0 +1). Moreover, if γ : [t0 , t1 ] → Tn is an extremal curve with t1 − t0 > 2, then −H(γ(t0 ), p(t0 ), t0 ) ∈ −∂2+ A(x, t0 , y, t1 ), −H(γ(t1 ), p(t1 ), t1 ) ∈ ∂4+ A(x, t0 , y, t1 ). Proof. Let T = 2, and set t00 = t1 − T . We apply Lemma 7.14 to the curve γ|[t00 , t1 ], using the bound κ ≤ D, to get A(γ(t00 ), t00 , γ(t1 ), t1 + ∆T ) − A(γ(t00 ), t00 , γ(t1 ), t1 ) ≤ −H(γ(t1 ), p(t1 ), t1 ) · (∆T ) + C(∆T )2

(7.2)

for h ∈ Rn and ∆T ∈ [−1, 1], and some constant C > 0 depending only on D. Since A(γ(t0 ), t0 , γ(t1 ) + h, t1 + ∆T ) − A(γ(t0 ), t0 , γ(t1 ), t1 ) ≤ A(γ(t00 ), t00 , γ(t1 ) + h, t1 + ∆T ) − A(γ(t00 ), t00 , γ(t1 ), t1 ), we obtain (7.2) with t00 replaced with t0 . This proves our corollary. √ Proof of Proposition 7.13. Since √ t1 − t0 > 1/ , similar to the proof of Corollary 7.15, we set t00 = t1 − 1/ , and apply Lemma 7.14 to H (x, p, t) = √ H1 (x, p) + H2 (x, p, t), the curve γ|[t00 , t1 ], T = 1/ , κ = O(), and ∆T ∈ [−T /2, T /2]. We get A(γ(t0 ), t0 , γ(t1 ) + h, t1 + ∆T ) − A(γ(t0 ), t0 , γ(t1 ), t1 ) ≤ A(γ(t00 ), t00 , γ(t1 ) + h, t1 + ∆T ) − A(γ(t00 ), t00 , γ(t1 ), t1 )   1 2 ≤ p(t1 ) · h − H(γ(t1 ), p(t1 ), t1 ) · (∆T ) + Ckhk + C + κT (∆T )2 , T which implies the S -semi-concavity of A(x, t0 , ·, ·) on the set Rn ×(t1 −T /2, t1 + T /2). The semi-concavity for the first two component can be obtained by applying the same argument to the Lagrangian L(x, −v, t) to the curve γ(−·) on the interval [−t1 , −t0 ]. Let w(x, s) be a weak KAM solution to L − c · v, and define w(x, ˇ s) = Tˇcs,s+t w(·, s + t). Proposition 7.13 implies that w is S -semi-concave, while w ˇ is (−S )-semi-convex. Proof of Theorem 7.10. By considering the Hamiltonian H (θ, p + c, t), we reduce to the case c = 0. This also makes the constants depend on kck. By Proposition 6.14, φ−t G0,w ⊂ Iew,wˇ . Note also for each (x, p, s) ∈ Iew,wˇ , −H(x, p, s) + αL (c) = ∂s w(x, s). Applying Proposition 7.13 and Lemma 7.12, we obtain that for any (x1 , p1 , t1 ), (x2 , p2 , t2 ) ∈

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Iew,wˇ ,



    

x2 − x1 p2 − p1 C(x 2 − x1 )

; √ − ≤

H(x2 , p2 , t2 ) − H(x1 , p1 , t1 ) C (t2 − t1 ) S −1 t2 − t1 S 

therefore

√ 1 √ |H(x2 , p2 , t2 ) − H(x1 , p1 , t1 ) − C (t2 − t1 )|  ≤ Ckx2 − x1 k + Ckt2 − t1 k,

and the proposition follows.



Chapter Eight Cohomology of Aubry-Mather type

8.1

AUBRY-MATHER TYPE AND DIFFUSION MECHANISMS

Let r ≥ 2, H : Tn × Rn × T$ → R be a C r periodic Tonelli Hamiltonian in the family H(D). Definition 8.1. We say that the pair (H∗ , c∗ ) is of Aubry-Mather type if it satisfies the following conditions: 1. There is an embedding χ = χ(x, y) : T × (−1, 1) → Tn × Rn , such that C(H) = χ(T × (−1, 1)) is a normally hyperbolic weakly invariant cylinder under the time-$ map φH . We require C(H) to be symplectic, i.e. the restriction of the symplectic form ω to C is non-degenerate. 2. There is σ > 0, such that the following hold for all Vσ (H∗ ) := {kH − H∗ kC 2 } < σ,

Bσ (c∗ ) := {kc − c∗ k < σ}.

(We then say that the property below holds robustly at (H∗ , c∗ ).) a) The discrete-time Aubry set Ae0 (c) ⊂ C(H). Moreover, the pull back of the Aubry set is contained in a Lipschitz graph, namely, the map πx : χ−1 Ae0H (c) ⊂ T × (−1, 1) → T is bi-Lipschitz. b) If χ−1 Ae0H (c) projects onto T, then there is a neighborhood V 0 of A0H (c) such that the strong unstable manifold W u (Ae0H (c)) ∩ πθ−1 V 0 is a Lipschitz graph over the θ component. If (H∗ , c∗ ) is of Aubry-Mather type, let h ∈ Zn ' H1 (Tn , Z) be the homology class of the curve χ(T × {0}), called the homology class of (H∗ , c∗ ). Remark 8.2. The definition of Aubry-Mather type includes a much simpler case, that is when the Aubry set AeH (c) is a hyperbolic periodic orbit, still contained in a normally hyperbolic invariant cylinder (NHIC) C. The condition 2(a) is

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satisfied since Ae0H (c) is discrete. Condition (b) is vacuous since the χ−1 Ae0H (c) never projects onto T. This holds, in particular, at double resonance when the energy is critical. Definition 8.3. We say that the pair (H∗ , c∗ ) is of bifurcation Aubry-Mather type if there exist σ > 0 and open sets V1 , V2 ⊂ Tn with V1 ∩ V2 = ∅, and a smooth bump function f : Tn → [0, 1],

f |V1 = 0,

f |V2 = 1,

such that for all c ∈ Bσ (c∗ ) and H ∈ Vσ (H∗ ): 1. Each of the Hamiltonians H 1 = H − f and H 2 = H − (1 − f ) satisfies AH 1 (c) ⊂ V1 × T,

AH 2 (c) ⊂ V2 × T.

It follows that AeH 1 (c), AeH 2 (c) are both invariant sets of H, called the local Aubry sets. 2. (H 1 , c∗ ) (resp. (H 2 , c∗ )) are of Aubry-Mather type, with the invariant cylinders C1 and C2 . Moreover, they have the same homology class h. 3. The Aubry set AeH (c) ⊂ AeH 1 (c) ∪ AeH 2 (c). According to item (3), the Aubry set AeH (c) may be contained in one of the local components, or both. Remark 8.4. Suppose H is such that AH (c) consists of two disjoint compact components A1H (c) and A2H (c). Choose open sets V1 , V2 such that V1 ∩ V2 = ∅, and A1H (c) ⊂ V1 , A2H (c) ⊂ V2 . Letting H 1 , H 2 as in Definition 8.3, we get LH 1 = LH + f,

LH 2 = LH + (1 − f ).

Lemma 6.17 then implies AjH (c) = AH j (c), j = 1, 2. In this sense, our definition says that each of the two local components of the Aubry set is of Aubry-Mather type. There is another type of bifurcation in which one component of the Aubry set is of Aubry-Mather type with an invariant cylinder C1 , and another is a hyperbolic periodic orbit. We call this the asymmetric bifurcation. This case appears at double resonance, when the shortest loop is simple non-critical. See Section 4.2. Definition 8.5. We say that the pair (H∗ , c∗ ) is of asymmetric bifurcation type if there exist σ > 0 and open sets V1 , V2 ⊂ Tn with V1 ∩ V2 = ∅, and a smooth bump function f : Tn → [0, 1],

f |V1 = 0,

f |V2 = 1,

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79

such that for all c ∈ Bσ (c∗ ) and H ∈ Vσ (H∗ ): 1. Item 1 of Definition 8.3 holds. 2. (H 1 , c∗ ) is of Aubry-Mather type, with the invariant cylinder C1 . The Aubry set AeH 2 (c∗ ) is a single hyperbolic periodic orbit. 3. The Aubry set AeH (c) ⊂ AeH 1 (c) ∪ AeH 2 (c). Theorem 8.6. Suppose a pair (H∗ , c∗ ) is of Aubry-Mather type, and let Γ 3 c∗ be a smooth curve in R2 . Then there is σ > 0 and a residual subset Rσ (H∗ ) of Vσ (H∗ ) such that for all c ∈ Γ1 := Bσ1 (c∗ ) ∩ Γ and H ∈ Rσ (H∗ ), at least one of the following holds: 0 1. The projected Ma˜ n´e set NH (c) is contractible as a subset of Tn . 2. There is a double covering map Ξ such that the set 0 0 eH◦Ξ eH N (ξ ∗ c) \ Ξ−1 N (c)

is totally disconnected. Theorem 8.7. Suppose (H∗ , c∗ ) is of either • bifurcation AM type, or • asymmetric bifurcation type, and c∗ ∈ Γ, where Γ is a smooth curve in R2 , then there is σ > 0 and residual R ⊂ Vσ (H∗ ) such that for c ∈ Bσ (c∗ ) ∩ Γ and H ∈ R, at least one of the following holds: e 0 (c)), and either 1. Ae0H (c) has a unique static class, (hence Ae0H (c) = N H 0 a) the projected Ma˜ n´e set NH (c) is contractible as a subset of Tn , or b) there is a double covering map Ξ such that the set 0 0 eH◦Ξ eH N (ξ ∗ c) \ Ξ−1 N (c)

is totally disconnected. 0 e 2. AH (c) has two static classes, and 0 eH N (c) \ Ae0H (c)

is a discrete set. Analogous result in the a priori unstable setting is due to Mather [59] and Cheng-Yan [26, 27]. Our definition is more general and applies, as will be seen, to both single- and double-resonant settings. The prove is based on the result of [13], which applies to our setting with appropriate changes. Here we describe the changes needed for the proof in [13] to apply.

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If (H∗ , c∗ ) is of Aubry-Mather type, let h ∈ Zn ' H1 (Tn , Z) be its homology class. Then any minimal measure contained in Ae0 (c) must have rotation vector λh, λ ∈ R. Moreover, similar to the case of twist map, all such measures have the same rotation vector λh. We say the rotation vector is rational/irrational if λ is rational/irrational. The following proposition is a consequence of the Hamiltonian Kupka-Smale theorem, see for example [71]. Proposition 8.8. There are σ1 , σ2 > 0, and a residual subset R1 of the ball Vσ2 (H∗ ), such that for all c ∈ Γ1 := Bσ1 (c∗ ) ∩ Γ, H ∈ R1 , if c has rational rotation vector, then πχ−1 Ae0 (c) 6= T. Let 1 ≤ j ≤ n, and ej be the jth coordinate vector in Rn . Suppose that ej ∦ h; define the covering map ξ : Tn → Tn by ξ(θ1 , · · · , θn ) = (θ1 , · · · , 2θj , · · · , θn ). • For H ∈ R1 , define 0 Γ∗ (H) = {c ∈ Γ1 : πχ−1 (NH (c)) = T}.

(8.1)

Proposition 8.8 implies that each c ∈ Γ∗ (H) has an irrational rotation vector, 0 and the Aubry set has a unique static class, and A0H (c) = NH (c). Each Γ∗ (H) is compact due to upper semi-continuity of the Ma˜ n´e set. • For H ∈ R1 and c ∈ Γ∗ (H), the lifted Aubry set Ae0H◦Ξ (ξ ∗ c) = Ξ−1 AeH (c) has two static classes, denote them Se1 , Se2 (and projections S1 , S2 ). We have the decomposition 0 eH◦Ξ e12 ∪ H e21 , N (ξ ∗ c) = Se1 ∪ Se2 ∪ H

eij consists of heteroclinic orbits from Sei to Sej . We will use the where H e notation Sei (H, c), H(H, c) to show dependence on the pair (H, c), and Hij for eij . projection of H • For H ∈ R1 and c ∈ Γ∗ (H), consider the (discrete-time) Peierl’s barrier (see (6.6)) h(ζ1 , ·), h(ζ2 , ·), h(·, ζ1 ), h(·, ζ2 ), ζi ∈ Si , i = 1, 2, where h = hH◦Ξ, ξ∗ c . These functions are independent of the choice of ζi except for an additive constant. We define b− H,c (θ) = h(ζ1 , θ) + h(θ, ζ2 ) − h(ζ1 , ζ2 ) ± and b+ H,c by switching ζ1 , ζ2 . The functions bH,c are non-negative and vanish on H12 ∪ S1 ∪ S2 and H21 ∪ S1 ∪ S2 , respectively. • Consider small neighborhoods V1 , V2 of S1 (H∗ , c∗ ), S2 (H∗ , c∗ ), and define K = Tn \ (V1 ∪ V2 ). By semi-continuity of the Aubry set, for sufficiently small

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σ1 , σ2 , K is disjoint from Si (H, c) for all c ∈ Γ1 and H ∈ R1 . Moreover, e12 (H, c) and H e12 (H, c). π −1 K intersects every orbit of H Lemma 8.9 (Lemma 5.2 of [13]). For each (H, c) ∈ R1 × Γ1 , the set eH◦Ξ (ξ ∗ c) \ Ξ−1 N eH (c) N is totally disconnected if and only if NH◦Ξ (ξ ∗ c) ∩ K = (H12 ∪ H21 ) ∩ K is totally disconnected. We will show that the set of H ∈ R1 with the following property contains a dense Gδ set: for each c ∈ Γ∗ (H), NH◦Ξ (ξ ∗ c) ∩ K is totally disconnected. The following lemma implies the Gδ property. Lemma 8.10 (Lemma 5.3 of [13]). Let K ⊂ Tn be compact, then the set of H ∈ R1 such that for all c ∈ Γ∗ (H), the set NH◦Ξ (ξ ∗ c) is totally disconnected, is a Gδ set. The following proposition allows local perturbations of b± H,c simultaneously for all c’s in a small ball. Let Bσ (x) denote the ball of radius σ at x in a metric space. Proposition 8.11 (Proposition 5.2 of [13]). Let H∗ ∈ R1 , c∗ ∈ Γ∗ (H), and K ∩ AH◦Ξ (ξ ∗ c) = ∅. Then there is σ > 0 such that for all H ∈ R1 ∩ Bσ (H∗ ),

θ0 ∈ K ∩ H12 (H∗ , c∗ ),

ϕ ∈ Ccr (Bσ (θ0 )) with kϕkC r < σ,

there is a Hamiltonian Hϕ such that: 1. For all c ∈ Bσ (c∗ ), the Aubry sets AeHϕ ◦Ξ (ξ ∗ c) coincides with AeH◦Ξ (ξ ∗ c) with the same static classes. In particular, Bσ (c∗ ) ∩ Γ∗ (H) = Bσ (c∗ ) ∩ Γ∗ (Hϕ ). 2. For all c ∈ Bσ (c0 ) ∩ Γ∗ (H), there exists a constant e ∈ R such that + b+ Hϕ ,c (θ) = bH,c (θ) + ϕ(θ) + e,

θ ∈ Bσ (θ0 ).

(8.2)

The same holds for θ0 ∈ K ∩ H21 (H∗ , c∗ ), with b+ replaced with b− in (8.2). Moreover, for each H ∈ R1 ∩ Bσ (H∗ ), kHϕ − HkC r → 0 when kϕkC r → 0. We will use Proposition 8.11 to locally perturb the functions b+ H,c , therefore, perturbing H12 (H, c) ∩ Bσ (θ0 ) = argmin b+ H,c ∩ Bσ (θ0 ). Similarly for b− . However, as observed by Mather and Cheng-Yan, this requires additional information on how b± H,c depends on c.

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Proposition 8.12 (Section 8.3). There is β > 0 such that for each H ∈ R1 , 0 n the maps c 7→ b± older. H,c from Γ∗ (H) to C (T , R) are β-H¨ ∗ As a result, the set {b± H,c : c ∈ Γ (H)} has Hausdorff dimension at most 1/β 0 n in C (T , R). The following lemma allows us to take advantage of this fact.

Lemma 8.13 (Lemma 5.6 of [13]). Let F ⊂ C 0 ([−1, 1]n , R) be a compact set of finite Hausdorff dimension. The following property is satisfied on a residual set of functions ϕ ∈ C r (Rn , R) (with the uniform C r norm): For each f ∈ F, the set of minima of the function f + ϕ on [−1, 1]n is totally disconnected. As a consequence, for each open neighborhood Ω of [−1, 1]n in Rn , there exist arbitrarily C r -small compactly supported functions ϕ : Ω → R satisfying this property. Proof of Theorem 8.6. Let (H∗ , c∗ ) be of Aubry-Mather type, and let σ > 0 be as in Definition 8.1. Let K be as in Lemma 8.9 and denote Γ1 = Bσ (c∗ ) ∩ Γ. Let R2 ⊂ R1 ∩ Vσ (H∗ ) be the set of Hamiltonians such that for all c ∈ Γ1 ∩ Γ∗ (H), e 0 (ξ ∗ c) \ N e 0 (c) is totally disconnected. According to Lemma 8.10, the set N H◦Ξ H this set is Gδ , we will to show that it is dense on Vσ2 (H∗ ) for some 0 < σ2 < σ. Consider c ∈ Γ1 ∩ Γ∗ (H∗ ), let σc > 0 be small enough so that Proposition ∗ e0 8.11 applies to the pair (H∗ , c) on the set K. For each θ0 ∈ N H∗ ◦Ξ (ξ c) ∩ K, define √ Dσc (θ0 ) = {θ : max |θi − θ0i | ≤ σc /(2 n)} ⊂ Bσc (θ0 ). i

Proposition 8.12 implies that the family of functions b± H,c , c ∈ Γ1 ∩ Γ∗ (H) has Hausdorff dimension at most 1/β, therefore we can apply Lemma 8.13 on the cube Dσc (θ0 ) for each H ∈ R1 . We find arbitrarily small functions ϕ compactly supported in Dσc (θ0 ) and such that each of the functions b± N,c + ϕ,

c ∈ Γ1 ∩ Γ∗ (H)

have a totally disconnected set of minima in Dσc (θ0 ). We then apply Proposition 8.11 to get Hamiltonians Hϕ approximating H. We obtain: • The set of Hamiltonians H such that NH◦Ξ (ξ ∗ c) ∩ Dσc (θ0 ) is totally disconnected for each c ∈ Γ∗ (H) is dense in R1 ∩ Vσc (H∗ ). By Lemma 8.10, it is also Gδ , and therefore residual. Sk Since K is compact, there is a finite cover K ⊂ i=1 Dσi (θi ), such that • For a residual set Ri (c) of H ∈ Bσc (H∗ ), the set NH◦Ξ (ξ ∗ c) ∩ Dσc (θi ) is totally disconnected for all i = 1, . . . , k and c ∈ Γ∗ (H).

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T ∗ Take Rc = i Ri , then for H ∈ Rc , the set NH◦Ξ (ξ S c) ∩ K is totally disconnected. Finally, we consider a finite covering Γ1 ⊂ j Bσcj (cj ) and repeat the preceding argument. The next two sections are dedicated to proving Proposition 8.12.

8.2

WEAK KAM SOLUTIONS ARE UNSTABLE MANIFOLDS

An important consequence of the Aubry-Mather type is that the local unstable manifold coincides with an elementary weak KAM solution. Proposition 8.14. Suppose (H∗ , c∗ ) is of Aubry-Mather type. Then for each (H, c) ∈ Vσ (H∗ ) × Bσ (c∗ ) such that χ−1 Ae0H (c) projects onto T, we have W u (Ae0H (c)) ∩ πθ−1 V 0 = {(θ, c + ∇u(θ)) :

θ ∈ V 0 ∩ A0H (c)},

where u(θ) = hH,c (ζ, θ) for some ζ ∈ A0H (c). Lemma 8.15. Suppose Ae ⊂ Tn × Rn is a closed 1-dimensional Lipschitz sube is a Lipmanifold, invariant under φH , partially hyperbolic, and that W u (A) n n u e schitz manifold in R × T . Then W (A) is isotropic in the sense that for e and every v, w ∈ Tz W u (A), e we have Lebesgue almost every z ∈ W u (A) (dθ ∧ dp)(v, w) = 0.

(8.3)

e is a Lipschitz graph over a neighborhood V 0 ⊂ Rn , then there exists If W u (A) 1,1 aC function u : V 0 → R such that e = {(θ, ∇u(θ)) : θ ∈ V 0 }. W u (A) e φ = φH and ω = dθ ∧ dp. Since W u is backward Proof. Denote W u = W u (A), e it suffices to prove invariant, and every φH backward orbit is asymptotic to A, e Since the (8.3) for v, w ∈ Tz W u , for a.e. z ∈ W u in a neighborhood U of A. e partially hyperbolic splitting of A extends to a neighborhood, for z ∈ U ∩ W u , there is a continuous splitting Tz W u = E c (z) ⊕ E u (z), k such that for any v ∈ Tz W u and v u ∈ E u (z), we have k(Dφ−k H )vk ≤ Cµ and −k u k −1 k(DφH )v k ≤ Cλ for some C > 0 and 0 < µ < λ . Moreover, dim E c = 1 since Ae is 1-dimensional. For a.e. z ∈ U ∩ W u such that Tz W u is well defined:

• If v, w ∈ E c (z), then ω(v, w) = 0 since E c (z) is 1-dimensional.

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• If v ∈ Tz W u and w ∈ E u (z), then ωz (v, w) = ωφ−k ((Dφ−k (z))v, (Dφ−k (z))w) ≤ C 2 (λµ)k → 0,

as k → ∞.

This implies ω(v, w) = 0 for all v, w ∈ Tz W u . Suppose now that W u is a graph over V 0 , then W u is Lagrangian. Fix e for any z ∈ W u , consider a curve ζ(t) = (θ, p)(t) such that ζ(0) = z0 , z0 ∈ A, Rt ˙ ζ(t) = z, then the path integral 0 p(t) · θ(t)dt is independent of the path and 0 defines a function u : V → R. For a different choice of z0 , u differs by a constant. By differentiating the integral, we see that (θ, p) ∈ W u ∩ V 0 if and only if p = ∇u(θ). Lemma 8.16. Under the same assumption as Proposition 8.14, consider the S time-periodic Aubry set AeH (c) = t∈[0,$(H)] φtH (Ae0H (c)) ⊂ Td × Rd × (R/$Z) and W u (AeH (c)) its strong unstable manifold. Then there exists a neighborhood V of AH (c) and a C 1,1 function w : V 0 × R → R, w(·, t + ϕ(H)) = w(·, t), such that w solves the Hamilton-Jacobi equation wt + H(θ, c + ∂θ w(θ, t), t) = αH (c), and (θ, ∂θ w(θ, t), t) = W u (AeH (c)) for all (θ, t) ∈ V . Assume, in addition, that AeH (c) has a unique static class. Then there is C ∈ R and (ζ, τ ) ∈ AH (c) such that w(θ, t) = hH,c (ζ, τ, θ, t) + C,

(θ, t) ∈ V.

Proof. By considering H(θ, c + p, t), we reduce to the case c = 0. Denote Ae0 = e ⊂ Ae0H (0), Aet = φtH (Ae0 ), W u,0 = W u (Ae0 ), W u,t = φt (W u ), W u = W u (A) d d d T × R × (R/$Z). Let w0 : T × R → R denote a weak KAM solution of H. We have: • For each (θ, p, t) ∈ Aet , p = ∂θ w0 (θ, t). • Applying Lemma 8.15 to each Aet with t ∈ [0, $(H)], we get that each W u,t is the gradient of a function ut (θ). Moreover, ut is uniquely determined by requiring ut (θ) = w0 (θ, t) for θ ∈ πθ Aet . Write w(θ, t) = ut (θ), then e in a neighborhood (θ, ∂θ w(θ, t), t) coincides with W u (A) e ⊂ Tn × Let V be a neighborhood of AeH (0) contained in the projection πθ W u (A) s (R/ϕZ). Then for each (θ, t) ∈ V , we have (θ(s), p(s), s) = φH (θ, ∂θ w(θ, t), t) ∈ W u for all sufficiently small s. It follows that θ˙ = ∂p H(θ, ∂θ w(θ, t), t),

d (∂θ w(θ(t), t)) = −∂θ H(θ, ∂θ w(θ, t), t). dt

Suppose (θ(t), t) is a point at which ∂θθ w(θ, t) exists (such points have full

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COHOMOLOGY OF AUBRY-MATHER TYPE

Lebesgue measure), then ˙ = −∂θ H(θ, ∂θ w(θ, t), t). ∂θ,t w(θ, t) + ∂θθ w(θ, t)θ(t) Combine the last two equations, we get 0 = ∂θt w + ∂θ H + ∂θθ w∂p H = ∂θ (∂t w(θ, t) + H(θ, ∂θ w(θ, t), t), t)) . Since this equality holds for a.e (θ, t), we conclude that there is a function g(t) : (R/$Z) → R such that wt + H(θ, c + ∂θ w(θ, t), t) = g(t). e and we conclude that g(t) = αH (0). Finally, note that g(t) = αH (0) on A, We now prove w coincides with a Peierl’s barrier function. According to Proposition 4.1.8 of [37], when u solves the Hamilton-Jacobi equation, u is dominated by L, and that for each (θ, t) ∈ V there exists  > 0 and a curve γ : [t − , t] → Td such that γ is calibrated by w (see Section 6.2). Theorem 4.2.6 of [37] implies that the orbit (γ(s), ∂θ w(γ(s), s), s) solves the Hamiltonian equation, and in particular, (γ(s), s) ∈ V for all s < t. Repeating the argument, we conclude that at each (θ, t) there exists a backward calibrated curve γ : (−∞, t] → Td such that γ(t) = θ. We note that the associated Hamiltonian e orbit is in the unstable manifold of A. Suppose Tn < 0 is such that (γ(−Tn ), −Tn ) → (ζ, τ ) ∈ AH (0). Note that z 0 = (ζ, ∂θ w(ζ, τ )τ ) is the unique point such that (θ, ∂θ w(θ, t), t) ∈ W u (z 0 ). Let γ : (−∞, t] → Td the backward calibrated orbit ending at (θ, t). Using the uniform Lipschitz property of the action function AH , we have w(θ, t) = lim w(γ(t − Tn ), t − Tn ) + AH (γ(t − Tn ), t − Tn , θ, t) n→∞

= w(ζ, τ ) + lim AH (ζ, t − Tn , θ, t) ≥ w(ζ, τ ) + hH (ζ, τ, θ, t). n→∞

On the other hand, for arbitrary Tn0 = $n + t − τ , by the domination property w(θ, t) ≤ w(ζ, τ ) + AH (ζ, τ − Tn , θ, t), we get w(θ, t) ≤ w(ζ, τ )+hH (ζ, τ, θ, t) by taking lim inf to both sides. Note that we have not proven what’s needed since (ζ, τ ) depends on (θ, t). When there is only one static class we have hH (ζ1 , τ1 , θ, t) = hH (ζ1 , τ1 , ζ2 , τ2 ) + hH (ζ2 , τ2 , θ, t),

(ζ1 , τ1 ), (ζ2 , τ2 ) ∈ AH (0),

which allows a consistent choice of (ζ, τ ) for all (θ, t). Proof of Proposition 8.14. By converting Definition 8.1, (2)(b), to its continuous counterpart, the condition of Lemma 8.16 is satisfied for AeH (c). The proposition follows by taking the zero section of the weak KAM solution.

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Figure 8.1: Local coordinates in the configuration space near S1 .

8.3

REGULARITY OF THE BARRIER FUNCTIONS

In this section we prove Proposition 8.12. Let c∗ ∈ Γ∗ (H∗ ), H ∈ Vσ (H∗ ), and c ∈ Bσ (c∗ ) ∩ Γ∗ (H), then according to our assumption, the Aubry set AeH (c) is contained in the cylinder C and has a unique static class. Let us now consider the lift Ae0H◦Ξ (ξ ∗ c), which has two components Se1 , Se2 . The cylinder C lifts to two disjoint cylinders C1 , C2 . For the rest of the discussion, we will consider only the static class Se1 as the other case is similar. Definition 8.1 ensures that there is a Lipschitz function y = g(x) ∈ (−1, 1), x ∈ T, such that Se1 (H, c) = {χ(x, g(x)) : x ∈ T} =: {Fc (x) = (Fcθ (x), Fcp (x)) : x ∈ T} ⊂ Tn × Rn . Now suppose c, c0 ∈ Γ∗ (H) ∩ Bσ (c∗ ). First we have: Lemma 8.17. There is C > 0 such that 1

sup kFc (x) − Fc0 (x)k ≤ Ckc − c0 k 2 . x

Proof. The proof is the same as the one in Lemma 5.8, [13] where the proof only used the symplecticity of the cylinder. The main idea is that supx kFc (x) − Fc0 (x)k is 12 H¨ older with respect to the area between the two invariant curves restricted to the cylinder, while the latter is equivalent to the symplectic area by assumption. A direct calculation shows the symplectic area is bounded by kc0 − ck. Let us now denote uc (θ) = hH◦Ξ,ξ∗ c (ζ1 , θ),

ζ1 ∈ S1 (H, c).

Before moving forward, we define a convenient local coordinate system near S1 . According to Proposition 8.14, the graph (θ, c + ∇uc (θ)) locally coincides

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with W u (Se1 (H, c)). The (strong) unstable bundle E u (z) of z ∈ Se1 (H, c) is tangent to W u (Se1 (H, c)) at every point and transverse to the tangent cone of Se1 (H, c). Since W u (Se1 (H, c)) is a Lipschitz graph over Tn , the projection N u (z) := dπθ E u (z) forms a non-zero section of the normal bundle to S1 (H, c) = πθ Se1 (H, c) within the configuration space Tn . By choosing an orthonormal basis e1 (z), · · · , en−1 (z), N u (z) naturally defines a coordinate system (xf , xs ) ∈ T × Rn−1 on the tubular neighborhood of S1 : ι(xf , xs ) = Fcθ (xf ) +

n−1 X

ei (Fcθ (xf ))xsi .

i=1

See Figure 8.1. We note that the coordinate system is only H¨older since the unstable bundle is only H¨ older a priori. Therefore, we consider C ∞ functions θ that approximate Fc and ei in the C 0 sense, the new coordinate system is still well defined near S1 , and the xs coordinate projects onto the unstable direction. In the sequel, we fix such a coordinate system using W u (Se1 (H∗ , c∗ )). Due to semi-continuity, for (H, c) close to (H∗ , c∗ ), the coordinate system is defined in a neighborhood of S1 (H, c). For a partially hyperbolic invariant set, the holonomy map for the strong unstable foliation is a priori only H¨ older (see [70]). The H¨older constant depends on the ratio between the unstable expansion rate and norm of the tangent map in the center direction. Assume that β0 ∈ (0, 1] is the H¨older exponent for the strong unstable foliation of W u (Se1 (H, c)). We obtain the following regularity result: Lemma 8.18. There is σ ∈ (0, 1) such that for H ∈ Vσ (H∗ ), c, c0 ∈ Γ∗ (H) ∩ Bσ (c∗ ), θ ∈ Bσ (S1 (H, c∗ )), there is β > 0, C1 > 0, C2 ∈ R such that 1. |∇uc (θ) − ∇uc0 (θ)| ≤ C1 kc − c0 kβ ; 2. |uc (θ) − uc0 (θ) − C2 | ≤ C1 kc − c0 kβ . Moreover, the same holds with S1 replaced with S2 . Proof. The proof follows essentially Lemma 5.9 in [13]. Let σ1 be small enough such that the local coordinates (xf , xs ) are defined for |xs | < σ. We then consider the weak KAM solution uc ◦ ι(xf , xs ) instead, which we still denote as uc (xf , xs ), abusing the notation. Let |xs | < σ1 , let y = (xf , xs , c + ∇uc (xf , xs )), and let z ∈ S1 (H, c) be such that y ∈ W u (z). We then define z 0 ∈ S1 (H, c) to be the unique such point with xf (z 0 ) = xf (z). Finally define y 0 ∈ W u (z 0 ) to be such that xs (y 0 ) = xs (y), which is possible since W u (z 0 ) is a graph over the xs coordinates. See Figure 8.2. We note that within the center unstable manifold W u (C1 ), the NHIC C1 on one hand, and xs = xs (y) on the other hand serve as two transversals to the strong unstable foliation {W u (·)}. Since the foliation is β0 -H¨older, there exists

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Figure 8.2: Proof of Lemma 8.18.

C > 0 (throughout the proof, C denotes a generic constant) such that ky − y 0 k ≤ Ckz − z 0 kβ0 ≤ Ckc − c0 k

β0 2

.

Denoting w = (θ, c0 + ∇uc0 (θ)) and noting y 0 ∈ W u (S1 (H, c0 )) = {(θ, ∇uc0 (θ))} which is locally a C 1 graph, we get for C > 0 kw − y 0 k ≤ Ckπθ (w) − πθ (y 0 )k = Ckπθ (y) − πθ (y 0 )k ≤ Cky − y 0 k, therefore kc + ∇uc (θ) − c0 − ∇uc0 (θ)k ≤ kw − yk ≤ kw − y 0 k + ky − y 0 k ≤ Cky − y 0 k ≤ Ckc − c0 k

β0 2

.

Since kc − c0 k < σ < 1, kc − c0 k < kc − c0 kβ0 /2 , we have k∇uc (θ) − ∇uc0 (θ)k ≤ kc − c0 k + Ckc − c0 k

β0 2

≤ (C + 1)kc − c0 k

β0 2

.

We now revert the local coordinate ι to obtain item 1 with β = β0 /2, and possibly changing σ and C1 . Item 2 is obtained from item 1 by direct integration.

8.4

BIFURCATION TYPE

We prove Theorem 8.7 in this section. Suppose (H∗ , c∗ ) is of bifurcation type, and let H 1 , H 2 be the Hamiltonians as defined in Definition 8.3, then there exist cylinders C1 , C2 containing the local Aubry sets AeH 1 (c) and AeH 2 (c). i Let us denote αH (c) = αH i (c), called the local alpha functions. Then AeH (c) 1 2 has two static classes if and only if αH (c) = αH (c). Moreover, for each c, the

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COHOMOLOGY OF AUBRY-MATHER TYPE

i rotation vector ρH i (c) is uniquely defined, as a result, αH is a C 1 function.

Proposition 8.19. Let H∗ , c∗ be of bifurcation Aubry-Mather type. There is σ > 0 such that for an open and dense subset of H ∈ Vσ (H∗ ), there are at most 1 2 finitely many c ∈ Bσ (c∗ ) ∩ Γ for which αH (c) = αH (c). 0 e For each such (bifurcation) c, we have AH 1 (c), Ae0H 2 (c) both supported on e 0 (c) \ Ae0 (c) is a discrete set. hyperbolic periodic orbits. Moreover, N H H Proof. Let σ be as in Definition 8.3. For the rest of the proof, we refer to Bσ (c∗ ) ∩ Γ as Γ to simplify notations. Consider the family of Hamiltonians Hλ = H(θ, p, t) − λ(1 − f ) (f is the mollifier function in Definition 8.3). Lemma 6.17 implies 1 1 αH (c) = αH (c) + λ, λ

2 2 αH (c) = αH (c). λ

By Sard’s theorem, there exists a full Lebesgue measure set E of regular values 1 2 λ of the function αH − αH , which implies for λ ∈ E, 0 is a regular value of 1 2 αHλ − αHλ (c), which implies the equation has only finitely many solutions on Γ, and at each solution, α1 and α2 has different derivatives when restricted to Γ. Note that this property is open in H, which implies the claim of our proposition is an open property. We only need to prove density. 1 2 Let ci (λ) ∈ Γ, i = 1, · · · , N be the set on which αH = αH ; then since λ λ d 1 d 2 α (c ) = 6 α (c ), each c (λ) is locally a monotone function in λ. i dλ Hλ i dλ Hλ i We now impose the assumption that H is a Kupka-Smale system; namely, all periodic orbits are non-degenerate, which holds on a residual set of H’s (see [72]). In this setting, all invariant measures of rational rotation vectors are supported on hyperbolic periodic orbits, and by a further perturbation, we can ensure for each c there is only one minimal periodic orbit. Since the Aubry set is upper semi-continuous when there is only one static class, and the hyperbolic periodic orbit is structurally stable, we obtain that each hyperbolic periodic orbit is the Aubry set for an open set of c’s. Let us denote by R(H, Γ) the set of all c ∈ Γ such that Ae0H (c) is a hyperbolic periodic orbit, then R(H 1 , Γ) and R(H 2 , Γ) are both open and dense in Γ. For each ci , there is di > 0 and an open and dense subset of λ ∈ Bdi (0) such that ci (λ) ∈ R(H 1 , Γ) ∩ R(H 2 , Γ). Let d = min di , then there is an open and dense set of λ ∈ Bd (0) such that for all i = 1, · · · , N , ci (λ) ∈ R(Γ). For these ci ’s, Ae0H 1 (ci ), Ae0H 2 (ci ) are both hyperbolic periodic orbits. Using the Kupka-Smale theorem again, it is an open and dense property such that the stable and unstable manifolds of Ae0H 1 (ci ) and Ae0H 2 (ci ) intersect transversally. Due to dimension considerations, the intersection is a eH (ci ) \ AeH (ci ) is a heteroclinic orbit between discrete set. Since each orbit in N 0 0 AeH 1 (ci ) and AeH 2 (ci ), it is also a discrete set. We have now proven the claim of our proposition holds for an arbitrarily small perturbation Hλ of H, and hence is dense on Vσ (H∗ ). An analogous statement holds for the asymmetric bifurcation case:

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Proposition 8.20. Let H∗ , c∗ be of asymmetric bifurcation type, then there is σ > 0 such that for an open and dense of H ∈ Vσ (H∗ ), such that there is a unique 1 2 c ∈ Bσ (c∗ ) ∩ Γ for which αH (c) = αH (c). Moreover, at such (bifurcation) c, 0 0 e e we have AH 1 (c), AH 2 (c) both supported on hyperbolic periodic orbits. Moreover, e 0 (c) \ Ae0 (c) is a discrete set. N H H Proof. The proof is nearly identical to Proposition 8.19 with the simplification that for σ small enough, AeH 2 (c) is always the same hyperbolic periodic orbit, and αH 2 (c) is a linear function. Since αH 1 (c) is convex on Γ, there is at most one bifurcation. The rest of the proof is identical. Proof of Theorem 8.7. Case 1 : Let H∗ , c∗ be of bifurcation Aubry-Mather type, and let σ > 0 be such that Proposition 8.19 holds on a open and dense subset R1 of Vσ (H∗ ). Then on Γ ∩ Bσ (c∗ ) there are at most finitely many bifurcations ci , i = 1, · · · , N . Moreover, for each bifurcation value ci , there is σi > 0 such that for c ∈ Bσi (ci ), the local Aubry sets are hyperbolic periodic orbits. This means e 0 (c) = Ae0 (c) are that for each (non-bifurcation) c ∈ Bσi (ci ) \ ci , the sets N H H e 0 (ci ) \ Ae0 (ci ) contractible (in fact finite). At the bifurcation values, the set N H H is discrete. We now consider the set 

N  [ Γ ∩ Bσ (c∗ ) \ Bσi (ci ), i=1

which is compact with finitely many connected components. On each of the components Ae0H (c) has a unique static class. We apply Theorem 8.6 to get that there is σ 0 > 0 such that the dichotomy of Theorem 8.6 holds for a residual subset R2 of H ∈ Vσ0 (H∗ ) on each of the connected components. The theorem follows by taking the intersection of R1 and R2 as well as the smallest value of σ and σ 0 . Case 2 : If (H∗ , c∗ ) is of asymmetric bifurcation type, the proof is the same with Proposition 8.20 replacing Proposition 8.19.

Part III

Proving forcing equivalence

Chapter Nine Aubry-Mather type at the single resonance In this chapter we prove that for a single-resonance normal form system which satisfies the non-degeneracy conditions [SR1λ ] or [SR2λ ], every c in the resonance curve Γ is of either Aubry-Mather or bifurcation Aubry-Mather type. The main results are Theorems 9.3 and 9.5, which restate Propositions 3.9 and 3.10 in Part I. At the end of this chapter we prove that the conditions hold on an open and dense set of Hamiltonians.

9.1

THE SINGLE MAXIMUM CASE

In this section we consider the Hamiltonian system N = H0 (p) + Z(θs , p) + R(θ, p, t), where θ = (θs , θf ) ∈ Tn−1 × T and p = (ps , pf ) ∈ Rn−1 × R. Consider the resonant curve Γ = {p ∈ B n :

∂ps H0 (p) = 0} = {p∗ (pf ) = (ps∗ (pf ), pf ) :

pf ∈ [a− , a+ ]}.

We first consider the case p ∈ Γ satisfy the condition [SR1λ ], namely for each p ∈ Bλ (p0 ) ∩ Γ, Z(θs , p∗ (pf )) has a unique global maximum at θ∗s (pf ), and 2 D−1 I ≤ ∂pp H0 ≤ DI,

λI ≤ −∂θ2s θs Z(θ∗s (pf ), p∗ (pf )) ≤ λI, (9.1) and that for K > 0, and p0 = p∗ (a0 ) ∈ Γ, kZkC 3 ≤ 1,

kRkCI2 (Tn ×BK √ (p0 )×T) ≤ δ. Note that we are using the rescaled norm CI2 (see (3.2)). We then set K1 = K/D and consider the local segment √ √ Γ(, p0 ) = {p∗ (pf ) : pf ∈ [a0 − K1 , a0 + K1 ]} ⊂ Γ ∩ BK √ (p0 ), which is contained in Bλ (p0 )∩Γ if 0 is small enough depending on K. Throughout this section, we write f = O(g) if |f | ≤ C|g| for C > 0 that may depend

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only on D, λ and n. Theorem 9.1 (See Sections 9.4 and 9.5, see also Theorem 3.1 of [13]). Suppose that the Hamiltonian N = H0 + Z + R satisfies (9.1), and that K1 > 2. There are δ0 , 0 √ > 0 and C = C(D, λ, n) > 1, such that if 0 < δ < δ0 and 0 <  < max{0 , δ}, there is a C 2 map √ √ (Θs , P s )(θf , pf , t) : T × [a0 − K1 /2, a0 + K1 /2] × T → Tn−1 × Rn−1 , such that C = {(θs , ps ) = (Θs , P s )(θf , pf , t)} is weakly invariant in the sense that the vector field is tangent to C. C is contained in the set V = {(θ, p, t) :

kθs − θ∗s (pf )k ≤ C −1 ,

kps − ps∗ (pf )k ≤ C −1 }

and it contains all the invariant sets contained in V . Moreover, we have

s f f

√

Θ (θ , p , t) − θ∗s (pf ) ≤ Cδ, P s (θf , pf , t) − ps∗ (pf ) ≤ Cδ , k∂pf Θs k ≤ C

p

δ/,

√ k∂(θf ,t) Θs k ≤ C ,

k∂pf P s k ≤ C,

√ k∂(θf ,t) P s k ≤ C .

The cylinder C is normally hyperbolic with its stable/unstable bundle projecting onto the θs direction. Theorem 9.2 (See Section 9.6, see also Theorems 4.1 and 4.2 of [13]). There are δ0 = δ0 (λ, n, D) > 0 and 0 = 0 (λ, n, D, δ) such that if 0 < δ < δ0 and 0 <  < 0 , the Ma˜ n´e set of the cohomology c satisfies eN (c) ⊂ Bδ1/5 (θs ) × T × B√·δ1/16 × T ⊂ Tn−1 × T × Rn × T. N ∗  e c) ⊂ Tn × Rn is If u is a weak KAM the set I(u, √ solution of N at c, then n contained in a 18 D-Lipschitz graph above T . The theorems as stated are analogous to the cited theorems in [13]. The main difference is that we now assume the much weaker assumption kRkCI2 ≤ δ. Nevertheless, we now check that the method in [13] applies in the same way and leads to the estimates as stated.

9.2

AUBRY-MATHER TYPE AT SINGLE RESONANCE √ √ Theorem 9.3. Let c = p∗ (pf ) with pf ∈ [a0 − K1 /4, a0 + K1 /4] and there are 0 , δ0 > 0 such that for N = H0 + Z + R with 0 <  < 0 and δ < δ0 , then (N , c) is of Aubry-Mather type with the hyperbolic cylinder given by the embedding χ(θf , pf ) = (Θs , P s )(θf , pf , 0), where Θs , P s are from Theorem 9.1.

AUBRY-MATHER TYPE AT THE SINGLE RESONANCE

95

We prove Theorem 9.3 assuming Theorems 9.1 and 9.2. Proof of Theorem 9.3. Let us denote C the cylinder in Theorem 9.1 and C 0 = C ∩ {t = 0}. Let δ0 , 0 be small enough such that 9.2 applies for 0 <  < 0 and 0 < δ < δ0 . In particular, these statements hold on a open set N and c, as required by Definition 8.1. The embedding as described is a weakly normally hyperbolic invariant cylinder. Moreover, Corollary 7.7 implies that √ for any two tangent vectors v, v 0 ∈ Tz C 0 , we have |dΘs ∧ dP s (v, v 0 )| ≤ C δ|dθf ∧ dpf (v, v 0 )|, and therefore for δ small enough √ (dΘs ∧ dP s + dθf ∧ dpf )(v, v 0 ) ≥ (1 − C δ) ≥ 1 |dθf ∧ dpf (v, v 0 )|. 2 Since the form dθf ∧ dpf is non-degenerate on C 0 , the symplectic form dθs ∧ dpf + dθf ∧ dpf is non-degenerate when restricted to C 0 . eN (c) is conTheorem 9.2 implies that for δ small enough, the Ma˜ n´e set N  tained in the neighborhood V described in Theorem 9.1. Since the Ma˜ n´e set is invariant, it must be contained in the maximally invariant cylinder C. Therefore e 0 (c) ⊂ C 0 . Ae0N (c) ⊂ N N We now show that Ae0N is a Lipschitz graph over θf . Let (θ1 , p1 ), (θ2 , p2 ) ∈ Ae0N (c), then by Theorem 9.2 we have  √ √  kp2 − p1 k ≤ 18D kθ2 − θ1 k ≤ 18D  kθ2s − θ1s k + kθ2f − θ1f k . By Theorem 9.1, kθ2s − θ1s k ≤ C(1 +

p

  δ/) kθ2f − θ1f k + kp2 − p1 k .

(9.2)

Combining everything, we get for some constant C1 depending on n, D,  √  √ √ √ 1 − C1 (  + δ) kp2 − p1 k ≤ C1 (  + δ)kθ2f − θ1f k, which implies kp2 − p1 k ≤ kθ2f − θ1f k if δ,  are small enough depending only on C1 . Combining with (9.2), we get k(θ2 , p2 ) − (θ1 , p1 )k ≤ 2kθ2f − θ1f k if , δ are small enough. Finally, let u be a weak KAM solution for N at cohomology c, and assume that Ae0N (c) projects onto the θf component. Since the strong unstable manifold depends C 1 on the base point, the unstable manifold W u (Ae0H (c)) is a Lipschitz manifold. Moreover, since the strong unstable direction projects onto the θs direction, W u (Ae0H (c)) is locally a graph over θ = (θf , θs ). This verifies 2b of Definition 8.1. We have verified all conditions in Definition 8.1 and therefore N , c is of Aubry-Mather type.

96 9.3

CHAPTER 9

BIFURCATIONS IN THE DOUBLE MAXIMA CASE

In this section we assume that for the Hamiltonian N = H0 + Z + R, where Z satisfies the condition [SR2λ ], namely for all p ∈ Bλ (p0 )∩Γ, there exist two local maxima θ1s (p) and θ2s (p) of the function Z(., p) in Tn−1 satisfying ∂θ2s Z(θ1s (p), p) < λI

,

∂θ2s Z(θ2s (p), p) < λI,


2 Z(θs , p) < max{Z(θ1f (p), p), Z(θ2f (p), p)} − λ min{d(θs − θ1s ), d(θs − θ2s )} . Let f : Tn−1 → R be a bump function satisfying the following conditions: f |Bλ (θ1s (p0 )) = 0,

f |Bλ (θ2s (p0 )) = 1,

and 0 ≤ f ≤ 1 otherwise. Define Z1 = Z − f,

Z2 = Z − (1 − f );

then Z1 (·, p0 ) has a unique maximum at θ1s (p0 ), while Z2 (·, p0 ) has a unique maximum at θ2s (p0 ). There is κ > 0 depending only on λ such that the same holds for p ∈ Bκ (p0 ). To the Hamiltonian Ni = N + Zi + R,

i = 1, 2

we may apply Theorems 9.1 and 9.2 to obtain the existence of the NHIC Ci which contains the local Aubry sets Ai (c) for c ∈ Bκ/2 (p0 ). Moreover, we may define the local alpha functions αi (c) = αNi (c) similar to Section 8.4. The cohomology c ∈ Γ ∩ Bκ/2 (p0 ) is of bifurcation type if and only if α1 (c) = α2 (c), and is of Aubry-Mather type if and only if α1 (c) 6= α2 (c). Moreover, in this case we have the following analog of Theorem 9.2. Theorem 9.4 (See Section 9.6, see also Theorem 4.5 of [13]). Suppose Z satisfies condition [SR2λ ] at p = c. Then there exists δ0 , 0 , κ > 0 depending only on λ, n, D such that if 0 <  < 0 and 0 < δ < δ0 , for every c ∈ Bκ/2 (p0 ) ∩ Γ the Aubry set satisfies
√ e ⊂ B(θ1s , δ 1/5 ) ∪ B(θ2s , δ 1/5 ) × T × B(c, δ 1/16 ) × T A(c) ⊂ Tn−1 × T × Rn × T. If, moreover, the projection θs (A0 (c)) ⊂ Tn−1 is contained in one of the (disjoint) balls B(θis , δ 1/5 ), then the projection θs (N 0 (c)) ⊂ Tn−1 of the Ma˜ n´e set is contained in the same ball B(θis , δ 1/5 ). Theorem 9.5. Suppose Z satisfies condition [SR2λ ] at p = c∗ . Then there

97

AUBRY-MATHER TYPE AT THE SINGLE RESONANCE

exists 0 , δ0 > 0, such that if 0 <  < 0 , 0 < δ < δ0 , and kRkCI2 < δ, N = H0 + Z + R is of bifurcation Aubry-Mather type. Proof. Let V1 = Bλ (θ1s (p0 ))×T ⊂ Tn−1 ×T, and V2 = Bλ (θ2s (p0 ))×T, and define the local Hamiltonians N1 and N2 as in Definition 8.3. Theorem 9.4 implies the Aubry set AeN (c) = AeN1 (c) ∩ AeN2 (c). We check that all the conditions of Definition 8.3 are satisfied.

9.4

HYPERBOLIC COORDINATES

The Hamiltonian flow admits the following equation of motion:   θ˙s = ∂ps H0 + ∂ps Z + ∂ps R    s   p˙ = −∂θs Z − ∂θs R θ˙f = ∂pf H0 + ∂pf Z + ∂pf R .    p˙f = −∂θf R    t˙ = 1

(9.3)

The system (9.3) is a perturbation of θ˙s = ∂ps H0 ,

p˙s = −∂θs Z,

θ˙f = ∂pf H0 ,

p˙f = 0,

t˙ = 1,

which admits a normally hyperbolic invariant cylinder {(θ∗s (pf ), θf , ps∗ (pf ), pf , t) :

θf , t ∈ T, }.

Set B(pf ) := ∂p2s ps H0 (p∗ (pf )),

A(pf ) := −∂θ2s θs Z(θ∗s (pf ), p∗ (pf ));

then as in [13], there is a positive definite matrix T (pf ) such that Λ(pf ) = T (pf )A(pf )T (pf ) = T −1 (pf )B(pf )T −1 (pf ). We lift the equation to the universal cover, and consider the change of variable x = T −1 (pf )(θs − θ∗s (pf )) + −1/2 T (pf )(ps − ps∗ (pf )) y = T −1 (pf )(θs − θ∗s (pf )) − −1/2 T (pf )(ps − ps∗ (pf )), I = −1/2 (pf − a0 ),

(9.4)

Θ = γθf ,

where 0 < γ < 1 is a parameter to be determined. Lemma 9.6 (Lemmas 3.1 and 3.2 of [13]). For each pf ∈ [a− , a+ ], we have

98 Λ(pf ) ≥

CHAPTER 9

p

λ/DI. We also have the following estimates: kT kC 2 , kT −1 kC 2 , k∂pf θ∗s kC 0 , kps∗ kC 2 = O(1), √ kθs − θ∗s kC 0 = O(ρ), kps − ps∗ kC 0 = O( ρ),

where ρ = max{kxk, kyk}. Note we are using the regular C 2 norm in Lemma 9.6 since T depends only on H0 and Z which are bounded in the regular norm. Lemma 9.7. The equation of motion in the new variables takes the form √ √ √ x˙ = Λ(a0 + I)x +  O(δ + ρ2 ), √ √ √ y˙ = − Λ(a0 + I)y +  O(δ + ρ2 ), √ and I˙ = O( δ). Proof. The last equation is straightforward. We only prove the equation for x˙ as the calculations for y˙ are the same. In the original coordinates, we have √ θ˙s = B(pf )(ps − ps∗ (pf )) + O(kps − ps∗ (pf )k2 ) + O(δ ), p˙s = A(pf )(θs − θ∗s (pf )) + O(kθs − θ∗s (pf )k2 ) + O(δ), 1

1

where we used k∂p Rk = − 2 k∂I Rk = O(− 2 δ). Differentiating (9.4), using p˙f = O(δ) and Lemma 9.6, we get √ 1 x˙ = T −1 θ˙s + − 2 T p˙s + O( δ) √ √ 1 = T −1 BT −1 · − 2 T (ps − ps∗ (pf )) + T AT · T −1 (θs − θ∗s ) √ √ + O( δ + ρ2 ) √ √ = Λ(pf )x +  O(δ + ρ2 ). √ Lemma 9.8. Suppose  ≤ δ; linearized system is given by √ Λ 0 √  0 − Λ  0 0 L=   0 0 0 0 where ρ = max{kxk, kyk}.

then in the new coordinates (x, y, Θ, I, t) the 0 0 0 0 0

0 0 0 0 0

 0 0  √ −1 0  + O(δγ + ρ + γ),  0 0

99

AUBRY-MATHER TYPE AT THE SINGLE RESONANCE

Proof. In the original coordinates, the linearized equation is given by the matrix   √ √ O( δ) B + O(δ) O( δ) ∂p2f ps H0 + O(δ) 0 √ √ A + O(ρ) O( δ) 0 O( δ) 0   √ √ ˜  L =  O( δ) O(1) O( δ) O(1) 0  + O(δ). √ √  0 O( δ) 0 O( δ) 0 0 0 0 0 0 The coordinate change matrix is  √ T /2    T −1 /2 s s f f  ∂(θ , p , θ , p , t) 0 =  ∂(x, y, Θ, I, t)  0 0

√T /2 − T −1 /2 0 0 0

0 0 γ −1 0 0

√

 √ O( ) 0 ∂pf ps∗ + O(ρ) 0  0 . √0  0 0 1

The product is   s s f f ˜ ∂(θ , p , θ , p , t) = O(δγ −1 + ρ) + L ∂(x, y, Θ, I, t) √  √ √ √ √ √ BT −1 /2 + O( δ) − BT −1 /2 + O( δ) O( δγ −1 ) O( δ) 0  AT AT 0√ 0   √/2 √/2 √ 0 −1  O( ) O( ) O( δγ ) O( ) 0  .  0 0 0 0 0 0 0 0 0 0 Most of the computations are straightforward, with the exception of the fourth row, first column, which contains the following cancellation:
∂p2s pf H0 ∂pf ps∗ + ∂p2f pf H0 = ∂pf ∂pf H0 (p∗ (pf )) = 0, since ∂pf H0 (p∗ (pf )) = 0 for every pf by definition. The differential of the inverse coordinate change is  −1  T −1/2 T 0 O(−1/2 λ−1/4 ) 0   T −1 −−1/2 T 0 O(−1/2 λ−1/4 ) 0   ∂(x, y, Θ, I, t) = 0 γ 0 0  0 . ∂(θs , ps , θf , pf , t)  0 0 0 −1/2 0 0 0 0 0 1

100

CHAPTER 9

Finally, the new matrix of the linearized equation is     s s f f ∂(x, y, Θ, I, t) ˜ ∂(θ , p , θ , p , t) L= L ∂(θs , ps , θf , pf , t) ∂(x, y, Θ, I, t)  √ Λ 0 0 0 √  0 − Λ 0 0  √ √ √ √ O( γ) O( γ) O( γ) 0 = O( (δγ −1 + ρ)) +    0 0 0 0 0 0 0 0

9.5

 0 0  0 . 0 0

NORMALLY HYPERBOLIC INVARIANT CYLINDER

We state the following abstract statement for existence of normally hyperbolic invariant manifolds, given in [13]. Let F : Rn → Rn be a C 1 vector field. We split the space Rn as Rnu × Rns × nc R , and denote by z = (u, s, c) the points of Rn . We denote by (Fu , Fs , Fc ) the components of F : F (x) = (Fu (z), Fs (z), Fc (z)). We study the flow of F in the domain Ω = B u × B s × Ωc , where B u and B s are the open Euclidean balls of radius ru and rs in Rnu and Rns , and Ωc = Ωc1 × Rc2 is a convex open subset of Rnc . We denote by   Luu (z) Lus (z) Luc (z) L(z) = dF (z) =  Lsu (z) Lss (z) Lsc (z)  Lcu (z) Lcs (z) Lcc (z) the linearized vector field at point z. We assume that kL(z)k is bounded on Ω, which implies that each trajectory of F is defined until it leaves Ω. We denote by W c (F, ω) the union of all full orbits contained in Ω, W sc (F, Ω) the set of points whose positive orbit remains inside Ω, and by W uc (F, Ω) the set of points whose negative orbit remains inside Ω. Let us further consider a positive parameter b > 0, and consider the set Ωcb2 = Bb (Ωc2 ) and Ωcb = Ωc1 × Ωcb2 . Proposition 9.9 (Proposition A.6 of [13]). Let F : Rnu × Rns × Ωcb → Rnu × Rns × Rnc be a C 2 vector field. Assume that there exist α, m, σ > 0 such that ¯s × Ω ¯ c. • Fu (u, s, c) · u > 0 on ∂B u × B b u s ¯ ¯ • Fs (u, s, c) · s < 0 on B × ∂B × Ωcb .

101

AUBRY-MATHER TYPE AT THE SINGLE RESONANCE

• Luu (z) > αI, Lss (z) 6 −αI for each x ∈ Ωb in the sense of quadratic forms. • kLus k + kLuc k + kLss k + kLsc k + kLcu k + kLcs k + kLcc k 6 m evaluated at each z ∈ Ωb . • kLus k + kLuc k + kLss k + kLsc k + kLcu k + kLcs k + kLcc k + 2kFc2 k/b 6 m evaluated at each z ∈ Ωb − Ω. Assume furthermore that K :=

m 1 6 , α − 2m 8

then there exist C 2 maps wsc : B s × Ωcb → B u ,

wuc : B u × Ωcb → B s ,

wc : Ωcb → B u × B s

satisfying the estimates kdwsc k 6 K,

kdwuc k 6 K,

kdwc k 6 2K,

the graphs of which respectively contain W sc (F, Ω), W uc (F, Ω), W c (F, Ω). Moreover, the graphs of the restrictions of wsc , wuc , and wc to, respectively, B s × Ωc , B u × Ωc , and Ωc , are tangent to the flow. There exists an invariant C 1 foliation of the graph of wuc whose leaves are graphs of K-Lipschitz maps above B u . The set W uc (F, Ω) is a union of leaves: it has the structure of an invariant C 1 lamination. Two points x, x0 belong to the same leaf of this lamination if and only if d(x(t), x0 (t))etα/4 is bounded on R− . If in addition there exists a group G of translations of Rnc1 such that F ◦ (id ⊗ id ⊗ g ⊗ id) = F for each g ∈ G, then the maps w∗ can be chosen such that wsc ◦ (id ⊗ g ⊗ id) = wsc ,

wuc ◦ (id ⊗ g ⊗ id) = wuc ,

wc ◦ (g ⊗ id) = wc (9.5)

for each g ∈ G. The lamination is also translation invariant. Proof of Theorem 9.1. Consider the equation of N in the (x, y, θf , pf , t) coordinates, and set B u = {kxk < ρ}, B s = {kyk < ρ},pΩc2 = {kI f k ≤ K1 /2}, Ωc1 = {(θf , t)} with the group translation Z2 , α = λ/4D, b =√ 1 < K1 /2. Note that our choice of b implies Ωcb2 ⊂ {kI f k < K1 }. We fix γ = δ According to Lemmas 9.6 and 9.7, we have for kxk = ρ,
√ √ √ x˙ · x ≥ αρ2 − ρ O(δ + ρ2 ) = ρ ρ − O(δ + ρ2 ) > 0 as long as ρ ≥ Cδ for C large enough. This verifies the first bullet point assumption. The second assumption is verified in the same way.

102

CHAPTER 9

By Lemma 9.8, we have for C1 > 1 depending on λ, D, n, kLus (z)k + kLuc (z)k + kLss (z)k + kLsc (z)k + kLcu (z)k + kLcs (z)k + kLcc (z)k √ √ √ = O(δγ −1 + ρ + γ) ≤ C1 ( δ + ρ) =: m/2, and Luu ≥ (2α − m)I ≥ αI as long as m < α. This is possible as √ long as −1 , δ,√ρ < √ C1 < α. Finally, we check that if z ∈ Ωb \ Ω, kFc2 (z)k = O( δ) ≤ C1 ( δ + ρ)/2 = m/4 if C1 is chosen large enough. These estimates ensure all the bullet point conditions are satisfied. By choosing C1 even smaller, we can ensure m < α/10 which implies K = m/(α − 2m) ≤ 1/8. To summarize, we have shown all conditions of Proposition 9.9 are satisfied if 0 <  < C1−1 , 0 < δ < C1−1 , Cδ ≤ ρ ≤ C1−1 . We apply the Proposition twice, once for ρ = C1−1 and once for ρ = Cδ. The first application shows the invariant cylinder is the maximal invariant set in the set Ωc = {kxk, kyk ≤ C1−1 }. The second application shows that the cylinder is in fact contained in the set {kxk, kyk we get the estimate √ ≤ Cδ}. Moreover, in the second application, √ m = O( δ), which allows the estimate K = O( δ). We now return to the original coordinates using the formula 1 Θs (θf , pf , t) = θ∗s (pf ) + T (pf ) · (wuc + wsc )(γθf , a0 + −1/2 pf , t) 2√  −1 f s f f s f P (θ , p , t) = p∗ (p ) + T (p ) · (wuc − wsc )(γθf , a0 + −1/2 pf , t). 2

(9.6)

All the estimates stated in Theorem 9.1 follow directly from √ these expressions, and from the fact that {kxk, kyk ≤ Cδ}, kdwc k 6 2K = O( δ).

9.6

˜E ´ SETS LOCALIZATION OF THE AUBRY AND MAN

Let D1 = 2D, the Hamiltonian N satisfies the following estimates if δ is small enough depending only on D: √ 2 2 2 D1−1 I ≤ ∂pp N ≤ DI, k∂θp N k ≤ 2 , k∂θθ N k ≤ 3. Let L(θ, p, t) be the Lagrangian associated to N , and L0 the Lagrangian of H0 . Lemma 9.10. The following estimates hold for the Lagrangian L. 1. (Lemma 4.1 of [13]) For K > 0, the image of the set Tn × BK √ × T under the diffeomorphism ∂p N contains the set Tn × BK1 √ (c) × T, where K1 = K/(4D1 ). 2. (Lemma 4.2 of [13]) The estimate √ 2 2 k∂θv LkC 0 ≤ 2D1 , k∂θθ LkC 0 ≤ 3

AUBRY-MATHER TYPE AT THE SINGLE RESONANCE

103

holds on Tn × BK1 √ (c) × T. 3. (Lemma 4.3 of [13]) For v ∈ BK1 √ (c), we have |L(θ, v, t) − (L0 (v) − Z(θs , c)) | ≤ 2δ. 4. (Lemma 4.4 of [13]) The alpha function αN (c) satisfies |αN (c) − (H0 (c) +  max Z(·, c))| ≤ 2δ. 5. (Lemma 4.5 of [13]) There is C > 1 depending only on D such that if  < C −1 δ, we have the estimates L(θ, v, t) − c · v + α(c) > kv − ∂H0 (c)k2 /(4D1 ) − Zˆc (θs ) − 4δ L(θ, v, t) − c · v + α(c) 6 D1 kv − ∂H0 (c)k2 − Zˆc (θs ) + 4δ

(9.7) (9.8)

for each (θ, v, t) ∈ Tn × Rn × R, where Zˆc (θs ) := Z(θs , c) − maxθs Z(θs , c). √ 2 Proof. We remark that in [13] it is assumed that k∂θp N k ≤ 2 instead of 2  as we assumed. However the same calculations as given in the cited lemmas prove Lemma 9.10, once appropriate changes are made. Proposition 9.11. For each c ∈ Rn , the weak KAM solution u of cohomology √ c is 3 D1 /2-semi-concave. Proof of Theorem 9.2. The proofs of Theorem 4.1 and 4.2 in [13] rely on the estimates 3, 4, and 5 in Lemma 9.10, as well as Proposition 9.11. We have arrived at the same estimates using a weaker assumption, the only change is that we replaced the constant D with D1 = 2D. The rest of the proofs are exactly identical. Proof of Theorem 9.4. This is similar to Theorem 9.2. The proof of Theorem 4.5, [13] applies once we take into account Lemma 9.10 and Proposition 9.11.

9.7

GENERICITY OF THE SINGLE-RESONANCE CONDITIONS

We prove Proposition 3.2 in this section, which is a direct consequence of the Lemma 9.12. We say a local maximum x0 of a function g(x) is non-degenerate 2 if ∂xx g(x0 ) is strictly negative definite. Lemma 9.12. Suppose r ≥ 4. Let a1 < a2 . Let R be the set of f = f (x, a) ∈ C r (T × [a1 , a2 ]) such that f (·, a) has unique non-degenerate global maximum for all but finitely many a’s, and that at those exceptional points there are two non-degenerate maxima. Then R is open and dense in C r (T × [a1 , a2 ]).

104

CHAPTER 9

Proof. Letting [a1 , a2 ] ⊂ (b1 , b2 ), we consider the subspace D of the jet space J 3 (T × (b1 , b2 )), defined by ∂x f (x, a) = ∂xx f (x, a) = ∂xxx f (x, a) = 0, and let D1 be the restriction of D to base points in T × [a1 , a2 ]. By Thom’s transverality theorem (see for example Theorem 2.8 of [48]), if r ≥ 4, the set R1 ⊂ C r (T × (b1 , b2 )) such that the jet map f 7→ J 3 f is transversal to D at D1 is open and dense. Moreover, since D has co-dimension 3 and T × (b1 , b2 ) has dimension 2, for each g ∈ R1 , J 3 g is disjoint from D1 , i.e., for all a ∈ [a1 , a2 ] such that ∂x f (x, a) = ∂xx f (x, a) = 0, we have ∂xxx f (x, a) 6= 0. It follows that any degenerate critical point cannot be a local maximum. We have proven that if g ∈ R1 , then g(·, a) has no degenerate local maxima for all a ∈ [a1 , a2 ]. Claim 1: Suppose g ∈ R1 . Let a0 ∈ [a1 , a2 ]. Suppose x1 , . . . , xk are the maxima of g(·, a0 ). By the implicit function theorem, there exist , δ > 0, such that for each g1 ∈ Vδ (g) (the neighborhood is taken in C r (T × (b1 , b2 ))), there exist smooth maps xgj 1 : [a0 − , a0 + ] → T, j = 1, . . . , k, such that xgj 1 is continuous in g1 in C 3 topology, with xgj (a0 ) = xj . xgj 1 (a) are the local maxima of g1 (·, a), and max g1 (x, a) = max g1 (xgj 1 (a), a), x

1≤j≤k

∀a ∈ [a0 − , a0 + ].

(9.9)

To prove Claim 1, note that there are at most finitely many global maxima of g(·, a0 ), since a maximum is isolated (since it’s non-degenerate) and cannot accumulate (the limit point is an non-isolated maximum). Since a limit point (in both a and in g) of maxima is a maximum, this implies for , δ sufficiently small, a ∈ (a0 − 2, a0 + 2), and g1 ∈ Vδ (g), any maximum of g1 (·, a) must be in a neighborhood of {xj : 1 ≤ j ≤ k}. Since xgj 1 is the unique local maximum in a neighborhood (again, due to non-degeneracy), we conclude that any maximum of g1 (·, a) must be taken among xgj 1 , 1 ≤ j ≤ k. Claim 2: Let g, a0 , , δ be as in Claim 1. The following property holds for g1 in an open and dense subset of Vδ : for each a ∈ [a0 − , a0 + ], g1 (·, a) has at most two maxima, each maximum is non-degenerate, and at each a for which there are two maxima, say xgi 1 (a) and xgj 1 (a), we have d d g1 (xgi 1 (a), a) 6= g1 (xgj 1 (a), a). da da As a result, there are at most finitely many points with two maxima. Before proving Claim 2, we note that our lemma follows from it by covering [a1 , a2 ] with finitely many intervals on which Claim 2 holds. To prove Claim 2, let Uj , 1 ≤ j ≤ k, be disjoint open sets, xgj 1 ∈ Uj for all g1 ∈ Vδ (g), ρj be compactly supported C ∞ functions such that ρj |Uj = 1, and that supp ρj are mutually disjoint. Fix a g1 ∈ Vδ (g), β = (β1 , . . . , βk ),

105

AUBRY-MATHER TYPE AT THE SINGLE RESONANCE

γ = (γ1 , . . . , γk ), define G(x, a, β, γ) = g1 (x, a) +

k X

βj ρ j +

j=1

k X

γj aρj ,

j=1

and for σ > 0 small enough, (the evaluation map) eG : [a0 − , a0 + ] × Bσ (0) × Bσ (0) → R2k , G(·,·,β,γ)

(eG(a, β, γ))2j−1,2j = (G(xj

, a, β, γ),

d G(·,·,β,γ) G(xj , a, β, γ)), da

1 ≤ j ≤ k. d Note that (eG(a, β))2j−1,2j = (g1 (xgj 1 (a), a)+βj , da g1 (xgj 1 (a), a)+γj ), and hence the map is transversal to R2k . Define

∆2,2 = {(yj , vj )kj=1 ∈ R2k : ∃j1 < j2 such that yj1 = yj2 , vj1 = vj2 }, ∆3 = {(yj , vj )kj=1 ∈ R2k : ∃j1 < j2 < j3 such that yj1 = yj2 = yj3 }. Since ∆2,2 and ∆3 are both co-dimension 2, Sard’s theorem implies that for a generic β, the image of eG(·, β) has no intersections with either set. This observation proves the density part of Claim 2. The condition is clearly open. This proves Claim 2 and our lemma.

Chapter Ten Normally hyperbolic cylinders at double resonance We prove Theorem 4.4 in this chapter. There are two cases: 0 1. (Simple critical homology) In this case we show the homoclinic orbit η±h can be extended to periodic orbits both in positive and negative energy. The union of these periodic orbits forms a C 1 normally hyperbolic invariant manifold (which is homotopic to a cylinder with a puncture). 2. (Non-simple homology) In this case we show that for positive energy, there exist periodic orbits shadowing ηh0 1 and ηh0 2 in a particular order.

Our strategy is to prove the existence of these periodic orbits as hyperbolic fixed points of composition of local and global maps. A main technical tool to prove the existence and uniqueness of these fixed points is the Conley-McGehee isolation block ([68]). The plan of this section is as follows. In Section 10.1 we state a standard normal form near the hyperbolic fixed point. In Section 10.2 we study the property of the Shil’nikov boundary value problem. In Section 10.3, we apply results of Section 10.2 to establish strong hyperbolicity of the local map Φ∗loc as well as the existence of unstable cones. Since the global maps Φ∗glob are Poincar´e maps of bounded transition time, they have bounded norms and the linearizations of the proper compositions Φ∗glob Φ∗loc are dominated by the local component. In Section 10.4, we define isolating blocks of Conley-McGehee [68] and use it to construct isolating blocks for the proper compositions of Φ∗glob Φ∗loc , assuming conditions [DR1c ] − [DR4c ]. We then apply the same analysis to Φ∗glob Φ∗loc · · · Φ∗glob Φ∗loc and construct families of shadowing orbits in the nonsimple case. In Section 10.5 we complete the proof of Theorem 4.4 by showing that periodic orbits constructed in the previous two sections form a normally hyperbolic invariant cylinder. Moreover, they coincide with the shortest geodesics for the Jacobi metric. In Section 10.6 we prove Lemma 4.2 which describes, in the non-simple case, the order at which the simple periodic orbits are shadowed.

NORMALLY HYPERBOLIC CYLINDERS AT DOUBLE RESONANCE

10.1

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NORMAL FORM NEAR THE HYPERBOLIC FIXED POINT

We describe a normal form near the hyperbolic fixed point (assumed to be (0, 0)) of the slow Hamiltonian H s : T2 × R2 → R. For the rest of this section, we drop the superscript s to abbreviate notations. In a neighborhood of the origin, there exists a symplectic linear change of coordinates under which the system has the normal form H(u1 , u2 , s1 , s2 ) = λ1 s1 u1 + λ2 s2 u2 + O3 (s, u). Here s = (s1 , s2 ), u = (u1 , u2 ), and On (s, u) stands for a function bounded by Ck(s, u)kn . By taking a standard straightening coordinate change, we get: Lemma 10.1. After an C r−1 symplectic coordinate change Φ, the Hamiltonian takes the form X N = H ◦ Φ = λ1 s1 u1 + λ2 s2 u2 + si uj O1 (s, u), i,j=1,2

and the equation is     s˙ −Λs + sO1 (s, u) = , u˙ Λu + uO1 (s, u)

(10.1)

where Λ = diag{λ1 , λ2 }. Proof. Since (0, 0) is a hyperbolic fixed point, for sufficiently small r > 0, there exist stable manifold W s = {(u = U (s), |s| ≤ r} and unstable manifold W u = {s = S(u), |u| ≤ r} containing the origin. All points on W s converge to (0, 0) exponentially in forward time, while all points on W u converge to (0, 0) exponentially in backward time. In a Hamiltonian system, the unstable/stable manifolds of equilibria is Lagrangian. Indeed, if z ∈ W u and v1 , v2 ∈ Tz W u , −t then ω(v1 , v2 ) = ω(Dφ−t H v1 , DφH v2 ) → 0 as t → ∞. We now claim that the coordinate change (s, u) 7→ (s+ , u+ ) = (s, u − U (s)) is symplectic. Indeed u = U (s) is an Lagrangian graph, and hence U (s) = ∇G(s) for a smooth function G. Then d(s+ ) ∧ d(u+ ) = ds ∧ du + (D2 G(s)ds) ∧ ds = ds ∧ du. Similarly, (s+ , u+ ) 7→ (s0 , u0 ) = (s+ − S(u+ ), u+ ) is also symplectic. Under the new coordinates, W s = {u0 = 0} and W u = {s0 = 0}. We abuse notation and keep using (s, u) to denote the new coordinate system. Under the new coordinate system, the Hamiltonian has the form H(s, u) = λ1 s1 u1 + λ2 s2 u2 + H1 (s, u), where H(s, u) = O3 (s, u) and H1 (s, u)|s=0 = H1 (s, u)|u=0 = 0. The Hamiltonian and the vector field take the desired form under these coordinates.

108 10.2

CHAPTER 10

SHIL’NIKOV’S BOUNDARY VALUE PROBLEM

The local maps Φloc are Poincar´e maps from a section Σs± transversal to s1 axis, to Σu± transversal to the u1 axis. The main tool for their study is the Shil’nikov boundary value problem. Proposition 10.2 (Shil’nikov, Lemmas 2.1, 2.2, 2.3 of [73]). There exist C > in 0, δ0 > 0 and T0 > 0 such that for each 0 < δ < δ0 , any sin = (sin 1 , s2 ), out out out u = (u1 , u2 ) with k(s, u)k ≤ δ, and any large T > T0 , there exists a unique solution (sT , uT ) : [0, T ] → BCδ of the system (10.1) with the property sT (0) = sin and uT (T ) = uout . Moreover, we have ksT (t)k ≤ Cδe−λ1 t/2 , and

T T

∂s (t) ∂u (t) −λ1 t/2



,

∂sin + ∂sin ≤ Ce

kuT (t)k ≤ Cδe−λ1 (T −t)/2 ,

T T

∂s (t) ∂u (t) −λ1 (T −t)/2



.

∂uout + ∂uout ≤ Ce

Corollary 10.3. Let (sT , uT ) be the solution in Proposition 10.2 with fixed boundary values sin , uout . Then



d T



(s (t), uT (t)) ≤ Ce−λ1 (T −t)/2 , d (sT (T − t), uT (T − t)) ≤ Ce−λ1 t/2 ,

dT

dT

(10.2) and d (sτ (T ), uτ (T )) = −XN (sT (T ), uT (T )) + O(e−λ1 T /2 ), dτ τ =T (10.3) d (sτ (T − τ ), uτ (T − τ )) = XN (sT (0), uT (0)) + O(e−λ1 T /2 ), dτ τ =T where XN denotes the Hamiltonian vector field of N . Proof. Consider two solutions (sT1 , uT1 ) and (sT2 , uT2 ) of the boundary value problem, we extend the definition of the solutions to R by solving the ODE. We note that (sT1 , uT1 ) : [0, T2 ] → R4 satisfies the boundary condition sin and uT1 (T2 ) on [0, T2 ], and by Corollary 10.3,

T T

(s 1 , u 1 )(t) − (sT2 , uT2 )(t) ≤ Ceλ1 (T2 −t)/2 |uT1 (T2 ) − uout |. Note that lim

T2 →T1

uT1 (T2 ) − uout uT1 (T2 ) − uT1 (T1 ) d = lim = uT1 (t), T2 →T1 T2 − T1 T2 − T1 dt t=T1

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NORMALLY HYPERBOLIC CYLINDERS AT DOUBLE RESONANCE

and as a result,



d

T T λ1 (T1 −t)



.

dT T =T1 (s , u )(t) ≤ Ce This proves the first half of (10.2) while the second half is similar. For (10.3), note d d d (sτ (T ), uτ (T )) = (sτ (τ ), uτ (τ )) − (sT (τ ), uT (τ )), dτ τ =T dτ τ =T dτ τ =T the first half of (10.3) follows. The second half is similar. Let (vs1 , vs2 , vu1 , vu2 ) denote the coordinates for the tangent space induced by (s1 , s2 , u1 , u2 ). For K > 0 and x ∈ Br , we define the strong unstable cone by u CK (x) = {K 2 |vu2 |2 > |vu1 |2 + |vs1 |2 + |vs2 |2 }

(10.4)

and the strong stable cone to be s CK = {K 2 |vs2 |2 > |vs1 |2 + |vu1 |2 + |vu2 |2 }.

The following statement follows from the hyperbolicity of the fixed point O via standard techniques. Lemma 10.4 (See for example [49]). For any 0 < κ < λ2 − λ1 , there exists δ = δ(κ, K) and C = C(κ, K) > 1 such that the following hold: • If φtN (x) ∈ B4δ (O) for 0 ≤ t ≤ t0 (φtN is the Hamiltonian flow of the normal u u form Hamiltonian N ), then DφtN (CK (x)) ⊂ CK (φtN (x)) for all 0 ≤ t ≤ t0 . u Furthermore, for any v ∈ CK (x), kDφtN (x)vk ≥ C −1 e(λ2 −κ)t ,

0 ≤ t ≤ t0 .

−t −t s s • If φ−t N (x) ∈ B4δ (O) for 0 ≤ t ≤ t0 , then DφN (CK (x)) ⊂ CK (φN (x)) for all s 0 ≤ t ≤ t0 . Furthermore, for any v ∈ CK (x), −1 (λ2 −κ)t kDφ−t e , N (x)vk ≥ C

0 ≤ t ≤ t0 .

Lemma 10.5. For any κ > 0 there is δ0 > 0 such that for any 0 < δ ≤ δ0 , out in out ksin 1 k = ku1 k = δ, and ks2 k, ku2 k ≤ κλ1 δ/(2λ2 ), we have





d T





u2 (0) ≤ κ d uT1 (0) , d sT2 (T ) ≤ κ d sT1 (T ) .

dT

dT

dT

dT

Moreover, by integrating in T , we get kuT2 (0)k ≤ κkuT1 (0)k,

ksT2 (T )k ≤ κksT1 (T )k.

Proof. Given κ > 0 we can choose δ0 small enough such that the backward flow

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Dφ−t preserves the cone u c (CK ) = {|vu2 | ≤ K −1 k(vs1 , vs2 , vu1 )k},

where K = κ−1 . Note that d T d (s (0), uT (0)) = Dφ−T (sτ (T ), uτ (T )). dT dτ τ =T We have d (sτ (T ), uτ (T )) = −XN (sT (T ), uT (T )) + O(e−λ1 T /2 ) dτ τ =T = (λ1 sT1 (T ), λ2 sT2 (T ), λ1 uT1 (T ), λ2 uT2 (T )) + O(δ 2 ) + O(e−λ1 T /2 ) 2 −λ1 T /2 u c = (0, 0, λ1 δ, λ2 uout ) ∈ (CK ) , 2 ) + O(δ ) + O(e

if |uout 2 | ≤ κλ1 δ/(2λ2 ), δ is small and T is large enough. By the invariance of d d T u c unstable cones, dT (sT (0), uT (0)) ∈ (CK ) . Keep in mind that dT s (0) = 0, the first half of our estimate follows. The second half is proven in a symmetric way.

10.3

PROPERTIES OF THE LOCAL MAPS

Consider first the simple critical case, where we have a unique shortest loop γh+ for the Jacobi matrix. The corresponding Hamiltonian orbit ηh+ is a nondegenerate homoclinic orbit. We drop the subscript h and call η − the timereversal of η + . Σs+ = {(δ, s2 , u1 , u1 ) : |s2 |, |u1 |, |u2 | ≤ δ}, Σu+ = {(s1 , s2 , δ, u1 ) : |s2 |, |u1 |, |u2 | ≤ δ}.

(10.5)

+ Denote q + = η + ∩ {s1 = δ} = {(σ, s+ = η + ∩ {u1 = δ} = 2 , 0, 0)}, and p + {(0, 0, δ, u2 )}. Let κ > 0 be a parameter to be defined in Proposition 10.7. Since + η + is tangent to the s1 , u1 axes, we can choose δ > 0 such that |s+ 2 |, |u2 | < κδ, + + s u therefore q , p are contained Σ+ and Σ+ respectively. Define

ls = {(δ, s2 , 0, 0) : −κδ ≤ s2 ≤ κδ} ⊂ W s (O) ∩ Σs+ , lu = {(0, 0, δ, u2 ) : −κδ ≤ u2 ≤ κδ} ⊂ W u (O) ∩ Σu+ . Let s Σs,E + = Σ+ ∩ {N (s, u) = E},

u Σu,E + = Σ+ ∩ {N (s, u) = E}

be the restriction of the sections to an energy E close to 0. We would like to

NORMALLY HYPERBOLIC CYLINDERS AT DOUBLE RESONANCE

111

Figure 10.1: Rectangles mapped under Φ++ loc s,E u,E study the domain of the restricted local map Φ++ loc : Σ+ → Σ+ . The following lemma allows us to parameterize both sections using the s2 , u2 variables.

Lemma 10.6. There exists κ ∈ (0, 1) such that for |E| < κδ, the set Σs,E + ∩ E,s E,s {|s2 |, |u2 | < κδ} is given by a graph u1 = u1 (s2 , u2 ) with kDu1 k = O(1). The same holds for Σu,E with the graph given by s1 = sE,u + 1 . Proof. We restrict to (s, u) ∈ Σs+ and set s1 = δ. Since N (s, u) = λ1 δu1 + λ2 s2 u2 +O(δ), for sufficiently small κ, N (s1 , δ, s2 , u2 )−E > 0, N (s1 , −δ, s2 , u2 )− E < 0, ∂u1 N > 0, implying the existence of a unique u1 solving N − E = 0, E,s which we denote uE,s is automatically C r−2 (since 1 (s2 , u2 ). As a function, u1 r−1 N is C ) by the implicit function theorem, and the norm estimates for its derivative follows a direct computation. The case for Σu,E is similar. + A rectangle R is a diffeomorphic image of the Euclidean rectangle in R2 . Let us label the vertices by 1, 2, 3, 4 in clockwise order, and call the four sides l12 , l34 , l14 , l23 . Proposition 10.7. There exist e, κ > 0 such that for each 0 < E < e, there is a rectangle R++ (E) ⊂ Σs,E + with sides lij (E), such that: ++ ++ 1. Φ++ (E), and its image Φ++ (E)) is also a rectloc is well defined on R loc (R 0 angle, it’s four sides denoted lij (E). 2. As E → 0, both l12 (E) and l34 (E) converge to lu in Hausdorff metric. Simi0 0 larly, both l14 (E) and l23 (E) converge to lu . s,E 0 0 3. If (s, u) ∈ Σ+ , (s0 , u0 ) ∈ Σu,E is such that Φ++ + loc (s, u) = (s , u ), with (s, u) ∈ Bκδ (q + ) and (s0 , u0 ) ∈ Bκδ (p+ ), then (s, u) ∈ R++ (E).

See Figure 10.1. We first prove a version of Proposition 10.2 using E as a parameter.

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Proposition 10.8. There is e, κ > 0 such that if sin and uout satisfies out |sin 2 |, |u2 | ≤

λ1 κδ, 2λ2

out sin = δ, 1 = u1

then for all E ∈ (0, e) there is T = TE > 0 and a unique orbit (sE , uE ) : [0, TE ] → B4δ (O), such that sE (0) = sin ,

uE (TE ) = uout .

Proof of Proposition 10.8. Let SE denote the energy surface {N (s, u) = E}. Given sin and sout and T > T0 , Proposition 10.2 implies the existence of a solution (sT , uT ) solving the boundary value problem. Moreover, as T → ∞, (sT , uT )(0) → (sin , 0) ∈ S0 . Writing E(T ) the energy of the orbit (sT , uT ), we have E(T ) → 0 as T → ∞. The conclusions of our proposition follow from Proposition 10.2, as long as we can show the following claims. Claim: (1) E(T ) is positive and strictly monotone; therefore E(T ) is oneto-one and onto for T ∈ [T0 , ∞); (2) There is e > 0 such that uniformly over all sin , uout , we have 0 < E(T ) < e if T ≥ T0 , and therefore the inverse TE is well defined for E ∈ (0, e]. By Lemma 10.5 |uT2 (0)| ≤ κ|uT1 (0)|. We first show N (sT (0), uT (0)) > 0. Setting E = 0, Lemma 10.6 implies there exists a graph u1 = u0,s 1 (s2 , u2 ) on which N = 0. This submanifold divides Σ+ ∩ {|s2 |, |u2 | < κδ} into two components, with • N (s, u) > 0 whenever u1 > u0,u 1 (s2 , u2 ), and N (s, u) < 0 whenever u1 < u0,u 1 (s2 , u2 ). c Moreover, u0,u 1 |u2 =0 = 0, and |∂u2 u1 | ≤ C, where C is a constant depending c on kN kC 2 . Therefore |u1 (s2 , u2 )| ≤ C|u2 |. This implies if (δ, u1 , s2 , u2 ) ∈ Σ satisfies u1 > C|u2 | ≥ |u0,u 1 |, we have N (δ, u1 , s2 , u2 ) > 0. In particular, this is satisfied for (sT (0), uT (0)) if κ < C −1 . E(T ) is a continuous function defined on T ≥ T0 , and limT →∞ E(T ) = 0. We now prove that it is monotone. Using Lemma 10.5,   d d d T d T T T E(T ) = N (s (0), u (0)) = ∇N · 0, 0, u (0), u (0) dT dT dT 1 dT 2 d T d T = (λ1 sin u (0) + (λ2 sin u (0) 6= 0 1 + O2 (s, u)) 2 + O2 (s, u)) dT 1 dT 2

if κ is sufficiently small. Since we have proved E(T ) > 0, claim (1) follows. Finally, Proposition 10.2 implies 0 < E(T ) ≤ Cδ 2 e−λ1 T /2 , and claim (2) follows.

NORMALLY HYPERBOLIC CYLINDERS AT DOUBLE RESONANCE

113

Figure 10.2: Local map and global map in s2 , u2 coordinates.

Proof of Proposition 10.7. Let κ > 0 be small enough such that Proposition 10.8 applies, and let us rename λ1 κ/(2λ2 ) into κ. Consider two parameters a ∈ [−κδ, κδ] and b ∈ [−κδ, κδ], and then there exist unique orbits sE a,b solving the boundary value problem sE a,b (0) = (δ, a),

sE a,b (Ta,b (E)) = (δ, b),

then

Φ++ loc

 E R++ (E) = (sE a,b (0), ua,b (0)) : a, b ∈ [δ/2, δ/2] ,
 E R++ (E) = (sE a,b (Ta,b (E)), ua,b (Ta,b (E))) : a, b ∈ [δ/2, δ/2] .

An illustration of the rectangles in the (s2 , u2 ) variables is in Figure 10.2. Note that two sides of the rectangle R++ (E) are graphs over s2 ∈ [−κδ, κδ]; since the u1 , u2 components converge to 0 exponentially fast as T → ∞ (and E → 0), we conclude that these two sides converge to ls . The same can be said ++ about the rectangle Φ++ (E)) and lu . loc (R Finally, note that by definition, R++ (E) contains the initial point of all orbits (s, u) : [0, T ] → SE such that s1 (0) = δ, |s2 (0)| ≤ κδ, u1 (T ) = δ, and |u2 (T )| ≤ κδ, which contains the orbits such that (s, u)(0) ∈ Bκδ (p) and (s, u)(T ) ∈ Bκδ (p+ ). For each of the other symbols −−, +−, and −+, statements analogous to Proposition 10.7 hold, after making appropriate changes. One important point is that we consider E ∈ (0, e) (positive energy) for the symbols ++ and −−, and E ∈ (−e, 0) (negative energy) for the symbols +− and −+. + We now turn to the global map. We have p+ ∈ lu , q + ∈ ls , and Φ+ glob (p ) = u s q + . Moreover, condition [DR4c ] implies that Φ+ glob (l ) intersects l transversally in the energy surface SE . Since the transition time for the Poincar´e map Φ+ glob is uniformly bounded, the restricted map Φ+ |S depends smoothly on E. By E glob ++ ++ Proposition 10.7, for sufficiently small E, the rectangle Φ+ ◦ Φ (R (E)) glob loc intersects R++ (E) transversally. Applying the same argument to the other

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symbols, we obtain the following corollary. Corollary 10.9. There exists e > 0 such that the following hold. ++ ++ 1. For 0 < E < e, Φ+ (E)) intersects R++ (E) transversally. This glob ◦ Φloc (R means the images of l12 and l34 intersect l12 and l34 transversally, and the images of γ14 and γ23 do not intersect R++ (E). −− −− 2. For 0 < E < e, Φ− (E)) intersects R−− (E) transversally. glob ◦ Φloc (R +− +− 3. For −e < E < 0, Φ− (E)) intersect R−+ (E) transversally, and glob ◦ Φloc (R + −+ −+ +− Φglob ◦ Φloc (R (E)) intersect R (E) transversally.

See Figure 10.2 for a demonstration.

10.4

PERIODIC ORBITS FOR THE LOCAL AND GLOBAL MAPS

The rectangles as constructed form isolating blocks of Conley and McGehee [68]. We call a rectangle R = I1 × I2 ⊂ Rd × Rk , I1 = {kx1 k ≤ 1}, I2 = {kx2 k ≤ 1} an isolating block for the C 1 diffeomorphism Φ, if the following hold: 1. The projection of Φ(R) to the first component covers I1 . 2. Φ|I1 × ∂I2 is homotopically equivalent to the identity restricted on the set I1 × (Rk \ int I2 ). See Figure 10.3. Proposition 10.10. Let R be an isolating block for the map Φ. Then the sets W + = {x ∈ R : Φk (x) ∈ R, k ≥ 0},

W − = {x ∈ R : Φ−k (x) ∈ R, k ≥ 0})

are non-empty, W + project onto I1 , W − project onto I2 , and W + ∩ W − is non-empty. We now apply the isolating block construction to the maps and rectangles obtained in Corollary 10.9. Proposition 10.11. There exists e > 0 such that the following hold. ++ + s ++ 1. For 0 < E < e, Φ+ (E). glob ◦ Φloc has a unique fixed point p (E) on Σ+ ∩ R − −− 2. For 0 < E < e, Φglob ◦ Φloc has a unique fixed point p− (E) on Σs− ∩ R−− (E). −+ − +− c 3. For −e < E < 0: Φ+ glob ◦ Φloc ◦ Φglob ◦ Φloc has a unique fixed point p (E) − +− on R+− (E) ∩ (Φglob ◦ Φloc )−1 (R−+ (E)).

Notice that the rectangle R++ (T ) has C 1 sides, and there exists a C 1 change of coordinates turning it to a standard rectangle. It’s easy to see that the

NORMALLY HYPERBOLIC CYLINDERS AT DOUBLE RESONANCE

115

Figure 10.3: Isolating block.

isolating block conditions are satisfied for the following maps and rectangles: ++ Φ+ glob ◦ Φloc

and R++ (E),

−+ − +− Φ+ glob ◦ Φloc ◦ Φglob ◦ Φloc

and

−− Φ− glob ◦ Φloc

and R−− (E),

+− −1 −+ (Φ− R (E) ∩ R+− (E). glob ◦ Φloc )

We will only prove item 1 of Proposition 10.11, as the rest are analogous. By Proposition 10.10, we obtain that \ \ ++ k ++ ++ −k ++ W + (E) = (Φ+ (E), W − (E) = (Φ+ R (E) glob ◦ Φloc ) R glob ◦ Φloc ) k≥0

k≥0

are non-empty. In order to prove that the intersection W + (E) ∩ W − (E) is a single point, we use the invariant cones introduced in (10.4). Lemma 10.12. There exists K > 1 and T0 > 0 such that the following hold. Assume that U ⊂ Σs+ ∩ Br is a connected open set on which the local map Φ++ loc is defined, and for each x ∈ U , inf{t ≥ 0 : φtN (x) ∈ Σu+ } ≥ T0 . ++ u Then the map D(Φ+ glob ◦ Φloc ) preserves the cone field CK , and the inverse + ++ −1 s D(Φglob ◦ Φloc ) preserves CK .

Recall that lu 3 p+ is the intersection of the unstable manifold W u (O) with the section Σu+ . Let T u be the tangent vector to lu at p+ , and T s the tangent vector to T s at q + . We will show that if Φ++ loc |SE (x) = y, then the image of the u u unstable cone DΦ++ loc (x)CK is very close to T . This happens because the flow of tangent vector is very close to that of a linear flow. The following lemma holds for a general flow φt on Rd × Rk , and a trajectory xt of the flow. Let v(t) = (v1 (t), v2 (t)) be a solution of the variational equation, i.e. v(t) = Dφt (xt )v(0). Lemma 10.13. With the above notations assume that there exist b2 > 0, b1
0 such that the variational equation    A(t) B(t) v1 (t) v(t) ˙ = C(t) D(t) v2 (t) satisfies A ≤ b1 I and D ≥ b2 I as quadratic forms, and kBk ≤ σ, kCk ≤ δ. Then for any c > 0 and κ > 0, there exists δ0 > 0 such that if 0 < δ, σ < δ0 , we have u u (Dφt ) CK ⊂ C1/β , t

βt = ce−(b2 −b1 −κ)t + σ/(b2 − b1 − κ).

Proof. Denote γ0 = c. The invariance of the cone field is equivalent to
d βt2 hv2 (t), v2 (t)i − hv1 (t), v1 (t)i ≥ 0. dt Computing the derivatives using the variational equation, and applying the norm bounds and the cone condition, we obtain 2βt (βt0 + (b2 − δβt − b1 )βt − σ) kv2 k2 ≥ 0. We assume that βt ≤ 2γ0 , then for sufficiently small δ0 , δβt ≤ κ. Denote b3 = b2 − b1 − κ and let βt solve the differential equation βt0 = −b3 βt + σ. It’s clear that the inequality is satisfied for our choice of βt . Solve the differential equation for βt and the lemma follows. Proof of Lemma 10.12. We will only prove the unstable version. By Assumption + uu + u 4, there exists c > 0 such that DΦ+ (q ) ⊂ CK (p+ ). Note that as glob (q )T + T0 → ∞, the neighborhood U shrinks to p and V shrinks to q + . Hence there u u exist β > 0 and T0 > 0 such that DΦ+ glob (y)C1/β (y) ⊂ CK for all y ∈ V . Let (s, u)(t)0≤t≤T be the trajectory from x to y. By Proposition 10.2, we have ksk ≤ e−λ1 T /4 for all T /2 ≤ t ≤ T . It follows that the matrix for the variational equation     A(t) B(t) − diag{λ1 , λ2 } + O(s) O(s) = (10.6) C(t) D(t) O(u) diag{λ1 , λ2 } + O(u) satisfies A ≤ −(λ1 −κ)I, D ≥ (λ1 −κ)I, kCk = O(δ), and kBk = O(e−(λ1 −κ)T /2 ). u As before, CK (x) = {kvs k ≤ ckvu k}, and Lemma 10.13 implies u u DφT (x)CK (x) ⊂ C1/β (y), T 0

where βT = O(e−λ T /2 ) and λ0 = min{λ2 − λ1 − κ, λ1 − κ}. Finally, note u u that DφT (x)CK (x) and DΦ++ loc (x)CK (x) differ by the differential of the local

NORMALLY HYPERBOLIC CYLINDERS AT DOUBLE RESONANCE

117

Poincar´e map near y. Since near y we have |s| = O(e−(λ1 −κ)T ), using the equation of motion, the Poincar´e map is exponentially close to identity on the (s1 , s2 ) components, and is exponentially close to a projection of u2 on the u (u1 , u2 ) components. It follows that the cone C1/β is mapped by the Poincar´e T map into a strong unstable cone with exponentially small size. In particular, for T ≥ T0 , we have u u DΦ++ loc (x)CK (x) ⊂ C1/β (y), and the lemma follows. ++ Proof of Proposition 10.11. We have proven that the map D(Φ+ glob ◦ Φloc ) preu serves the unstable cone field CK . Similar to the argument in Lemma 10.5, this u c implies any two points z1 , z2 ∈ W + (E) must satisfy z1 ∈ z2 + (CK ) , otherwise, we can use the expansion in the u2 direction to show that the distance between the forward orbits of z1 and z2 must grow exponentially, contradicting the fact that z1 , z2 ∈ W + (E). Moreover, in Lemma 10.6 we showed that Σs,E can be + parameterized by s2 , u2 , this implies W + (E) is a K −1 -Lipschitz graph over s2 . By the same argument, W − (E) is a K −1 -Lipschitz graph over u2 . Since K > 1, there is a unique intersection in W + (E) ∩ W − (E), which we denote by p+ (E). This concludes the proof of item 1. The other cases are analogous.

In the case of the non-simple homology, there exist two rectangles R1 and R2 , whose images under Φglob ◦Φloc intersect themselves transversally, providing a “horseshoe” type picture. Proposition 10.14. There exists e > 0 such that the following hold: 1. For all 0 < E ≤ e, there exist rectangles R1 (E), R2 (E) ∈ Σs,E such that for + i = 1, 2, Φiglob ◦ Φ++ loc (Ri ) intersects both R1 (E) and R2 (E) transversally. 2. Given σ = (σ1 , · · · , σn ), there exists a unique fixed point pσ (E) of 1  Y

 i Φσglob ◦ Φ++ loc |Rσi (E)

(10.7)

i=n

on the set Rσ1 (E). 3. The curve pσ (E) is a C 1 graph over the u1 component with uniformly bounded derivatives. Furthermore, pσ (E) approaches pσ1 and for each 1 ≤ j ≤ n − 1, 1  Y

 σ i Φσglob ◦ Φ++ loc (p (E))

i=j

approaches pσj+1 as E → 0. The proof is analogous to that of Proposition 10.11.

118 10.5

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NORMALLY HYPERBOLIC INVARIANT MANIFOLDS

We prove Theorem 4.4 in this section. Proof. Non-simple case. If the homology h is non-simple, then h = n1 h1 + n2 h2 with h1 , h2 being simple homologies. Let (σ1 , . . . , σn ) be the sequence determined by Lemma 4.2. Applying Proposition 10.14, we obtain the fixed points pσ (E) for all 0 < E < e. The fixed points correspond to hyperbolic periodic orbits that we call ηhE . Let γhE be the projection of ηhE to the configuration space, we now prove that they must be identical to the shortest curve in Jacobi metric gE , after a reparametrization. According to the condition [DR3c ], γh0 is the unique shortest curve for the Jacobi metric g0 , and there exists c0 such eH s (c0 ). Any gE shortest curve γ 0 corresponds to the that ηh0 = AeH s (c0 ) = N E 0 Aubry set of cohomology cE , which lifts to an orbit ηE in phase space. Using 0 semi-continuity, ηE must be contained in a neighborhood of ηh0 in the phase space. In particular, it must intersect the sections Σs+ and Σu+ sufficiently close to q + and p+ . According to Proposition 10.8, item 3, the intersection with Σs+ must be contained in the rectangle R++ (E). Since the fixed point pσ (E) is the 0 unique fixed point for the map (10.7), we conclude that ηE = ηhE . Simple critical case. The existence of the periodic orbits follows from Proposition 10.11 in the same way as the non-simple case. Also, by the same reasoning, E we know that the orbits ηhE , η−h must coincide with the minimal geodesics of the Jacobi metric. It suffices to show that [ [
E 0 M= ηhE ∪ η−h ∪ ηh0 ∪ η−h ∪ ηcE −e 0. As a result, the unstable bundle of ηhE projects onto a bundle transversal to γhE , and the unstable manifold projects onto the θ component. We now consider the Hamiltonian √ Hs (ϕ, I, τ ) = H s (ϕ, I) + P (ϕ, I, τ ), where kP kC 2 ≤ 1.

ϕ ∈ T2 , I ∈ R2 , τ ∈ T√ ,

(11.2)

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¯ there is 0 > 0 depending on Proposition 11.3. Given any κ > 0, 0 < e < E ¯ there is E1 < E0 < E2 , H s , e, κ, such that the following hold. For all E0 ∈ [e, E], such that for all 0 <  < 0 , there is an embedding χ (x, y, τ ) : T × (E1 , E2 ) × T√ → T2 × R2 × T√ ,

πτ χ = id

such that C is a weakly invariant normally hyperbolic cylinder for the Hamiltonian flow of Hs (see (11.2)). Moreover, we have kχ (ϕ, I, τ ) − χ0 (ϕ, I)kC 1 < κ, and C contains all the invariant sets in V = Bκ (C0 ) × T√ . √ √ Proof. Since k P kC 2 ≤ , this is a regular perturbation of the vector field, compared to the singular perturbation we see in Chapter 9. The proof follows from standard theory, see for example [70, 31]. We will denote by χ0 (x, y) = χ (ϕ, I, 0) the zero-section of the embedding, and χ0 (T × (E1 , E2 )) = C0 , √ √  which is invariant under the time-  map φ = φHs . The following Lipschitz estimate is a crucial part of establishing Aubry-Mather type. Proposition 11.4. Given R > 0, there is C > 0 depending only on k∂ 2 Kk, kH s kC 2 , and R, such that if u is any weak KAM solution of Hs at cohomology c with kck ≤ R, then for any \ e u) = (ϕ1 , I1 ), (ϕ2 , I2 ) ∈ I(c, φ−k (Gc,u ) , k≥−1

we have

1

|H s (ϕ1 , I1 ) − H s (ϕ2 , I2 )| ≤ C 4 kϕ1 − ϕ2 k. (11.3) T −n e u) = In particular, (11.3) holds on the sets I(c, (Gc,u ) and Ae0Hs . n∈N φ √ Proof. We apply Theorem 7.10 to the system Hs = H s + P and a continuous √ 12 1 time weak KAM solution w(x, t). We then obtain a O(  ) = O( 4 ) Lipschitz √ −t estimate (note that the small parameter is ), on the set φHs Gc,w , as long √ √  as t > 1/ . Note that the corresponding discrete dynamics is φ = φHs , and √ √ therefore we need to take k > 1/ to have t = k  > 1/ . Proof of Theorem 11.1. Part (1): Let c0 = c¯h (E0 ), and γhE0 is the unique shortest curve, we show (Hs , c0 ) is of Aubry-Mather type. By Proposition 11.3 and

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eH s (c0 ) ⊂ the semi-continuity of the Ma˜ n´e set, for sufficiently small  and σ, N  √ e e Bκ (C0 ) × T  for all for c ∈ Bσ (c0 ). As a result, AHs (c) ⊂ AHs (c) ⊂ C . We now prove the graph property. Suppose (x1 , y1 ),

(x2 , y2 )

∈ (χ0 )−1 (Ae0Hs (c))

where c ∈ Bσ (c0 ). Observe that H s ◦χ0 (x, y) = y. Then due to Proposition 11.4, kH s ◦ χ0 (x, y) − ykC 1 = kH s ◦ χ0 − H s ◦ χ0 k ≤ kH s kC 2 kχ0 − χ0 kC 1 ≤ Cκ, where C will be used to denote a generic constant. We have ky2 − y1 k ≤ kH s ◦ χ0 (x2 , y2 ) − H s ◦ χ0 (x1 , y1 )k + Cκk(x2 − x1 , y2 − y1 )k 1

≤ C 4 kπϕ ◦ χ0 (x2 , y2 ) − πϕ ◦ χ0 (x1 , y1 )k + Cκk(x2 − x1 , y2 − y1 )k 1

≤ C(κ +  4 )k(x2 − x1 , y2 − y1 )k. (11.4) 1 For , κ small enough, we have ky2 − y1 k ≤ 2C(κ +  4 )kx2 − x1 k. Finally, assume that (χ0 )−1 Ae0Hs (c) projects onto the x component. We will show that W u (Ae0 s (c)) is a graph over the θ component. According to Lemma H

11.2, W u (Ae0H s (c)) is a smooth graph over θ component, and the projection of the unstable bundle is normal to γhE . Let us note that (11.4) implies Ae0Hs (c) is close to Ae0H s (c) in Lipschitz norm, and the unstable bundle depends smoothly on perturbation. Therefore the unstable bundle of Ae0 s (c) is also transversal to H

AHs (c). We still need to prove that W u (Ae0Hs (c)) is a Lipschitz graph over the θ component near the Aubry set A0Hs (c), if the Aubry set is an invariant circle (see Definition 8.1, 2b). Letting G denote the pseudograph associated to the unique weak KAM solution of Hs at c, we will show that G and W u (Ae0Hs (c)) coincide on some neighborhood of A0Hs (c). Let us extend the coordinates χ0 (x, y) to C 1 coordinates Λ(x, y, s, u) in a neighborhood U of C0 such that the s coordinate axis is transversal to the unstable bundle, and the u coordinate axis is transversal to the stable bundle. We similarly extend χ0 (x, y) to Λ (x, y, s, u), and we can do this keeping the estimate kΛ − Λ kC 1 < Cκ with the norm defined on U (C). √ Let z ∈ G ∩ U , then φ−k (z) accumulates to A0Hs (c) ⊂ C (φ is the time-  map of Hs ). It follows that there exists ξ ∈ C such that z ∈ W u (ξ). We claim that ξ ∈ Ae0Hs (c) (which implies z ∈ W u (Ae0Hs (c))). Assume this does not hold. For a sufficiently large k, due to normal hyperbolicity, there exist µ > 1 and λ < µ such that kφ−k z − φ−k ξk < Cµ−k ,

dist(φ−k ξ, Ae0Hs (c)) > C −1 λ−k .

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Let ζk denote the unique point in Ae0Hs (c) such that ζk and φ−k ξ has the same x component in the local coordinates (this is possible because Ae0 s (c) is a Lipschitz H

graph over x component). Choosing k > 1/, we have: • kπ(x,y) (φ−k ξ − ζk )k ≥ dist(φ−k z, Ae0Hs (c)) > C −1 λ−k . 1

• kπy (φ−k z − ζk )k ≤ 2C(κ +  4 )kπxT (φ−k z − ζk )k ≤ kπx (φ−k z − ζk )k. Since −k both φ z and ζk are contained in j≥−1 φ−j G, Proposition 11.4 and (11.4) apply. • kπx (φ−k z − ζk )k = kπx (φ−k z − φ−k ξ)k < Cµ−k . Combining all the inequalities, we get C −1 λ−k ≤ kπ(x,y) (φ−k ξ − ζk )k ≤ kπ(x,y) (φ−k z − ζk )k + Cµ−k ≤ 2kπx (φ−k z − ζk )k + Cµ−k ≤ 3Cµ−k , but this is a contradiction when k is large enough. We have proven that G ∩ U is contained in W u Ae0Hs (c). By Proposition 6.14, φ−1 G is a Lipschitz graph. Taking U smaller, we have φ−1 G ∩ U = W u Ae0 s (c) ∩ U is also a Lipschitz graph. H

This proves 2b) of Definition 8.1. Part (2): Suppose c0 = c¯h (E0 ) is such that AH s (c) = γhE0 ∪ γ¯hE0 are two shortest curves. Let V1 , V2 be neighborhoods of γ E0 , γ¯ E0 such that πϕ η E ⊂ V1 , πϕ η¯E ⊂ V2 for all E ∈ (E0 − δ, E0 + δ), and let f : Tn → [0, 1],

f |V1 = 0,

f |V2 = 1

be a smooth bump function. Then for H 1 = H s − f,

H 2 = H s − (1 − f ),

AeH 1 (c0 ) = γ E0 and √ AeH 2 (c0 ) = γ¯ E0 . We then apply part (1) to obtain that for i each i = 1, 2, H + P, c0 is of Aubry-Mather type. This implies Hs , c0 is of bifurcation Aubry-Mather type (see Definition 8.3).

11.2

SIMPLE NON-CRITICAL CASE

Suppose h is a simple non-critical homology, which means that for the energy E = 0, there is a unique shortest curve γh0 in the homology h corresponding to a hyperbolic periodic orbit ηh0 of the Hamiltonian system. In this case, however, we have, for c0 = c¯h (0) AH s (c0 ) = γh0 ∪ O, where O is the origin (which is where U attains its minimum). If we consider V1 , V2 disjoint open sets containing γh0 and 0 respectively, and define the local

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Aubry sets AH 1 (c) and AH 2 (c), it follows that AH 1 (c) = γh0 and AH 2 (c) = O. Theorem 11.5. Suppose h is simple non-critical, and let c0 = c¯h (0). Then there are 0 , δ > 0 depending only on K, U such that for 0 <  < 0 and U 0 ∈ Vδ (U ), for √ Hs = K(I) − U 0 (ϕ) + P, kP kC 2 ≤ 1, the pair (Hs , c0 ) is of asymmetric bifurcation type. Proof. The fact that H 1 , c0 is of Aubry-Mather type follows the same proof as Theorem 11.1. On the other hand, AeH s (c0 ) is a hyperbolic periodic orbit, which is robust under perturbation. Therefore Definition 8.5 is satisfied.

11.3 11.3.1

SIMPLE CRITICAL CASE Proof of Aubry-Mather type using local coordinates

Theorem 11.6. Suppose h ∈ H1 (T2 , Z) is a simple homology for H s , and consider the cohomology class c0 = c¯h (0). Then there exist 0 , e, δ > 0 depending only on H s , h such that for each 0 <  < 0 , U 0 ∈ Vδ (U ) and c ∈ Be (c0 ), the pair (Hs , c) is of Aubry-Mather type. Moreover, the same holds for (H s , λc) for all 0 ≤ λ ≤ 1. We have the following (See Chapter 10): • ηh0 = AeH s (c0 ) contains the hyperbolic fixed point (0, 0), and is a homoclinic orbit to (0, 0). s/u s/u • (0, 0) admits eigenvectors −λ2 < −λ1 < λ1 < λ2 . Let v1 and v2 denote the eigendirections of the eigenvalues ±λ1 , ±λ2 . Let Inv(ϕ, I) = (ϕ, −I) denote the involution of the Hamiltonian system. Since the flow is timereversible, we have Inv(vis ) = ±viu , i = 1, 2. Without loss of generality, we assume Inv(vis ) = viu . As a result, there exist vi , wi ∈ R2 such that vis = (vi , wi ),

viu = (vi , −wi ),

i = 1, 2.

(11.5)

• There is a C 1 normally hyperbolic invariant manifold M containing ηh0 . In particular, M must contain (0, 0) and it’s tangent to the plane Span{v1s , v1u } = Rv1 ⊕ Rw1 ⊂ R2 × R2 at (0, 0). The projection πϕ ηh0 = γh0 then is a C 1 curve in T2 , since 0 ∈ T2 is the only possible discontinuity of the tangent direction, but at 0 the curve is tangent to v1 = πϕ v1s = πϕ v1u . Let πv1 : R4 → R and πw1 : R4 → R be the orthogonal projections to v1 , w1 directions. We also need the following analog of Lemma 11.2. Lemma 11.7. W u (ηh0 ) is a Lipschitz graph over the θ component on a neigh-

AUBRY-MATHER TYPE AT THE DOUBLE RESONANCE

127

Figure 11.1: Coordinates near the homoclinic orbit: vertical lines indicate the level curves of the x coordinate.

borhood of γh0 . Proof. In the proof of Lemma 11.2, the Lipschitz constant of the Green bundles is uniform over all energy E. The lemma follows by taking the limit E → 0 in the space of Lipschitz graphs. We require a suitable parametrization of the cylinder C(H s ) near the homoclinic ηh0 . An illustration of the parametrization is given in Figure 11.1. Proposition 11.8. For each κ > 0 there exist δ1 , δ2 > 0, and a smooth embedding χ0 = χ0 (x, y) : T × (−δ2 , δ2 ) → M, such that the following hold: 1. C(H s ) := χ0 (T × (−δ2 , δ2 )) ⊃ ηh0 . (We use the notation C, since the image of χ0 is a cylinder. ) 2. The cylinder C(H s ) is symplectic. 3. χ0 is “almost vertical” near x = 0, namely πϕ ◦ χ0 (x, y) is κ-Lipschitz in y for all |x| < δ1 . 4. The vertical coordinate is given by the energy function away from x = 0, namely for all |x| ≥ δ1 /2 we have H s (χ0 (x, y)) = y. We now prove Theorem 11.6 assuming Proposition 11.8. Proof of Theorem 11.6. Let us consider the system √ Hs = K(I) + U 0 (ϕ) + P. where kP kC 2 ≤ 1 and kU − U 0 kC 2 ≤ δ. Then for 0 , δ small enough depending only on K, U , standard perturbation theory implies Hs admits a normally hyperbolic weakly invariant manifold C(Hs ), let us denote√by C its zero sec√  tion. Then C is invariant under the time-  map Φ = ΦHs , and it admits a

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parametrization χ : T×(−δ2 , δ2 ) → C , and kχ −χ0 kC 1 = o(1) as , δ → 0. The cylinder is symplectic, since symplecticity is open under perturbations. MoreeH s (c) ⊂ C(H s ) for all c ∈ Be (c0 ), and over, for e > 0 small enough, the set N e eH s (c) is since N is upper semi-continuous under Hamiltonian pertubations, N  s s close to C(H ) for small . Since C(H ) contains all the invariant sets in its eH s (c) ⊂ C(Hs ). neighborhood, we conclude that AeHs (c) ⊂ N  e We now show χ−1  AHs (c) is a Lipschitz graph for c ∈ Be (c0 ). Let (ϕi , Ii ) = χ (xi , yi ), i = 1, 2 be two points in AeHs (c). The proof consists of two cases. In the first case we use the almost verticality of the cylinder, and the idea is similar to the proof of Theorem 9.3. In the second case we use the strong Lipschitz estimate for the energy H s , and the idea is similar to the proof of Theorem 11.1. Case 1. |x1 |, |x2 | < δ1 . In this case, we apply the a priori Lipschitz estimates for the Aubry sets: there is C > 0 depending only ∂ 2 K, kH s kC 2 such that kI2 − I1 k ≤ Ckϕ2 − ϕ1 k. Let κ be as in Proposition 11.8, and let 0 be small enough such that πϕ χ (x, y) is 2κ-Lipschitz in y. Since χ0 is an embedding, there is C > 1 depending only on H s such that for all 0 small enough, kϕ2 − ϕ1 k + kI2 − I1 k ≥ C −1 (kx2 − x1 k + ky2 − y1 k) . Let (ϕ3 , I3 ) = χ (x2 , y1 ) then kϕ3 − ϕ2 k ≤ kχ (x2 , y1 ) − χ (x2 , y2 )k ≤ 2κky2 − y1 k, kϕ3 − ϕ1 k ≤ kχ (x2 , y1 ) − χ (x1 , y1 )k ≤ Ckx2 − x1 k. Combining all estimates, we get C −1 (kx2 − x1 k + ky2 − y1 k) ≤ (1 + C)kϕ2 − ϕ1 k ≤ (1 + C) (2κky2 − y1 k + Ckx2 − x1 k) , or

C −1 − 2(1 + C)κ ky2 − y1 k ≤ C −1 + (1 + C)C kx2 − x1 k, which is what we need if κ < 1/(4C(1 + C)). Case 2. |x1 |, |x2 | > δ1 /2. In this case we apply Proposition 11.4, to get 1

kH s (ϕ2 , I2 ) − H s (ϕ1 , I1 )k ≤ C 4 kϕ2 − ϕ1 k. Assume that  is small enough such that kχ − χ0 kC 1 < κ. Then a computation 1 identical to (11.4) implies ky2 −y1 k ≤ 2C(κ+ 4 )kx2 −x1 k if , κ is small enough depending only on C. e0 We obtain the Lipschitz property of χ−1  AHs (c) after combining the two

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cases. We now check that the unstable bundle E u is uniformly transverse to the projection A0Hs (c). In case 1, the almost verticality of the cylinder implies A0Hs (c) differs from AH s (c0 ) = γh0 by O(κ) in Lipschitz norm. Given that s/u

the tangent vector of γh0 is close to the weak directions v1 in case 1, while s/u the projection of E u is close to v2 , the desired transversality holds. In case 2, similar to the proof of Theorem 11.1, AeHs (c) is also close to AeH s (c0 ) in Lipschitz norm (for a different reason, i.e (11.4)). The claim follows, similar to the proof of Theorem 11.1, using Lemma 11.7. e0 Finally, we show W u (Ae0Hs (c)) is a graph over ϕ when χ−1  AHs (c) projects onto the x component. Similar to the proof of Theorem 11.1, it suffices to show that if G is a weak KAM solution of Hs at cohomology c, there is a neighbor U of G such that any z ∈ U ∪ U and ζ ∈ C such that z ∈ W u (ζ), we can conclude that z ∈ Ae0Hs (c). Arguing by contradiction as before, we suppose ζ ∈ / Ae0Hs (c) and let ξk be the unique point on Ae0Hs (c) with the same x component as φ−k ζ. Just as before, we can extend the coordinate χ to a tubular neighborhood of C. We obtain in the same fashion • kπ(x,y) (φ−k ξ − ζk )k ≥ dist(φ−k z, Ae0Hs (c)) > C −1 λ−k . • kπx (φ−k z − ζk )k = kπx (φ−k z − φ−k ξ)k < Cµ−k . The remaining inequality to prove is kπy (φ−k z − ζk )k ≤ Ckπx (φ−k z − ζk )k, which we split into Case 1 and 2 as before. Case 2 is identical to Theorem 11.1. Case 1 follows the same computation, with the observation that we can define the extension in such a way that πϕ Λ(x, y, s, u) is still 2κ-Lipschitz in y. 11.3.2

Construction of the local coordinates

This is done separately near the hyperbolic fixed point (local) and away from it (global). Furthermore, the local coordinate requires a preliminary step. Lemma 11.9. For each κ > 0 there is δ > 0, and a smooth embedding χpre = χpre (x, y) : (−δ, δ) × (−δ, δ) → M ∩ B2δ (0, 0) ⊂ T2 × R2 , satisfying kχpre ◦ (πv1 , πw1 ) − idkC 1 < κ.

(11.6)

0 −1 There is 0 < δ1 < δ such that the curve χ−1 pre ηh ∩ πx {(−δ1 , δ1 )} is given by a Lipschitz graph {(x, g(x)) : x ∈ (−δ1 , δ1 )}.

Proof. The existence of the local coordinate follows from T(0,0) M = Rv1 ⊕ Rw1 , the fact that M is C 1 , and the implicit function theorem. Since γh0 = πϕ ηh0 is a C 1 curve tangent to v1 at 0 ∈ T2 , γh0 can be reparametrized using its projection to the v1 direction. Since ηh0 is a Lipschitz graph over γh0 , the second claim

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follows. Lemma 11.10. For each κ > 0 there is δ1 > δ2 > 0, a smooth embedding χloc = χloc (x, y) :

(−δ1 , δ1 ) × (−δ1 , δ1 ) → M ∩ B2δ (0, 0),

a neighborhood V of the local homoclinic ηh0 ∩ πx−1 {(−δ1 , δ1 )} on which the cylinder is “almost vertical” in the sense that there is C > 0 such that πϕ χloc (x, y) is Cκ − Lipschitz in y. 0 −1 The pull back χ−1 loc ηh ∩ πx {(−δ1 , δ1 )} is a Lipschitz graph over x, and in addition,

H s (χloc (x, y)) = y,

for all

δ1 /2 < |x| < δ1 , |y| < δ2 .

(11.7)

The manifold χloc (−δ1 , δ1 ) × (−δ1 , δ1 )) is symplectic. Proof. First we show the image of χpre is symplectic. To see this, note that 2 2 H s (ϕ, I) = 12 AI · I − 12 Bϕ · ϕ + O3 (I, ϕ), where A = ∂II K and B = ∂ϕϕ U (0) are both positive definite. Moreover, we have λ1 w1 = Av1 and λ1 v1 = Bw1 , where v1 , w1 are the vectors from (11.5). From (11.6), we have ∂x χpre (x, y) = v1 + O(κ),

∂y χpre (x, y) = w1 + O(κ),

Let ω be the standard symplectic form; it follows that ω(∂x χpre , ∂y χpre ) = ω((v1 , 0), (0, w1 )) + O(κ) = λ1 v1 · Av1 + O(κ) is uniformly bounded away from 0 if κ is small enough. Let δ, δ1 , χpre and g be from Lemma 11.9. We claim that after possibly shrinking δ1 , there is δ2 > 0 and C > 1 such that for all (x, g(x) + y) ∈ R2 , |y| < δ2 , δ1 /2 < |x| < δ1 we have Cδ1 > |∂y (H s ◦ χpre (x, g(x) + y)) | > C −1 δ1 > 0. We will only prove this claim for the case δ1 /2 < x < δ, as the other half is symmetric. s/u Since ηh0 is tangent to the stable/unstable vectors v1 , we have χpre (x, g(x)) = xv1u + O(x2 ) = x(v1 , w1 ) + O(x2 ), χpre (x, g(x) + y) = x(v1 , w1 ) + O(x2 ) + O(y),

x > 0.

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AUBRY-MATHER TYPE AT THE DOUBLE RESONANCE

We have
∂y (H s ◦ χpre (x, g(x) + y)) = x(Bv1 , Aw1 ) + O(x2 ) + O(y) · (w1 + O(κ)) = xAw1 · w1 + O(x2 ) + O(y) + O(κ(|x| + |y|)) ≥ 4C −1 δ1 + O(x2 ) + O(y) + O(κ(|x| + |y|)) > C −1 δ1 , if 8C −1 = kAw1 · w1 k, δ1 /2 < x < δ1 , δ1 < C −1 , |y| < δ2 < C −1 δ1 , and κ < C −1 /2. The upper bound can be obtained similarly. This proves our claim. We consider the function F = F (x, y) =

δ1−1 , ∂y (H s ◦ χpre (x, g(x) + y))

δ1 /2 < |x| < δ1 , |y| < δ2 .

and for each fixed x, let YF (x, y) denote the solution to the ODE d YF = F (x, y), dy

YF (0) = g(x),

then ∂y H s ◦ χpre (x, YF (x, y)) = 1, and therefore H s ◦ χpre (x, YF (x, y)) = y. Finally, let us define the vector field G(x, y) via ( (0, F (x, y)), δ1 /2 < |x| < δ1 , |y| < δ2 ; G(x, y) = (0, 1), |x| < δ1 /4, or |y| > 2δ2 ; and smoothly interpolated (keeping the first coordinate 0) in between. Since C −1 < |F1 | < C, this can be done keeping kGkC 1 = O(δ1−1 ). Let g1 (x) be a mollified version of g such that g1 (x) = g(x) for all |x| ≥ δ1 /4, and |g1 (x) − g(x)| ≤ δ1 /2. Finally define χloc (x, y) = χpre ◦ ΦyG (x, g1 (x)), where ΦG (x0 , y0 ) is the time-y-flow of G(x, y). The modification g1 is to ensure the coordinate system is smooth. See Figure 11.2. We have: • χloc (x, y) = χpre (x, g1 (x) + y) when |x| < δ1 /4 and |y| > 2δ1 . • H s ◦χloc (x, y) = H s ◦χpre (x, YF (x, y)) = y when δ1 /2 < |x| < δ1 and |y| < δ2 . • The function χpre is κ-Lipschitz in y (see (11.6)). Therefore (11.7) holds when |x| < δ1 /4 and |y| > 2δ1 . Moreover, given that the flow ΦyG is vertical, we have πx ΦyG (x0 , y0 ) = x0 and |∂y0 ΦyG | ≤ 1 + |y| · kGkC 1 = 1 + O(δ1−1 |y|). Therefore
|∂y (πϕ χpre ) ◦ ΦyG (x, g1 (x))| ≤ k∂y (πϕ χpre )k · 1 + O(δ1−1 |y|) ≤ 2κ if |y| < δ2 is sufficiently small. As a result, (11.7) holds on the set V := {|y| < δ2 }, which is the gray area in Figure 11.2.

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Figure 11.2: Construction of local coordinate.

• The curve ηh0 = {χpre (x, g(x)) : |x| < δ1 } coincides with χloc (x, 0) when |x| ≥ δ1 /4 and coincides with χloc (x, y−g1 (x)+g(x)) when |x| ≤ δ1 /4, and therefore is a Lipschitz graph under χ−1 loc .

Proof of Proposition 11.8. Consider the χloc coordinate system as constructed. By construction, the sections Σ± = χloc ({|y| ≤ δ2 , x = ±δ1 }) are transversal to the Hamiltonian flow, and therefore there is a Poincar´e map Φ : Σ+ → Σ− . Moreover, we must have χ−1 loc ◦ Φ ◦ χloc (δ1 , y) = (−δ1 , y), since the flow preserves energy. Let T (y) denote the time it takes for φtH s to flow from χloc (δ1 , y) to χloc (−δ1 , y). We now define   0 ≤ x < δ1 ; χloc (x, y), χ(x, y) = χloc (x − 1, y), 1 − δ1 ≤ x ≤ 1;   s(x,y) x−δ1 φH s (δ1 , y), s(x, y) = 1−2δ T (y), δ1 ≤ x ≤ 1 − δ 1 . 1 Then χ(x, y) is an embedding T × (−δ2 , δ2 ) → T2 × R2 satisfying item 1, 3, and 4 of Proposition 11.8. It remains to prove (2), namely symplecticity. For |x| ≤ δ1 , this is covered in Proposition 11.8. For |x| ≥ δ1 , note that the tangent plane to the cylinder is spanned by two vector fields, one being XH s and the other is ∂y χ. We have ω(XH , ∂y χ) = ∇H · ∂y χ = 1 by construction, therefore the manifold is symplectic.

Chapter Twelve Forcing equivalence between kissing cylinders In this section we prove Theorem 5.9, i.e. the jump mechanism. We assume that h is a non-simple homology class which satisfies h = n1 h1 + n2 h2 , where h1 , h2 are simple homologies. They are associated with curves c¯h (E) and c¯µh1 (E) in the corresponding channels, where we assume

c¯h (0) − c¯µ (0) < µ. h1 Our goal is to prove forcing equivalence of the cohomologies Φ∗L (¯ ch (E1 )) and Φ∗L (¯ cµh1 (E2 )) for some E1 , E2 ∈ (e, e + µ), where e > 0 is sufficiently small. We will construct a variational problem which proves forcing equivalence for the original Hamiltonian H using Definition 6.18. The proof consists of the following steps. 1. We construct a special variational problem for the slow mechanical system H s . A solution of this variational problem is an orbit “jumping” from one homology class h to the other h1 . The same can be done with h and h1 switched. 2. We then modify this variational problem for the fast time-periodic perturbas tion , i.e. for the perturbed slow system Hs (ϕs , I s , τ ) = K(I s )−U (ϕs )+ √ of H √ s s P (ϕ , I , τ ) with τ ∈  T. This is achieved by applying the perturbative results established in Chapter 7. Recall the original Hamiltonian system H near a double resonance can be brought to a normal form N = H ◦ Φ and this normal form, in turn, is related to the perturbed slow system through coordinate change and energy reduction (see Chapter 14). The variational problem for Hs can then be converted to a variational problem for the original H . 3. Using this variational problem we prove the forcing equilvalence between Φ∗L (¯ ch (E1 )) and Φ∗L (¯ cµh1 (E2 )).

12.1

VARIATIONAL PROBLEM FOR THE SLOW MECHANICAL SYSTEM

The slow system is given by H s (ϕ, I) = K(I) − U (ϕ), where we assumed that the minimum of U is achieved at 0. Given m ∈ T2 , a > 0 and a unit vector

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ω ∈ R2 , define S(m, a, ω) = {m + λω : λ ∈ (−a, a)}. S(m, a, ω) is a line segment in T2 and we will refer to it as a section (see Figure 12.1). Given c1 , c2 ∈ R2 , we say that c1 , c2 have a non-degenerate connection for s H , at the section S(m, a, ω) ∈ T2 , if the following conditions hold. [N1A] (c1 − c2 ) ⊥ S,

αH (c1 ) = αH (c2 ).

[N2A] There exists a relatively compact set K ⊂ S such that for all x ∈ AH (c1 ) and z ∈ AH (c2 ), the minimum of the variational problem min{hH s ,c1 (x, y) + hH s ,c2 (y, z)} y∈S

is never achieved outside of K. [N3A] Suppose the above minimum is achieved at y0 ; let p1 − c1 be any superdifferential of hH s ,c1 (x, ·) at y0 , and −p2 +c2 a super-differential of hH s ,c2 (·, z) at y0 , then ∂p H s (y0 , pi ) · S ⊥ , i = 1, 2 have the same signs; here S ⊥ denotes a normal vector to S. Remark 12.1. • The “A” in [N1A]–[N3A] stands for “autonomous.” • Conditions of the type [N1A]–[N3A] are common in variational construction of shadowing orbits, called the “no corners” conditions (see for example [15]). They imply that the minimizers of the hH s ,c1 (x, y) and hH s ,c (y, z) concatenate to a smooth trajectory of the Euler-Lagrange equation. We take advantage of this fact to prove the forcing equivalence between c1 and c2 using the defintion. • In the sequel, the sections S chosen will always be an affine manifold without boundary of co-dimension 1, and hence the perpendicular direction S ⊥ is well defined. We will sometimes abbreviate the phrase “the minimum is never reached outside of a relatively compact subset K of S” as “has interior minimum.” Recall that the cohomology classes c¯h (E), c¯µh1 (E) are chosen in the channels E of h and h1 , i.e. LF(λE h h) and LF(λh1 h1 ). Since h is non-simple, Proposition 15.11 implies the channel will pinch to a point, and c¯h (0) is uniquely chosen. The channel of h1 has positive width at E = 0, and c¯h (0) is at the boundary of this segment. In particular, since the channel for h1 is parallel to h⊥ 1 , we always have c¯µh1 (0) − c¯h (0) ⊥ h1 . (12.1) We will choose c¯h1 (0) sufficiently close to c¯h (0) according to the following proposition.

FORCING EQUIVALENCE BETWEEN KISSING CYLINDERS

135

Figure 12.1: Jump from one cylinder to another in the same homology.

Proposition 12.2. Suppose the slow mechanical system H s satisfies conditions [A0]–[A4] Then there is µ0 > 0 depending only on H s such that if 0 < µ < µ0 , we have: There exists a section S(0) = S(m(0), a(0), ω(0)), such that conditions [N1A]-[N3A] are satisfied for c¯h1 (0), c¯h (0) and section S(0). Moreover, the same holds with c¯h1 (0), c¯h (0) switched. Proof. We first show that [N1A]–[N3A] are satisfied when c1 = c2 = c¯h (0), then use perturbation arguments. Note that [N1A] is trivially satisfied when c1 = c2 . Recall that AH s (¯ ch (0)) = γh01 ∪ γh02 , and the curves γh01 and γh02 are tangent to a common direction at m0 , which we will call v0 . By the choice of h1 , v0 is not parallel to h1 . We now choose ω(0) = khh11 k , m(0) sufficiently close to 0, and a(0) = a > 0, such that S(m(0), ω(0), a(0)) intersects AH s (¯ ch (0)) transversally and is disjoint from 0 (see Figure 12.1). The Aubry set A˜H s (¯ ch (0)) supports a unique minimal measure, and therefore has a unique static class. For any x, z ∈ AH s (¯ ch (0)), the function hc¯h (0) (x, ·) + hc¯h (0) (·, z) reaches its global minimal at NH s (¯ ch (0)), which coincides with AH s (¯ ch (0)). Hence the minimum in  min hc¯h (0) (x, y) + hc¯h (0) (y, z) (12.2) y∈S(0)

is reached at S(0) ∩ AH s (¯ ch (0)), which consists of finitely many points, and hence is compactly contained in S(0). This implies [N2A] is satisfied for c1 = c2 = c¯h (0) along the section S(0). Moreover, for any y0 reaching the minimum in (12.2), according to the above analyis y0 also reaches the global minimum of hc¯h (0) (x, ·) + hc¯h (0) (·, z). Then using Proposition 6.2, the associated super-gradients p1 − c¯h (0) and −p2 + c¯h (0) (in [N3A]) satisfy p1 = p2 , and ∂p H s (y0 , p1 ) = ∂p H s (y0 , p2 ) is the velocity of the unique backward minimizer at y0 . To show that [N3A] holds, we only need to show ∂p H s (y0 , p1 ) 6= 0. This is the case because 0 is the only equilibrium in

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AH s (¯ ch (0)), and S(0) is disjoint from 0. We now perturb c1 away from c¯h (0) while keeping c2 = c¯h (0). We choose c1 = c¯µh1 (0) at the bottom of the h1 channel, then (12.1) and ω(0) = h1 means [N1A] is still satisfied. Proposition 7.4 implies that [N2A] and [N3A] are both robust under the perturbation of c1 , and therefore [N1A]–[N3A] holds if k¯ ch1 (0)−¯ ch (0)k is small enough. To prove the same with c¯h1 (0), c¯h (0) switched, we perform the same perturbation argument keeping c1 = c¯h (0) and c2 = c¯µh1 (0). We now perform one more step of perturbation by taking E > 0. Proposition 12.3. Suppose the slow mechanical system H s satisfies conditions [DR1c ] − [DR4c ], then there exists e > 0 such that the following hold. For each 0 ≤ E ≤ e, there exists a section S(E) := S(m(E), a(E), ω(E)), with S(E) → S(0) in Hausdorff distance, and the conditions [N1A]–[N3A] are satisfied for c¯µh1 (E), c¯h (E) at S(E). Moreover, the same conditions are satisfied with c¯h (E) and c¯µh1 (E) switched. Proof of Proposition 12.3. Since the Aubry sets AH s (¯ ch1 (0)) and AH s (¯ ch (0)) both have a unique static class, Proposition 7.4 implies that [N2A] and [N3A] are both robust under the perturbation of c1 and c2 . As a result, it suffices to construct a section S(E) = S(m(E), ω(E), a(E)) such that [N1A] holds, and S(E) → S(0) as E → 0. To do this, we choose ω(E) to be a unit vector orthogonal to c¯h1 (E) − c¯h (E), and by continuity of the functions c¯, ω(E) → ω(0) as E → 0. Since αH s (¯ ch1 (E)) = αH s (¯ ch (E)) = E, [N1A] is satisfied. We then choose m(E) = m(0) and a(E) = a(0). Clearly S(E) → S(0) in Hausdorff metric. The proposition follows.

12.2

VARIATIONAL PROBLEM FOR ORIGINAL COORDINATES

The original Hamiltonian H is reduced via coordinate change, and time change, to √ Hs (ϕs , I s , τ ) = K(I s ) − U (ϕs ) +  P (ϕs , I s , τ ), s 2 2 2 with kP √kC ≤ C1 , see Section 14.2. The system H is defined on T × R × R, but is  periodic in√τ , i.e. it is a periodic Tonelli Hamiltonian introduced in √ √ Section 6.1. Denote T = R/( Z), then Hs projects to T2 × R2 × T. We now define a variational problem for the perturbed system. First, we will adjust the cohomologies so that they have the same alpha function.

Lemma 12.4. Fix e > 0. There exist C > 0, and 0 > 0 depending only on K(I), kU kC 0 and e such that for any 3e ≤ E ≤ 2e 3 and 0 ≤  ≤ 0 , there exists

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FORCING EQUIVALENCE BETWEEN KISSING CYLINDERS

0 < E  < e such that µ αHs (¯ ch (E)) = αH ch1 (E  )), s (¯ 

√ |E − E  | ≤ C .

Proof. We note that αH s (ch (E)) = E = αH s (cµh1 (E). By Lemma 7.9,

αH s (ch (E)) − E , 

√

αH s (cµ (E)) − E ≤ C . h1 

√ The Lemma easily follows if we choose 0 such that e > 3C 0 . We define a section S  (E) = S(m(E), a(E), ω  (E)), by keeping m(E), a(E) the same as before, with ω  (E) to be a unit vector orthogonal to c¯h (E)−¯ cµh1 (E  ). √ √ It is natural to study the extended section S  × T ⊂ T2 × T, and condition [N1] becomes   √ c1 − c2 ⊥ S  × T, −αHs (c1 ) + αHs (c2 ) and the variational problem for [N2] is: For (x, 0) ∈ AHs (¯ ch (E)) and (z, 0) ∈ AHs (¯ cµh1 (E  )), consider the minimization min √

(y,t)∈S  × T

n hHs ,¯ch (E) (x, 0; y, t) + hHs ,¯cµh

1

o

(E  ) (y, t; z, 0)

.

(12.3)

We will not, however, study conditions [N1]–[N3] for Hs directly. √ Instead, we transform the cohomology c¯h (E) and c¯h1 (E  ), the section S  × T, and the variational problem (12.3) directly to the original system H . Recall that the original Hamiltonian H can be brought into a normal form system N . N is related to Hs via the coordinate ΦL , and an energy reduction, see section 14.2. Denote      
ϕ√ 1 √0 θ 1 −1 1 −1 ΦL (ϕ, τ ) = B , ΦL (θ, t) = B , τ/  0  t this is the angular component of the affine coordinate change ΦL (see (14.17)). The 3 × 3 matrix B is defined in (14.10). Given c¯ ∈ R2 , we define  √  ¯ c  T c = p0 + B0 , (12.4) −αHs (¯ c) where B0 is the first two rows of B, given precisely by k1T , k2T . According to Proposition 14.9, we have    √  c − p0  c¯ T =B = Φ∗L (¯ c, αH s (¯ c)), (12.5) −αH (c ) + H0 (p0 ) − αHs (¯ c) where Φ∗L is from (5.8), and coincides with the action component of ΦL . In

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particular, (12.4) is the first row of (12.5). Let us denote the cohomologies ch (E) and ch1 (E) the image of c¯ = c¯h (E) and c¯ = c¯eh1 (E) under (12.4). We define a section Σ = Σ(θ0 , a, Ω, l) ⊂ T2 × T (for the original Hamiltonian H : T2 × R2 × T → R) by Σ(θ0 , a, Ω, l) = {(θ0 + λΩ + lt, t) ∈ T2 × T : −a < λ < a, t ∈ T},

(12.6)

where Ω ∈ R3 , l ∈ Z3 . √ √ The section S(m, a, ω) ×  T ⊂ T2 ×  T is mapped under Φ1L to Σ(θ0 , a, Ω, l) with       0 m ω θ0 = B −1 ∈ T3 , Ω = B −1 ∈ R3 , l = B −1 0 ∈ Z3 . (12.7) 0 0 1 Note that for√(ci , αH (ci )) = Φ∗L (¯ ci , αHs (¯ ci )), i = 1, 2, and using (12.4) and Σ = Φ1L (S × T), we have     √ c¯1 − c¯2 c1 − c2 ⊥ S × T ⇐⇒ ⊥ Σ. (12.8) −αHs (¯ c1 ) + αHs (¯ c2 ) −αH (c1 ) + αH (c2 ) We say that c1 , c2 ∈ R2 have non-degnerate connection along a section Σ(θ, a, Ω, l), if the following conditions hold. [N1] We have 

 c1 − c2 ⊥ Σ. −αH (c1 ) + αH (c2 )

[N2] There exists a compact set K ⊂ Σ such that: For each (x, 0) ∈ AH (c1 ) and (z, 0) ∈ AH (c2 ) , the minimum in min {hH ,c1 (x, 0; y, t) + hH ,c2 (y, t; z, 0)}

(y,t)∈Σ

is never achieved outside of K. [N3] Assume that the minimum [N2] is reached at (y0 , t0 ), and let p1 −c1 and −p2 + c2 be any super-differentials of hc1 (x, 0; ·, t) and hc2 (·, t0 ; z, 0) respectively. Then (∂p H(y0 , p1 , t0 ), 1) · Σ⊥ , (∂p H(y0 , p2 , t0 ), 1) · Σ⊥ have the same signs, where Σ⊥ is a normal vector to Σ. Proposition 12.5. Consider ch (E), ch1 (E) be defined from c¯h (E), c¯h1 (E) using (12.4). Let E  be as in Lemma 12.4, and let the √ section S  (E) be as in (12.3),   and the section Σ (E) be obtained from S (E) × T via (12.7).

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Then for each 0 <  < 0 , we have ch (E), ch1 (E  ), Σ (E) satisfy [N1]–[N3]. Moreover, the same holds with ch (E) and ch1 (E  ) switched. The following statement is a direct application of Theorem 12.8, which holds for general Tonelli Hamiltonians. Proposition 12.6. Assume that the conclusions of Proposition 12.5 hold. In addition, assume that both AH (ch (E)) and AH (ch1 (E  )) admit a unique static class. Then  ch (E) a` ce,µ h1 (E ). Proof of Theorem 5.9. By Proposition 12.5, for the system H , which is a perturbation of H , all conditions of Proposition 12.6 are satisfied, except the condition of uniqueness of static classes. For this we consider again the residual condition that all rational minimal periodic orbits are hyperbolic, and have transversal homoclinic and heteroclinic intersections. Under this condition all cohomologies ch (E) and ce,µ h1 (E) have a unique static class (using the fact that they are of Aubry-Mather type). We prove Proposition 12.5 in Section 12.3 and Proposition 12.6 in Section 12.4.

12.3

SCALING LIMIT OF THE BARRIER FUNCTION

In this section we prove Proposition 12.5. Using the choice of the cohomology classes and the sections, together with (12.8), we get [N1] is satisfied for ch (E), ch1 (E  ), and Σ (E). It suffices to prove [N2] and [N3]. We will show that the variational problem in [N2] is a scaling limit of the variational problem (12.3). √ Proposition 12.7. The family of functions hH ,ch (E) (χE, , 0; ·, t)/  is uniformly semi-concave, and   √ θ lim sup hH ,ch (E) (χE, , 0; θ, t)/  − hH s ,¯ch (E) (xE , B0 ) = 0 t →0+ (θ,t)∈T2 ×T uniformly over (χE, , 0) ∈ AH (ch (E)),

(xE , 0) ∈ AH (¯ ch (E)).

The same conclusions apply to the barrier function hH ,ceh

1

(E) (·, ·; χ

E,

√ , 0)/ .

Proof. The uniform semi-concavity of barrier functions follows from Proposi-

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tion 7.6. Moreover, according to Proposition 14.9, item 2, the families of functions
√ hH ,ch (E) (χE, , 0; θ, t)/ , hHs ,¯ch (E) Φ1LS (χE, , 0); Φ1L (θ, t) share the same limit points as  → 0. Moreover, Proposition 14.9, item 3 implies Φ1L (χE, , 0) ∈ AHs (¯ ch (E)). The functions Hs form a uniform family of periodic Hamiltonians, and Hs → H s in C 2 (T2 × R2 × R). By construction, AH s (¯ ch (E)) has unique static class; therefore Proposition 7.4 applies, and for any (xE , 0) ∈ AH s (¯ ch (E)),

hHs ,¯ch (E) Φ1LS (χE, , 0); ϕ, τ ) → hH s ,¯ch (E) xE , 0; ϕ, τ uniformly. Finally, noticing            θ B Id 0 θ Id 0 θ B 0 √ B √ Φ1L (θ, t) = = → 0 t  T 0  t 0  k3 t 0

(12.9)

√ as  → 0, we obtain the desired limit. The case for hH ,ceh (E) (·, ·; χE, , 0)/  is 1 symmetric and we omit it. Proof of Proposition 12.5. Condition [N1] is satisfied by the choice of section. Observe that       √ Id 0 Id 0 Id 0 √ −1 (S  (E) × T) B0 Σ (E) = BΣ (E) = BB −1 0 0 0 0 0    √ Id 0 = (S  (E) × T) = S  (E). 0 0 Since S  (E) → S(E) as E → 0, combing with Proposition 12.3 we get [N2] still holds for 0 ≤  < 0 , with 0 depending only on H s . [N3] follows from the same limiting argument and we omit it.

12.4

THE JUMP MECHANISM

In this section we show that the [N1]–[N3] condition combined with uniqueness of static class imply c1 ` c2 , which is Proposition 12.6. The discussions here apply to general Tonelli Hamiltonians. Denote A˜H,c (x, s, y, t) = AH,c (x, s, y, t) + αH (c)(t − s). The subscript H may be omitted.

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141

Theorem 12.8. Assume that c1 , c2 and Σ satisfy the conditions [N1]–[N3], and in addition, AH (c1 ), AH (c2 ) both have unique static class, then the following hold. 1. (Interior Minimum) There exist N < N 0 , M < M 0 ∈ N and a relatively compact set K 0 ⊂ Σ, such that for any semi-concave function u on T2 , the minimum in v(z) := min{u(x) + A˜c1 (x, 0; y, t + n) + A˜c2 (y, t + n; z, n + m)},

(12.10)

where the minimum is taken in x ∈ T2 , (y, t) ∈ Σ, N ≤ n ≤ N 0 , M ≤ m ≤ M 0 , is never achieved for (y, t) ∈ / K 0. 2. (No Corner) Assume the minimum in (12.10) is achieved at (y, t) = (y0 , t0 ), (n, m) = (n0 , m0 ), and the minimizing curves are γ1 : [0, t0 + n0 ] → T2 and γ2 : [t0 + n0 , t0 + n0 + m0 ] → T2 . Then γ1 and γ2 connect to an orbit of the Euler-Lagrange equation, i.e. γ˙ 1 (t0 + n0 ) = γ˙ 2 (t0 + n0 ). 3. (Connecting Orbits) The function v is semi-concave, and its associated pseudograph satisfies [ Gc2 ,v ⊂ φt Gc1 ,u . 0≤t≤N 0 +M 0

As a consequence, c1 ` c2 . Remark 12.9. We only need item 3 of the theorem for our purpose, the first two items are stated to illuminate the idea. Item 1 can be seen as a finite time version of the variational problem in [N2]. It holds for sufficiently large M , N , due to uniform convergence of the Lax-Oleinik semi-group. Item 2 implies the minimizers concatenate to a real orbit of the system, which is crucial in proving item 3. Lemma 12.10. 1. Let u be a continuous function on T2 . The limit lim

lim

min

N →∞ N 0 →∞ x∈T2 , N ≤n≤N 0

{u(x) + A˜c (x, 0; y, t + n)} = min2 {u(x) + hc (x, 0; y, t)} x∈T

is uniform in u and (y, t).

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2. The limit lim

lim

min

N →∞ N 0 →∞ N ≤n≤N 0

A˜c (y, t; z, n) = hc (y, t; z, 0)

is uniform in y, t, z. Proof. The proof of the first item is similar to the proof of Proposition 6.3 of [11] with some auxiliary facts proven in Appendix A there. The proof of the second item is similar to that of Proposition 6.1 from [11]. In both cases the action function, defined in (2.4) and (6.1) of [11], is restricted an to have integer time increment. For non-integer time increments the same argument applies. Using the representation formula (Proposition 6.16) and Lemma 7.5, we have the following characterization of the barrier functions. Lemma 12.11. Assume that A(c) has only one static class. For each point (y, t) ∈ T2 × T and each z ∈ T2 1. There exist x0 ∈ T2 and x1 ∈ A(c) such that min {u(x) + hc (x, 0; y, t)} = u(x0 ) + hc (x0 , 0; x1 , 0) + hc (x1 , 0; y, t).

x∈T2

2. There exists z1 ∈ A(c) such that hc (y, t; z, 0) = hc (y, t; z1 , 0) + hc (z1 , 0; z, 0). Proof of Theorem 12.8. According to Lemma 12.10, (12.10) converges uniformly as N, M → ∞ to min {u(x) + hc1 (x, 0; y, t) + hc2 (y, t; z, 0)}, x,y,t

which is equal to min {u(x0 ) + hc1 (x0 , 0; x1 , 0) + hc1 (x1 , 0; y, t) + hc2 (y, t; z1 , 0) + hc2 (z1 , 0; z, 0)} (y,t)

= min {const + hc1 (x1 , 0; y, t) + hc2 (y, t; z1 , 0) + hc2 (z1 , 0; z, 0)}. (y,t)

by Lemma 12.11. Since the above variational problem has a interior minimum due to condition N2, by uniform convergence, the finite-time variational problem (12.10) also has an interior minimum for sufficiently large N, M . We now prove the second conclusion. Let γ1 and γ2 be the minimizers for A˜c1 (x0 , 0; y0 , t0 + n0 ) and A˜c2 (y0 , t0 + n0 ; z, n0 + m0 ), and let p1 and p2 be the associated momentum. We will show that p1 (t0 + n0 ) = p2 (t0 + n0 ),

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which implies γ˙ 1 (t0 + n0 ) = γ˙ 2 (t0 + n0 ). To abbreviate notations, we write p01 = p1 (t0 + n0 ) and p02 = p2 (t0 + n0 ) for the rest of the proof. Note that u1 (x0 ) + A˜c1 (x0 , 0; y0 , t0 + n0 ) = min2 {u1 (x) + A˜c1 (x, 0; y0 , t0 + n0 )}. x∈T

By semi-concavity, the function u1 (x) + A˜c1 (x, 0; y0 , t0 + n0 ) is differentiable at x0 and the derivative vanishes. By Proposition 6.2 part 3, dx u(x0 ) = p1 (0) − c1 .

(12.11)

By a similar reasoning, we have A˜c1 (x0 , 0; y0 , t0 + n0 ) + A˜c2 (y0 , t0 + n0 ; z, n0 + m0 ) = min {A˜c (x0 , 0; y, t + n0 ) + A˜c (y, t + n0 ; z, n0 + m0 )}. (y,t)∈Σ

1

(12.12)

2

By Proposition 6.2, we know (p01 − c1 , αH (c1 ) − H(y0 , t0 , p10 )),

(−p02 + c1 , −αH (c2 ) + H(y0 , t0 , p20 ))

are super-differentials of A˜c1 (x0 , 0; y0 , t0 + n0 ) and A˜c2 (y0 , t0 + n0 ; z, n0 + m0 ) at (y0 , t0 + n0 ), since the minimum in (12.12) is obtained inside of Σ, we have (p01 − p02 , −H(y0 , t0 , p01 + H(y0 , t0 , p02 )) + (−c1 + c2 , αH (c1 ) − αH (c2 )) is orthogonal to Σ. Since the second term is orthogonal to Σ by [N1], we obtain (p01 − p02 , −H(y0 , t0 , p01 ) + H(y0 , t0 , p02 )) ∈ RΣ⊥ . We proceed to prove p01 = p02 . Write Σ⊥ = (v, w) ∈ R2 × R, then there exists λ0 such that p01 − p02 = λ0 v,

−H(y0 , t0 , p01 ) + H(y0 , t0 , p02 ) = λ0 w.

Define f (λ) = H(y0 , t0 , p01 + λv) − H(y0 , t0 , p1 ) + λw; then f (λ0 ) = 0. Since f (λ) is strictly convex on R, there are at most two solutions to f (λ) = 0, one of them is λ = 0. Suppose λ0 6= 0, by strict convexity again, f 0 (0) and f 0 (λ0 ) must have different signs. Since f 0 (0) = (∂p H(y0 , t0 , p01 ), 1) · (v, w),

f 0 (λ0 ) = (∂p H(y0 , t0 , p02 ), 1) · (v, w),

have the same signs from [N3], we get a contradiction. As a result, 0 is the only solution to f (λ) = 0, indicating p01 = p02 . Moreover, this implies p01 = p02 is uniquely defined and the functions A˜c1 (x0 , 0; ·, ·), Ac2 (·, ·; z0 , n0 + m0 ) are

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differentiable at (y0 , t0 + n0 ). As a consequence, (γ1 , p1 ) and (γ2 , p2 ) connect as a solution of the Hamiltonian flow. Using (12.11), we have φn0 +m0 (x0 , dux (x0 ) + c1 ) = (z, p2 (n0 + m0 )). Note that p2 (n0 + m0 ) − c2 is a super-differential to v at z. If v is differentiable at z, then p2 = dv(z) + c2 . This implies [ Gc2 ,v ⊂ φt Gc1 ,u 0≤t≤N 0 +M 0

and the forcing relation.

Part IV

Supplementary topics

Chapter Thirteen Generic properties of mechanical systems on the two-torus This chapter is devoted to proving generic properties of the minimizing orbits of the slow mechanical system. In the first three sections, we prove Theorem 4.5 concerning non-critical but bounded energy. In Section 13.4 we prove Proposition 4.6 concerning the very high energy, and in Section 13.5 we prove Proposition 4.7 concerning the critical energy. The proof of Theorem 4.5 consists of three steps. 1. In Section 13.1, we prove a Kupka-Smale-like theorem about non-degeneracy of periodic orbits. For a fixed energy surface, generically, all periodic orbits are non-degenerate. This fails for an interval of energies. We show that while degenerate periodic orbits exist, there are only finitely many of them. Moreover, there could be only a particular type of bifurcation for any family of periodic orbits crossing a degeneracy. 2. In Section 13.2, we show that a non-degenerate locally minimal orbit is always hyperbolic. Using step 1, we show that for each energy, the globally minimal orbits is chosen from a finite family of hyperbolic locally minimal orbits. 3. In Section 13.3, we finish the proof by proving the finite local families obtained from step 2 are “in general position,” and therefore there are at most two global minimizers for each energy.

13.1

GENERIC PROPERTIES OF PERIODIC ORBITS

We simplify notations and drop the supscript “s” from the notation of the slow mechanical system. Moreover, we treat U as a parameter, and write H U (ϕ, I) = K(I) − U (ϕ),

ϕ ∈ T2 , I ∈ R2 , U ∈ C r (T2 ).

(13.1)

We shall use U as an infinite-dimensional parameter. Denote by G r = C r (T2 ) the space of potentials, x denotes (ϕ, I), W denotes T2 × R2 , and either φU t or Φ(·, t, U ) denotes the flow of (13.1). We will use χU (x) = (∂K, ∂U )(x) to denote the Hamiltonian vector field of H U and use SE to denote the energy surface {H U = E}. We may drop the superscript U when there is no confusion. By the invariance of the energy surface, the differential map Dx φU t defines a

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map Dx φU SH(x) . t (x) : Tx SH(x) → TφU t (x) Since the vector field χ(x) is invariant under the flow, Dx φ induces a factor map ¯ x φU (x) : Tx SH(x) /Rχ(x) → TφU (x) SH(x) /Rχ(φt (x)). D t t Given U0 ∈ G r , x0 ∈ W and t0 ∈ R, let V = V (x0 ) × (t0 − a, t0 + a) × V (U0 ) 0 be a neighborhood of (x0 , t0 , U0 ), V (x0 , t0 ) of (x0 , t0 ), and V (φU t0 (x0 )) a neighU0 borhood of φt0 (x0 ), such that

U0 φU t (x) ∈ V (φt0 (x0 )),

(x, t, U ) ∈ V.

By fixing the local coordinates on V (x0 ) and V (φU t0 (xx )), we define e x Φ : V → Sp(2), D e x Φ(x, t, U ) is the 2 × 2 symplectic matrix associated to D ¯ x ϕU (x) under where D t the local coordinates. The definition depends on the choice of coordinates. 0 Let {φU t (x0 )} be a periodic orbit with minimal period t0 . The periodic e x Φ(x0 , t0 , U0 ). orbit is non-degenerate if and only if 1 is not an eigenvalue of D Furthermore, a degenerate periodic orbit necessarily falls into one of two types: e x Φ(x0 , t0 , U0 ) = Id, 1. A degenerate periodic orbit (x0 , t0 , U0 ) is of type I if D the identity matrix; e x Φ(x0 , t0 , U0 ) is conjugate to the matrix [1, µ; 0, 1] for 2. It is of type II if D µ 6= 0. Denote  1 N= 0

  µ : µ ∈ R \ {0} , 1

O(N ) = {BAB −1 : A ∈ N, B ∈ Sp(2)}.

e x Φ(x0 , t0 , U0 ) ∈ O(N ). Then (x0 , t0 , U0 ) is of type II if and only if D Lemma 13.1. The set O(N ) is a 2-dimensional submanifold of Sp(2). Proof. Any matrix  a c

in O(N ) can be expressed by      b 1 µ d −b 1 − acµ a2 µ = , d 0 1 −c a −c2 µ 1 − acµ

where ad − bc = 1 and µ 6= 0. Writing α = a2 µ and β = acµ, we can express

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149

any matrix in O(N ) by 

1−β 1 − β 2 /α

 α . 1−β

(13.2)

The standard Kupka-Smale theorem (see [69, 71]) no longer holds for an interval of energies. Generically, periodic orbits appear in one-parameter families and may contain degenerate ones. However, while degenerate periodic orbits may appear, generically, a family of periodic orbits crosses the degeneracy “transversally”. This is made precise in the following theorem. Theorem 13.2. Suppose r ≥ 3. There exists a residual subset G 0 of G r , such that for all U ∈ G 0 , the following hold: 1. The set of periodic orbits for φU t form a submanifold of dimension 2. Since a periodic orbit itself is a 1-dimensional manifold, distinct periodic orbits form one-parameter families. 2. There are no degenerate periodic orbits of type I. 3. The set of periodic orbits of type II forms a 1-dimensional manifold. Factoring out the flow direction, the set of type II degenerate orbits are isolated. 4. For U0 ∈ G r , let ΛU0 ⊂ W × R+ denote the set of periodic orbits for φU t , U0 0 and ΛU denote the set of type II degenerate ones. Then for any N ⊂ Λ 0 (x0 , t0 ) ∈ ΛU N , let V (x0 , t0 ) be a neighborhood of (x0 , t0 ). Then e x Φ|U =U : ΛU0 ∩ V (x0 , t0 ) → Sp(2) D 0 is transversal to O(N ) ⊂ Sp(2). Remark 13.3. Statement 4 of the theorem can be interpreted in the following way. Let A(λ) be the differential of the Poincar´e return map on associated to a family of periodic orbits. Then if A(λ0 ) ∈ O(N ), then the tangent vector A0 (λ0 ) is transversal to O(N ). We can improve the set G 0 to an open and dense set, if there is a lower and upper bound on the minimal period. Corollary 13.4. 1. Given 0 < T0 < T1 , there exists an open and dense subset G 0 ⊂ G r , such that the set of periodic orbits with minimal period in [T0 , T1 ] satisfies the conclusions of Theorem 13.2. 2. For any U0 ∈ G 0 , there are at most finitely many type II degenerate periodic orbits. Moreover, there exists a neighborhood V (U0 ) of U0 , such that the set of type II degenerate periodic orbits depends smoothly on U . (This means the number of such periodic orbits is constant on V (U0 ), and each periodic orbit depends smoothly on U .) We define F : W × R+ × G r → W × W,

(13.3)

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F (x, t, U ) = (x, Φ(x, t, U )). F is a C r−1 -map of Banach manifolds. Define the diagonal set by ∆ = {(x, x)} ⊂ 0 W ×W . Then {φU t (x0 )} is a period orbit of period t0 if and only if (x0 , t0 , U0 ) ∈ −1 F ∆. Proposition 13.5. Assume that (x0 , t0 , U0 ) ∈ F −1 ∆ or, equivalently, x0 is periodic orbit of period t0 for H U and that t0 is the minimal period, then there exists a neighborhood V of (x0 , t0 , U0 ) such that the map ⊥ dπ∆ D(x,t,U ) F : T(x,t,U ) (W × R+ × G r ) → TF (x,t,U ) (W × W )/T ∆ ⊥ has co-rank 1 for each (x, t, U ) ∈ V, where dπ∆ T (W × W ) → T (W × W )/T ∆ is the standard projection along T ∆.

Remark 13.6. If the aforementioned map has full rank, then the map is called transversal to ∆ at (x0 , t0 , U0 ). However, the transversality condition fails for our map. Given δU ∈ G r , the directional derivative DU Φ · δU is ∂ ∂ |=0 Φ(x, t, U + δU ). The differential DU Φ then defines a TΦ(x,t,U ) W . The following hold for this differential map:

defined as follows map from T G r to

Lemma 13.7. [69] Assume that there exists  > 0 and t0 ∈ (0, τ ) such that the orbit of x has no self-intersection for the time interval (t0 − , t0 + ); then the map DU Φ(x, τ, U ) : G r → TΦ(x,τ,U ) W generates a subspace orthogonal to the gradient ∇H U (Φ(x, τ, U )) and the Hamiltonian vector field χU (Φ(x, τ, U )) of H U . Proof. We refer to Lemmas 16 and 17 of [69]. We note that while the proof was written for a periodic orbit of minimal period τ , the proof holds for non-selfintersecting orbit. Proof of Proposition 13.5. We note that if {φU t (x)} is a periodic orbit of minimal 0 0 period τ , then the orbit {φU t (x )} satisfies the assumptions of Lemma 13.7. It follows that the matrix   ⊥ dπ∆ ◦ DU F = Dx Φ − I Dt Φ DU Φ has co-rank 1, since the last two components generates the subspace Image (DU Φ) + RχU , which is a subspace complementary to ∇H U . Proposition 13.5 allows us to apply the constant rank theorem in Banach spaces.

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151

Proposition 13.8. The set F −1 ∆ as a subset of a Banach space is a subman−1 ifold of codimension 3. If r ≥ 3, then for generic U ∈ G r , F −1 ∆ ∩ πU {U } is a 2-dimensional manifold. Proof. We note that the kernel and cokernel of the map dπ ◦ DU F has finite codimension; hence the constant rank theorem (see Theorem 2.5.15 of [1]) applies. As a consequence, we may assume that locally, ∆ = ∆1 × (−a, a) and that the map π1 ◦ F has full rank. Since the dimension of ∆1 is 3, F −1 ∆ is a submanifold of codimension 2n − 1. The mapping πU : F −1 ∆ → G r is Fredholm with index 2. Sard’s theorem applies if r ≥ 3, and at every regular value of πU , the set F −1 (∆) ∩ π −1 {U } is a submanifold of dimension 2. Denote Λ = F −1 ∆. On a neighborhood V of each (x0 , t0 , U0 ) ∈ Λ, we define the map e x Φ : Λ ∩ V → Sp(2), D e x Φ(x, t, U ) = Dφ e U D (13.4) t (x). First we refer to the following lemma by Oliveira: Lemma 13.9 (Theorem 18 of [69]). For each (x0 , t0 , U0 ) ∈ Λ such that t0 is the minimal period, let Ge be the set of tangent vectors in T(x0 ,t0 ,U0 ) Λ of the form (0, 0, V ). Then the map e x Φ : Ge → T e DU D Dx Φ(x0 ,t0 ,U0 ) Sp(2) has full rank. Corollary 13.10. The map (13.4) is transversal to the submanifold {Id} and O(N ) of Sp(2). Denote e x Φ)−1 ({Id}) and ΛN = Λ ∩ D e x Φ)−1 (O(N )). ΛId = Λ ∩ D Note that the expression is well defined because both preimages are defined independent of local coordinate changes. Proof of Theorem 13.2. The first statement of the theorem follows from Proposition 13.8. As the subset {Id} has codimension 3 in Sp(2), ΛId has codimension 3 in Λ, and hence has codimension 6 in W × R+ × G r . By Sard’s lemma, for a generic −1 U ∈ G r , the set ΛId ∩ πU is empty. This proves the second statement of the theorem. Since the set O(N ) has codimension 1, ΛN has codimension 1 in Λ, and hence has codimension 4 in W × R+ × G r . As a consequence, generically, the −1 set ΛU N = ΛN ∩ πU has dimension 1. This proves the third statement. −1 U0 0 Fix U0 ∈ G 0 , the set ΛU0 = Λ ∩ πU (U0 ) has dimension 2, while ΛU N ⊂ Λ U0 has dimension 1. It follows that at any (x0 , t0 ) ∈ ΛN , there exists a tangent

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vector 0 (δx, δt) ∈ T(x0 ,t0 ) ΛU0 \ T(x0 ,t0 ) ΛU N

such that (δx, δt, 0) ∈ T(x0 ,t0 ,U0 ) Λ \ T(x0 ,t0 ,U0 ) ΛN . It follows that e x Φ|U =U (x0 , t0 ) = D(x,t,U ) D e x Φ(x0 , t0 , U0 ) · (δx, δt, 0) D(x,t) D 0 is not tangent to O(N ). Since O(N ) has codimension 1, this implies that the e x Φ|U =U is transversal to O(N ). This proves the fourth statement. map D 0 Proof of Corollary 13.4. If a potential U ∈ G 0 , then by Theorem 13.2 conditions 1–4 are satisfied. In particular, all periodic orbits are either non-degenerate or degenerate satisfying conditions 3 and 4. Non-degenerate periodic orbits of period bounded both from zero and infinity form a compact set. Therefore, they stay non-degenerate for all potential C r -close to U . By condition 3 degenerate periodic orbits are isolated. This implies that there are finitely many of them. Condition 4 is a transversality condition, which is C r open for each degenerate orbit. Fix U ∈ G 0 as in Corollary 13.4. It follows that periodic orbits of φU t form one-parameter families. We now discuss the generic bifurcation of such a family at a degenerate periodic orbit. Proposition 13.11. Let (xλ , tλ ) be a family of periodic orbits such that (x0 , λ0 ) e x φU is degenerate. On one side of λ = λ0 , say λ > λ0 , the matrix D tλ (xλ ) has a pair of distinct real eigenvalues; if λ < λ0 , it has a pair of complex eigenvalues. e x φU Proof. Write A(λ) = D tλ (xλ ) for short. By choosing a proper local coordinates we may assume that A(λ0 ) = [1, µ; 0, 1]. The tangent space to Sp(2) at [1, µ; 0, 1] is given by the set of traceless matrices [a, b; c, −a]. Using (13.2), we have a basis of the tangent space to O(N ) at [1, µ; 0, 1] given by     0 1 −1 0 and . −β 2 /α2 0 −2β/α −1 An orthogonal matrix to this space, using the inner product tr(AT B), is given by [0, 0; 1, 0]. As a consequence, a matrix [a, b; c, −a] is transversal to O(N ) if and only if c 6= 0. The eigenvalues of the matrix   1 + ah µ + bh ch 1 − ah are given by λ = 1 ±

p

a2 h2 − bch2 − µch. Using µ 6= 0 and c 6= 0 we obtain

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GENERIC PROPERTIES OF MECHANICAL SYSTEMS ON THE TWO-TORUS

that a2 h2 − bch2 − µch changes sign at h = 0. This proves our proposition.

13.2

GENERIC PROPERTIES OF MINIMAL ORBITS

In this section, we analyze properties of families of minimizing orbits. It has been known since Poincar´e that a non-degenerate minimizing geodesic on a surface is hyperbolic. (We give a proof in Proposition 13.12.) However, we have shown in the previous section that degenerate ones do exist. The main idea to avoid degenerate minimizing orbits is the following: When one extends a family of hyperbolic minimal orbits, the family must terminate at a degenerate orbit of type II. We then can “slide” different families against each other so that the degenerate orbit is never the shortest. Let dE denote the metric derived from the Riemannian metric gE (ϕ, v) = 2(E +U (ϕ))K −1 (v). We define the arclength of any continuous curve γ : [t, s] → T2 by N −1 X lE (γ) = sup dE (γ(ti ), γ(ti+1 )), i=0 −1 where the supremum is taken over all partitions {[ti , ti+1 ]}N i=0 of [t, s]. A curve γ is called rectifiable if lE (γ) is finite. A curve γ : [a, b] → T2 is called piecewise regular, if it is piecewise C 1 and γ(t) ˙ 6= 0 for all t ∈ [a, b]. A piecewise regular curve is always rectifiable. We write lE (h) = inf lE (ξ), E ξ∈Ch

where ChE denote the set of all piecewise regular closed curves with homology h ∈ H1 (T2 , Z). A curve realizing the infimum is the shortest geodesic curve in the homology h, which we will also refer to as a global (gE , h)-minimizer. It is well known that for any E > − minϕ U (ϕ), a global gE -minimizer is a closed gE -geodesic. Hence, it corresponds to a periodic orbit of the Hamiltonian flow. It will also be convenient to consider the local minimizers. Letting V ⊂ T2 be an open set, a closed continuous curve γ : [a, b] → V is a local minimizer if lE (γ) =

inf

E ,ξ⊂V ¯ ξ∈Ch

lE (ξ).

(13.5)

Since γ is compact, there is an open set V1 such that γ ⊂ V1 ⊂ V1 ⊂ V2 . Then by modifying the metric outside of V1 , we can ensure that γ is a global minimizer of the new metric. Proposition 13.12. Assume that γ is a (gE , h) (local) minimizer, and assume that the associated Hamiltonian periodic orbit η is nondegenerate. Then η is

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hyperbolic. Proof of Proposition 13.12. A non-degenerate periodic orbit η : [0, T ] → T2 × R2 either is hyperbolic or the associated Poincar´e map has eigenvalues on the unit circle excluding {1}. We now invoke the Green bundles introduced in Section 6.7. According to Proposition 6.19, the tangent map of the flow preserves one of the (2-dimensional) Green bundles; hence the Poincar´e map preserves a 1-dimensional subspace at the periodic orbit. But this is impossible if the differential of the map has a complex eigenvalue. ¯ there exists an open and dense subset Theorem 13.13. Given 0 < e < E, G 0 of G r , such that for each U ∈ G 0 , the Hamiltonian H(ϕ, I) = K(I) − U (ϕ) satisfies the following statements. There exist finitely many smooth families of local minimizers ξjE ,

aj − σ ≤ E ≤ bj + σ, j = 1, · · · , N,

and σ > 0, with the following properties. 1. All ξjE for aj − σ ≤ E ≤ bj + σ are hyperbolic. S ¯ 2. j [aj , bj ] ⊃ [E0 , E]. ¯ any global minimizer is contained in the set of all ξ E ’s 3. For each E0 ≤ E ≤ E, j such that E ∈ [aj , bj ]. Proof of Theorem 13.13 occupies the rest of this section. Lemma 13.14. Assume that γE0 is a hyperbolic local (ρE0 , h)-minimizer. The following hold. 1. There exists a neighborhood V of γE0 , such that γE0 is the unique local (gE , h)minimizer on V . 2. There exists δ > 0 such that for any U 0 ∈ C r (T2 ) with kU − U 0 kC 2 ≤ δ and |E 0 − E0 | ≤ δ, the Hamiltonian H 0 (ϕ, I) = K(I) − U 0 (ϕ) admits a hyperbolic local minimizer in V . 3. There exists δ > 0 and a smooth family γE ⊂ V , E0 − δ ≤ E ≤ E0 + δ, such that each of them is a hyperbolic local minimizer. Proof. Let ηE0 denote the Hamiltonian orbit of γE0 . Inverse function theorem implies that if ηE0 is hyperbolic, then it is locally unique, and it uniquely extends to hyperbolic periodic orbit ηE,U 0 if E is close to E0 and U 0 is close to U . Let γE be the projection of ηE , then they must be the unique local minimizers. We now use the information obtained to classify the set of global minimizers. • Consider the Hamiltonian H(ϕ, I) = K(I)−U (ϕ)+minϕ U (ϕ). Since we only consider periodic orbits that correspond to rational homology h, the associ¯ ated closed geodesic have finite length (in the Jacobi metric). For 0 < E0 < E, the time change connecting the geodesic flow to the Hamiltonian system is

GENERIC PROPERTIES OF MECHANICAL SYSTEMS ON THE TWO-TORUS

•

•

•

• •

155

uniformly bounded. As a result, all the global minimizers we consider have uniformed bounded period. Corollary 13.4 applies. In particular, there are only finitely many degenerate periodic orbits and the number of them is constant under small perturbation of U . It follows from the previous item that all but finitely many global minimizers are non-degenerate. The non-degenerate minimizers are hyperbolic by Proposition 13.12. Since a global minimizer is always a local minimizer, using Lemma 13.14, it extends to a smooth one parameter family of local minimizers. The extension can be continued until either the orbit is nolonger locally minimizing, or if the orbit becomes degenerate. Once a local minimizer becomes degenerate, this family can no longer be extended as local minimizers as by Proposition 13.11, it must bifurcate to a periodic orbit of complex eigenvalues. It is well known that for a fixed energy, any two global minimizers do not cross (see for example, [66]). We can locally extend the global minimizers for a small interval of energy without them crossing. There are at most finitely many families of local minimizers, because they are isolated and do not accumulate (Lemma 13.14). There are at most finitely many energies on which the global minimizer may be a degenerate periodic orbit. We have proved the following statement.

¯ there exists an open and dense subset Proposition 13.15. Given 0 < e < E, G 0 of G r , such that for each U ∈ G 0 , the following hold for H(ϕ, I) = K(I) − U (ϕ) + minϕ U (ϕ). 1. There are at most finitely many (maybe none) isolated global minimizers E ,d γj j , j = 1, · · · , M . 2. There are finitely many smooth families of local minimizers γjE ,

a ¯j ≤ E ≤ ¯bj , j = 1, · · · , N,

¯ ⊃ S[¯ with [e, E] aj , ¯bj ], such that γjE are hyperbolic for a ¯j < E < ¯bj . The set E γ¯j for E = a ¯j , ¯bj may be hyperbolic or degenerate. S 3. For a fixed energy surface E, the sets {γjE,d } and a¯j ≤E≤¯bj γjE are pairwise disjoint. ¯ the global minimizer is chosen among the set of all γ cj ,d 4. For each e ≤ E ≤ E, j with E = cj , or one of the local minimizers γjE with E ∈ [¯ aj , ¯bj ]. Proof of Theorem 13.13. We first show that the set of potentials U satisfying the conclusion of Theorem 13.13 is open. By Lemma 13.14, the family of local minimizers persists under small perturbations of the potential U . It suffices to show that for sufficiently small perturbation of U satisfying the conclusion, the global minimizer is still taken at one of the local families. Assume, by

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contradiction, that there is a sequence Un approaching U , and for each Hn = K − Un , there is some global minimizer γnEn not from these families. By picking a subsequence, we can assume that it converges to a closed curve γ∗ , which belong one of the local families γjE . Using local uniqueness from Lemma 13.14, γnEn must belong to one of the local families as well. This is a contradiction. To prove density, it suffices to prove that for a potential U satisfying the conclusion of Proposition 13.15, we can make an arbitrarily small perturbation, such that there are no degenerate global minimizers. Our strategy is to eliminate the degenerate global minimizers one by one d using a sequence of perturbations. Suppose γE is an isolated degenerate minimizer. Then by ([28], proof of Theorem D, page 40-42), there is a perturbation d d δU with the property that: δU ≥ 0, U |γE = 0, such that γE is still a global d minimizer, but the associated Hamiltonian orbit ηE becomes hyperbolic (i.e. the d perturbation keeps the orbit ηE intact but changes it’s linearization). Repeating this process, we can remove all isolated degenerate minimizers. d Assume the previous step has been completed, suppose γE is the terminal E d ¯ point of one of the local families γj : E ∈ [¯ aj , bj ], let’s say γE = γjE (¯bj ). E ¯ Then since γj cannot not be extended to E > bj , the global minimizer for E ∈ (b¯j , ¯bj +δ) must be contained in a different local branch, say γiE : E ∈ [¯ ai , b¯i ]. In particular, γiE (¯bj ) is another global minimizer. E Let V be a neighborhood of γj j , such that V¯ is disjoint from the set of other global minimizers with the same energy. For δ > 0 we define Uδ : T2 → R such E that Uδ |γj j = δ and supp Uδ ⊂ V . Let Hδ = K − U − Uδ , and let lE,δ be the perturbed length function. We have Z q E E lEj ,δ (γj j ) = E 2(Ej + U + δ)K > lEj (γj j ) = lEj (h) = lEj ,δ (h). γj

j

E

As a consequence, γj j is no longer a global minimizer for the perturbed system. Moreover, for sufficiently small δ, no new degenerate global minimizer can be created. Hence the perturbation has decreased the number of degenerate global minimizers strictly. By repeating this process finitely many times, we can eliminate all degenerate global minimizers.

13.3

NON-DEGENERACY AT HIGH-ENERGY

In this section we complete the proof of Theorem 4.5. This amounts to proving that finite local families of local minimizers, obtained from the previous section, are “in general position”. The proof is similar to that of Proposition 3.2 (see Section 9.7). We assume that the potential U0 ∈ G r satisfies the conclusions of Theorem 13.13. Let γjE,U denote the branches of local minimizers from Theo-

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rem 13.13, where we have made the dependence on U explicit. There exists an neighborhood V (U0 ) of U0 , such that the local branches γjE,U are defined for E ∈ [aj − σ/2, bj + σ/2] and U ∈ V (U0 ). Define a set of functions fj (E, U ) = lE (γjE,U ).

fj : [aj − σ/2, bj + σ/2] × V (U0 ) → R, Then γiE,U is a global minimizer if and only if

fi (E, U ) = fmin (E, U ) := min fj (E, U ), j

where the minimum is taken over all j’s such that E ∈ [aj , bj ]. The following proposition implies Theorem 4.5. Proposition 13.16. There exists an open and dense subset V 0 of V (U0 ) such that for every U ∈ V 0 , the following hold: ¯ there are at most two j’s such that fj (E, U ) = fmin (E, U ). 1. For E ∈ [E0 , E], ¯ such that there are two j’s with 2. There are at most finitely many E ∈ [E0 , E] fj (E, U ) = fmin (E, U ). ¯ and j1 , j2 such that fj (E, U ) = fj (E, U ) = fmin (E, U ), 3. For any E ∈ [E0 , E] 1 2 we have ∂ ∂ fj1 (E, U ) 6= fj (E, U ). ∂E ∂E 2 Proof. We claim that it suffices to prove the theorem under the additional assumption that all functions fj ’s are defined on the same interval (a, b) with ¯ into finitely many fmin (E, U ) = minj fj (E, U ). Indeed, we may partition [E0 , E] intervals, on which the number of local branches is constant, and prove proposition on each interval. We define a map f = (f1 , · · · , fN ) : (a, b) × V (U0 ) → RN , and subsets ∆i1 ,i2 ,i3 = {(x1 , · · · , xn ); xi1 = xi2 = xi3 }, ∆i1 ,i2 = {(x1 , · · · , xn ); xi1 = xi2 },

1 ≤ i1 < i2 < i3 ≤ N, 1 ≤ i1 < i2 ≤ N

of RN × RN . We also write f U (E) = f (E, U ). The following two claims imply our proposition: 1. For an open and dense set of U ∈ V (U0 ), for all 1 ≤ i1 < i2 < i3 ≤ N , the set (f U )−1 ∆i1 ,i2 ,i3 is empty. 2. For an open and dense set of U ∈ V (U0 ), and all 1 ≤ i1 < i2 ≤ N , the map f U : (a, b) → RN is transversal to the submanifold ∆i1 ,i2 .

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Indeed, the first claim implies the first statement of our proposition. It follows from our second claim that there are at most finitely many points in (f U )−1 ∆i1 ,i2 , which implies the second statement. Furthermore, using the second claim, we have for any E ∈ (f U )−1 ∆i1 ,i2 , the subspace (Df U (E))R is transversal to T ∆i1 ,i2 . This implies the third statement of our proposition. For a fixed energy E and (v1 , · · · , vN ) ∈ RN , let δU : T2 → R be such that δU (ϕ) = vj on an open neighborhood of γjE for each j = 1, · · · N . Let lE, and γjE, denote the arclength and local minimizer corresponding to the potential U + δU . For any ϕ in a neighborhood of γjE , we have E + U (ϕ) + δU (ϕ) = E + U (ϕ) + vj ; E+v

j hence for sufficiently small  > 0, ξjE, = ξj . The directional derivative d d ∂ E+vj DU f (E, U ) · δU = lE, (ξjE, ) = lE+vj (ξj )= fj (E, U )vj . d =0 d =0 ∂E

It follows from a direct computation that each fj is strictly increasing in E and the derivative in E never vanishes. As a consequence, we can choose (v1 , · · · , vN ) in such a way that DU f (E, U ) · δU takes any given vector in RN . This implies the map DU f : (a, b) × T V (U0 ) → RN has full rank at any (E, U ). As a consequence, f is transversal to any ∆i1 ,i2 ,i3 and ∆i1 ,i2 . Using Sard’s lemma, we obtain that for a generic U , the image of f U is disjoint from ∆i1 ,i2 ,i3 and f U is transversal to ∆i1 ,i2 .

13.4

UNIQUE HYPERBOLIC MINIMIZER AT VERY HIGH ENERGY

In this section we prove Proposition 4.6, namely for the mechanical system H s (ϕ, I) = K(I) − U (ϕ), the shortest loop in the homology h ∈ H1 (T2 , Z) for Jacobi metric gE (ϕ, v) = 2(E +U (ϕ))K −1 (v), where K −1 (v) is the Legendre dual of K. Since the metrics √ gE (ϕ, v) = 2(1 + E −1 U (ϕ))K −1 ( Ev), g¯E (ϕ, v) = 2(1 + E −1 U (ϕ))K −1 (v) differ by a constant multiple, it suffices to study the shortest loops for the metric g¯E . Denote δ = E −1 , this is then equivalent to studying the mechanical system Hδ (ϕ, I) = K(I) − δU (ϕ)

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159

with energy E = 1. Without loss of generality, we may assume h = (1, 0). Let us write U (ϕ) = U1 (ϕ1 ) − U2 (ϕ1 , ϕ2 ), R1 where 0 U2 (ϕ1 , s)ds = 0. We will show an averaging effect that eliminates U2 . Let us denote by ρ0 the unique positive number such that K −1 (ρ0 h) = 1, and let c0 = ∂K −1 (ρ0 h). Lemma 13.17. There is C > 1 depending only on k∂ 2 Kk, k∂ 2 K −1 k such that then associated Hamiltonian orbit ηh1 = {(ϕ, I)(t)} satisfies kϕ˙ − ρ0 hk, kI − c0 k ≤ Cδ. Proof. Let us note the shortest curve γh1 corresponds to the orbit ηh1 = (ϕh , Ih ) of the Hamiltonian Hδ , satisfying ϕh (T ) = ϕh (0) + h. Note: 1. ϕ¨h (t) = δ∇U (ϕh (t)) = O(δ). 2. Since the energy is 1, K(Ih (t)) = 1+O(δ), K −1 (ϕ˙ h (t)) = K −1 (∂I K(Ih (t))) = K(Ih (t)) = 1 + O(δ). The first item implies ϕ˙ h (t) = T1 h + O(δ), when combined with the second item, implies ϕ˙ h (t) = ρ0 h + O(δ). Applying ∂K −1 , we get Ih (t) = c0 + O(δ). Note that the condition ∂K(I) = ρh corresponds precisely to the resonance segment Γ = I ∈ R2 , ∂K(I) · (1, 0) = 0, in other words, we are in precisely in the same situation as in single resonance. In fact, since there are fewer degrees of freedom, the normal form is much stronger. Proposition 13.18. Suppose r ≥ 4. For any M > 0, there is δ0 = δ0 (K, M ) and C = C(K, M ) > 0 such that for every Hδ with 0 < δ < δ0 , there is a C ∞ coordinate change Φ Hamiltonian isotopic to identity, such that Hδ ◦ Φ = K(I) − δU1 (ϕ1 ) − δR(ϕ, I), √ with kRkCI2 ≤ δ, and kπθ (Φ − Id)kC 2 ≤ Cδ and kπp (Φ − Id)kC 2 ≤ Cδ, where the norms are measured on the set I ∈ BM √δ (c0 ). If U1 (ϕ1 ) has a unique non-degenerate minimum at ϕ1 = ϕ∗1 , and satisfies U (ϕ1 ) − U (ϕ∗1 ) ≥ λkϕ1 − ϕ∗1 k2 . Then by choosing δ0 = δ0 (K, M, λ) > 0 smaller, the system Hδ ◦Φ has a normally hyperbolic invariant cylinder C given by (ϕ1 , I1 ) = (F (ϕ2 , I2 ) + ϕ∗1 , G(ϕ2 , I2 )),

ϕ2 ∈ T, I2 ∈ Γ ∩ BM √δ (c0 ),

√ satisfying kF kC 0 ≤ Cκ, kGkC 0 ≤ Cκ δ. C contains all the invariant sets in kϕ1 − ϕ∗1 k < C −1 ,

dist(I, Γ ∩ BM √δ ) < C −1 .

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The minimal periodic orbit ηh1 is contained in C and is a graph over ϕ2 . It is therefore unique and hyperbolic. Proof. The first part is very similar to Theorem 3.4, but in the much simpler case of 2-degrees of freedom. In this case, we can give out the coordinate change explicitly. Define Z 1 W (ϕ1 , ϕ2 ) = ρ−1 sU2 (ϕ1 , ϕ2 + s)ds, 0 0

R1 and note that W is well defined on T2 due to 0 U2 (ϕ1 , s)ds = 0. Note that ∂ϕ2 W = ρ−1 0 U2 . Consider the coordinate change I 7→ I + δ∇W . Then the new Hamiltonian is K(I + δ∇W ) − δU1 − δU2 = K(I) + δ∂K(I) · ∇W − δU1 − δU2 + δ 2 K(∇W ) = K(I) − δU1 + δ(ρ0 h · ∇W − U2 ) + δ 2 K(∇W ) + δ(∂K(I) − ρ0 h) · ∇W. The main observation is that the second term in the last equality vanishes, the 3 third term is O(δ 2 ) in C 2 norm, and the last term is O(δ 2 ) in CI2 norm. The other parts of the proposition is identical to Theorems 9.1 9.2, and are omitted. Proposition 4.6 follows immdediately from the above proposition.

13.5

GENERIC PROPERTIES AT THE CRITICAL ENERGY

We prove Proposition 4.7 that the conditions [DR1c ]–[DR4c ] holds on an open and dense set of U . Since all conditions are C 2 open, it suffices to prove denseness. First, we perturb U near the origin to achieve properties [DR1c ]. Let W 0 be a ρ-neighborhood of the origin in R2 for small enough ρ > 0 so that it does not intersect sections Σs± and Σu± . Consider ξ(ϕ) a C ∞ -bump function so that P ξ(ϕ) ≡ 1 for |ϕ| < ρ/2 and ξ(ϕ) ≡ 0 for |ϕ| > ρ. Let Qζ (ϕ) = ζij ϕi ϕj be a symmetric quadratic form. Consider Uζ (ϕ) = U (ϕ) + ξ(ϕ)(Qζ (θ) + ζ0 ). In W 0 × R2 we can diagonalize both: the quadratic form K(p) = hAp, pi and the Hessian ∂ 2 U (0). Explicit calculation shows that choosing properly ζ one can make the minimum of U at 0 being unique and eigenvalues to be distinct. Note that minimizing loops of the Jacobi metric may not intersect except possibly at 0, any two minimizing loops must be isolated away from a neighborhood of 0. We can apply localized perturbation to one of the geodesics to reduce its length, making it the global minimizer. Since the minimal loop is generically unique, its type is also determined uniquely. This proves [DR2c ].

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161

Denote wu = W u ∩ Σu+ an unstable curve on the exit section Σu+ and ws = W s ∩ Σs+ a stable curve on the enter section Σs+ . Denote on wu (resp. ws ) the point of intersection Σu+ (resp. Σs+ ) with strong stable direction q ss (resp. q uu ). Recall that q + (resp. p+ ) denotes the point of intersection of γ + with Σu+ (resp. Σs+ ). [DR3c ] is equivalent to conditions u s ss Φ+ glob w ∩ w 6= q ,

−1 (Φ+ (ws ) ∩ wu 6= q uu . glob )

By Lemma 13.7 the differential map DU Φ generates a subspace orthogonal to the gradient ∇H U and the Hamiltonian vector field χU (·) of H U . This means + u s we can perturb the global map Φ+ glob to move the intersection Φglob w ∩w away ss uu c from q , and similarly for q . [DR3 ] follows. Finally, a simple critical closed loop corresponds to a homoclinic orbit η contained in the intersection W u ∩W s of the unstable and stable manifold of the s/u equilibrium (0, 0). Moreover, if we set Wa = {z ∈ W s/u : dist(z, (0, 0)) < a}, where dist is the distance on the surface, then there exists a > 0 such that η ⊂ Wau ∩ Was . As part of the Kupka-Smale theorem (Lemma 9 of [69]), this intersection is transversal for an open and dense set of U . This proves [DR4c ] in the simple critical and non-simple case. In the simple critical case, η is a regular periodic orbit, and the statement again follows from the Kupka-Smale theorem.

Chapter Fourteen Derivation of the slow mechanical system In this chapter we derive the slow mechanical system at a maximal resonance. The discussions here applies to arbitrary degrees of freedom, and indeed we will consider H (θ, p, t) = H0 (p) + H1 (θ, p, t),

θ ∈ Tn , p ∈ Rn , t ∈ T,

and let p0 ∈ Rn be an n-resonance, namely, there are linearly independent k1 , · · · , kn ∈ Zn+1 such that ki · (∂H0 (p0 ), 1),

i = 1, · · · , n.

Since the resonance depends only on the hyperplane containing k1 , · · · , kn , we may choose k1 , · · · , kn to be irreducible, namely SpanZ {k1 , · · · , kn } = SpanR {k1 , · · · , kn } ∩ Zn+1 . Results in this chapter apply to the proof of our main theorem by restricting to the case n = 2. First, we will reduce the system near an n-resonance to a normal form. After that, we perform a coordinate change on the extended phase space, and an energy reduction to reveal the slow system. In Section 14.1, we describe a resonant normal form. In Section 14.2, we describe the affine coordinate change and the rescaling, revealing the slow system. In Section 14.3 we discuss variational properties of these coordinate changes.

14.1

NORMAL FORMS NEAR MAXIMAL RESONANCES

Write ω0 := ∂p H0 (p0 ), then the orbit (ω0 , 1) t is periodic. Let T = inf {t(ω0 , 1) ∈ Zn+1 } t>0

be the minimal period, then there exist a constant T∗ = T∗ (k1 , · · · , kn ) > 0, such that T ≤ T∗ (k1 , · · · , kn ).

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DERIVATION OF THE SLOW MECHANICAL SYSTEM

Given a function f : Tn × Rn × T → R, we define 1 [f ]ω0 (θ, p, t) = T

Z

T

f (θ + ω0 s, p, t + s)ds. 0

[f ]ω0 corresponds to the resonant component related to the maximal resonance. P Writing H1 (θ, p, t) = k∈Zn+1 hk (p)e2πik·(θ,t) and letting Λ = SpanZ {k1 , · · · , kn } ⊂ Zn+1 , then [H1 ]ω0 (θ, p, t) =

X

hk (p)e2πik·(θ,t) .

k∈Λ

We define a rescaled differential in the action variable by √ ∂I f (θ, p, t) = ∂p f (θ, p, t).

(14.1)

Note that the I in the subscript is only a notation and does not refer to a change of variable. Let k · kCIr denote the C r norm with ∂p replaced by ∂I . For a vector or a matrix valued function, we take the C r (or CIr ) norm to be the sum of the norm of all elements. Let Brn (p) denote the r-neighborhood of p0 in Rn . Theorem 14.1. Assume that r ≥ 5. Then for any M1 > 0, there exists 0 = 0 > 0, C1 > 1 depending only on kH0 kC r , T∗ (k1 , · · · , kn ), n, M1 such that for any 0 <  < 0 , there exists a C ∞ symplectic coordinate change n √ n √ Φ : Tn × BM (p0 ) × T → Tn × B2M (p0 ) × T, 1  1 

which is the identity in the t component, such that N (θ, p, t) := H ◦ Φ (θ, p, t) = H0 (p) + Z(θ, p, t) + Z1 (θ, p, t, ) + R(θ, p, t, ), where Z = [H1 ]ω0 , [Z1 ]ω0 = Z1 , and √ kZ1 kCI2 ≤ C1 ,

3

kRkCI2 ≤ C1  2 ,

(14.2)

and kΦ − IdkCI2 ≤ C1 . Moreover, Φ admits the following extensions: 1. Φ can be extended to Tn × Rn × T such that Φ is identity outside of Tn × U4M1 √ (p0 ) × T. e  (θ, p, t, E) on Tn × Rn × 2. The extension in item 1 can be further extended to Φ

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T × R, such that e  (θ, p, t, E) = (Φ (θ, p, t), E + E(θ, e p, t)), Φ e  is exact symplectic. and Φ e 3. kΦ − IdkCI2 ≤ C1  holds. Remark 14.2. • Our normal form is related to the classical “partial averaging,” see for example expansion (6.5) in [2, Section 6.1.2]. Our goal here is to obtain precise control of the norms with minimal regularity assumptions. In particular, the norm estimate of Φ − Id is stronger than the usual results, and is needed in the proof of Proposition 14.10. • For readers familiar with averaging theory, we perform two steps of resonant normal form transformations, and the term Z1 groups together the resonant components of both steps. • It is possible to lower the regularity assumption to r ≥ 4, and use the weaker estimate kRkCI2 ≤ C1 . This, however, requires more technical discussions in the next few sections, and we choose to avoid it. • Due to existence of the extension, we consider N = H ◦Φ as defined on all of Tn × Rn × T; however, the normal form holds only on the local neighborhood. The rest of this section is dedicated to proving Theorem 14.1. Denote Πθ (θ, p, t) = θ, Πp (θ, p, t) = p the natural projections. For a map Φ : Tn × U × T → Tn × Rn × T, which is the identity in the last component, denote Φ = (Φθ , Φp , Id), where Φθ = Πθ ◦ Φ and Φp = Πp ◦ Φ. Lemma 14.3. We have the following properties of the rescaled norm. 1. 2. 3. 4.

kf kCIr ≤ kf kC r , kf kC r ≤ −r/2 kf kCIr . k∂θ f kC r−1 ≤ kf kCIr , k∂p f kC r−1 ≤ √1 kf kCIr . I I There exists cr,n > 1 such that kf gkCIr ≤ cr,n kf kCIr kgkCIr . Let Φ be as before, and Id denote the identity map. There exists cr,n > 1 such that if √ max{kΠθ (Φ − Id)kCIr , kΠp (Φ − Id)kCIr / } < 1 we have kf ◦ ΦkCIr ≤ cr,n kf kCIr .

Proof. The first two conclusions follow directly from the definition. For the third conclusion, we have kfekC r = kf kCIr , where √ fe(θ, I) = f (θ, I). The conclusion then follows from properties of the C r -norm.

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DERIVATION OF THE SLOW MECHANICAL SYSTEM

For the fourth conclusion, we note e f ◦ Φ = fe ◦ Φ, e I) = (Φθ (θ, √I), Φp (θ, √I)/√). Moreover, let where fe is as before, and Φ(θ, e = Φ e − Id. Then there exists c > 0 us denote Ψ = Φ − Id, and note that Ψ depending only on dimension such that e C r−1 ≤ c + kD(Φ e − Id)kC r−1 ≤ c + max{kΠθ Ψk e C r , kΠp Ψk e Cr } kDΦk √ ≤ c + max{kΠθ (Φ − Id)kCIr , kΠp (Φ − Id)kCIr / } ≤ c + 1, To estimate the C r norm of composition, we use the following estimate for f, g ∈ C r , which is a consequence of the Faa-di Bruno formula (see for example, Appendix C of [33]): kf ◦ gkC r ≤ C (kf kC 1 kgkC r + kf kC r kgkrC r−1 ) .

(14.3)

We have   e C r ≤ dr kfekC r 1 + kDΦk e r r−1 kf ◦ ΦkCIr = kfe ◦ Φk C ≤ dr kf kCIr (1 + (c + 1)r ) ≤ cr,n kf kCIr , where cr,n = dr (1 + (c + 1)r ). For ρ > 0, denote Dρ = Tn × Uρ (p0 ) × T × R. We require another lemma for estimating the norm of a Hamiltonian coordinate change. This is an adaptation of Lemma 3.15 from [33]. Lemma 14.4. For 0 < ρ0 < ρ < ρ0 , r ≥ 4, 2 ≤ ` ≤ r − 1,  > 0 small enough, and a C r Hamiltonian G : Dρ0 → R, let ΦG t denote the Hamiltonian flow defined by G. Let cr,n > 1 be the constant from item 4, Lemma 14.3. Assume that √ 1 1 √ kGkC l+1 < √ min , ρ − ρ0 , I  cr,n  then for 0 ≤ t ≤ 1, the flow ΦG t is well defined from Dρ0 to Dρ . Moreover, kΠθ (ΦG t − Id)kCIl ≤ cr,n k∂p GkCIl ,

kΠp (ΦG t − Id)kCIl ≤ cr,n k∂θ GkCIl . (14.4)

Proof. Define  At = max

0≤τ ≤t

1 G kΠθ (ΦG τ − Id)kCIl , √ kΠp (Φτ − Id)kCIl 

 .

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Let t0 be the largest t ∈ [0, 1] such that the following conditions hold. (a) At ≤ 1. G (b) ΦG −t (Dρ ) ⊃ Dρ0 , in other words, Φt : Dρ0 → Dρ is well defined. We first show (14.4) holds on 0 ≤ t ≤ t0 , then we show t0 = 1. By Lemma 14.3, item 4, we have Z t




Πθ ΦG

∂p G ◦ ΦG

− Id ≤ t τ C l dτ ≤ cr,n k∂p GkCIl Cl I

and




Πp ΦG

t − Id C l ≤ I

I

0

Z 0

t



∂θ G ◦ ΦG

τ C l dτ ≤ cr,n k∂θ GkCIl . I

We now show t0 = 1. By the estimates obtained, 1 cr,n At0 ≤ cr,n max{k∂p GkCIl , √ k∂θ GkCIl } ≤ √ kGkC l+1 < 1. I   Moreover, we have
√ cr,n kΠp ΦG  √ kGkC l+1 < ρ − ρ0 , t0 − Id kCIl ≤ cr,n k∂θ GkCIl < I  implying Φ−t0 Dρ ) Dρ0 . If t0 6= 1, we can extend conditions (a) and (b) to t > t0 , contradicting the maximality of t0 . We now state our main technical lemma, which is an adaptation of an inductive lemma due to Bounemoura ([19]). Lemma 14.5. There exists a constant K > 2 depending only on r such that the following hold. Assume the parameters r ≥ 4, ρ > 0, µ > 0 satisfy   1 ρ 2 √ . 0 <  ≤ µ , KT µ < min , 4 2(r − 2)  Assume that H : Tn × Uρ (p0 ) × T × R → R,

H(θ, p, t, E) = l + g0 + f0 ,

where l(p, E) = (ω0 , 1)·(p, E) is linear, g0 , f0 are C r and depend only on (θ, p, t), and √ kg0 kCIr ≤ µ, kf0 kCIr (ρ0 ) ≤ , k∂p f0 kC r−1 ≤ . (14.5) I

ρ ρ − j 2(r−2)

Then for j ∈ {1, · · · , r − 2} and ρj = > ρ/2, there exists a collection of C r−j -symplectic maps Φj : Dρj → Dρ , of the special form e p, t)). Φj (θ, p, t, E) = (Θ(θ, p, t), P (θ, p, t), t, E + E(θ,

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DERIVATION OF THE SLOW MECHANICAL SYSTEM

The maps Φj have the properties kΠθ (Φj − Id)kC r−j (Dρ I

j

kΠp (Φj − Id)kC r−j (Dρ I

j

) )

≤ jK j+1 (T µ)2 , √ ≤ jK j (T µ) ,

(14.6)

and H ◦ Φj = l + gj + fj , for each j ∈ {1, · · · , r − 2} satisfying gj = gj−1 + [fj−1 ]ω0 and √ kgj kC r−j (Dρj ) ≤ (2 − 2−j ) µ, kfj kC r−j (Dρj ) ≤ (KT µ)j .

(14.7)

Proof. The proof is an adaptation of the proof of Proposition 3.2 in [19]. Define χj =

1 T

Z

T

s(fj − [fj ]ω0 )(θ + ω0 s, p, t + s)ds,

(14.8)

0

Φ0 = Id, and

χ

Φj+1 = Φj ◦ Φ1 j , χ

where Φs j is the time-s map of the Hamiltonian flow of χj . χ Using the fact that χj is independent of E, we have the map Φχj := Φ1 j is independent of E in the (θ, p, t) components. Furthermore, Φχj is the identity in the t component, and ΠE Φχj − E is independent of E. Hence Φj takes the special format described in the lemma. The special form of Φj implies that gj and fj are also independent of E, allowing the induction to continue. First of all, assuming the step j of the induction is complete, we prove the norm estimate (14.6). Using (14.7), k[fj ]ω0 kC r−j ≤ kfj kC r−j ≤ (KT µ)j , I

I

kχj kC r−j ≤ 2T kfj kC r−j < 2T (KT µ)j . I

I

We will choose K such that K ≥ 2cr,n . Then √ 1 1 1 √ kχj kC r−j ≤ (2T cr,n )(KT µ)j ≤ (KT µ)j+1 I cr,n cr,n  1 1 ρ < max{ , √ }, cr,n 4 2 (r − 2) therefore Lemma 14.4 applies with ρ = ρj , ρ0 = ρj+1 , G = χj . For j ≥ 1, using the inductive assumption and cr,n > 1, 1 k∂p χj kC r−j−1 ≤ √ kχj kC r−j ≤ (KT µ)j+1 , I I 

(14.9)

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while for j = 0, the initial assumption on f0 implies k∂p χ0 kC r−1 ≤ 2T k∂p f0 kC r−1 ≤ 2T  ≤ 2T µ2 < I

I

1 cr,n

(KT µ)2 ,

since T ≥ 1 and K ≥ 2cr,n . Applying Lemma 14.4, we get Φχj are well defined maps from Dρj+1 to Dρj , and kΠθ (Φχj − Id)kC r−j−1 ≤ cr,n k∂p χj kC r−j−1 ≤ (KT µ)2 , I

I

√ kΠp (Φχj − Id)kC r−j−1 ≤ cr,n k∂θ χj kC r−j−1 ≤ cr,n kχj kC r−j < (KT µ)j+1  I

I

I

Φχ1 0 ,

using (14.9). Since Φ1 = we obtain (14.6) for j = 1. For each j ≥ 2, using the bound on Φχj − Id above, Lemma 14.3, item 4 applies for Φ = Φχj . Then kΠθ (Φj−1 − Id) ◦ Φχj−1 kC r−j ≤ cr,n kΠθ (Φj−1 − Id)k. i

We have kΠθ (Φj − Id)kC r−j I

= kΠθ (Φj−1 ◦ Φχj−1 − Φχj−1 + Φχj−1 − Id)kC r−j I

≤ kΠθ (Φj−1 − Id) ◦ Φχj−1 kC r−j + kΠθ (Φχj−1 − Id)kC r−j I

I

≤ cr,n kΠθ (Φj−1 − Id)kC r−j + kΠθ (Φχj−1 − Id)kC r−j I

≤

j−1 X

I

cj−i−1 kΠθ (Φχi − Id)kC r−i−1 < jK j−1 (KT µ)2 , r,n I

i=0

noting cr,n < K and kΠ(Φχj − Id)kCrr−j−1 ≤ (KT µ)2 . By the same reasoning, we get kΠp (Φj − Id)kC r−j ≤

j−1 X

√ cj−i−1 kΠp (Φχi − Id)kC r−i−1 ≤ jK j−1 (KT µ) . r,n I

I

i=0

We now proceed with the induction in j. The inductive assumption (14.7) holds for j = 0 due to (14.5), which is in fact stronger. For the inductive step, define gj+1 = gj + [fj ]ω0 . √ j Since kfj kC r−j ≤ (KT µ)  < 4−j µ, we get I

√ √ kgj+1 kC r−j−1 ≤ kgj kC r−j + kfj kC r−j ≤ (2 − 2−j + 4−j ) µ < (2 − 2j+1 ) µ. I

I

I

169

DERIVATION OF THE SLOW MECHANICAL SYSTEM

By a standard computation, Z fj+1 =

1

{fjs , χj } ◦ Φχs j ds,

0

where fjs = gj + sfj + (1 − s)[fj ]ω0 . Estimate (14.7) implies kfjs kC r−j ≤ I √ 3kgj kC r−j ≤ 6 µ. Using Lemma 14.3, items 2 and 3, there exists an absoI

lute constant d > 1 such that for any f, g ∈ C l (Dρ ), we have d k{f, g}kCIl = k∂θ f · ∂p g + ∂p f · ∂θ gkCIl ≤ √ kf kC l+1 kgkC l+1 . I I  Therefore d k{fjs , χj }kC r−j−1 ≤ √ kfjs kC r−j kχj kC r−j I I I  √ d ≤ √ · 3 µ · 2T kfj kC r−j ≤ (6dT µ)(KT µ)j . I  Furthermore, by Lemma 14.3, items 4, kfj+1 kC r−j−1 ≤ max k{fjs , χj } ◦ Φχs j kC r−j−1 ≤ cr,n max k{fjs , χj }kC r−j−1 I

0≤s≤1

I

0≤s≤1

I

≤ (6cr,n d)T µ(KT µ)j  ≤ (KT µ)j+1  if we choose K ≥ 6cr,n d. The induction is complete. Proof of Theorem 14.1. First, we show that H(θ, p, t, E) = H (θ, p, t) − H0 (p0 ) + E = l + g0 + f0 , where l(p, E) = (ω0 , 1) · (p, E), g0 (θ, p, t) = H0 (p) − H0 (p0 ) − ω0 · p and f0 = H1 . √ Define ρ = 2M1 . We have the following estimates: ∂θ g0 = 0, √ k∂p g0 kC 0 = k∂p H0 (p) − ∂p H0 (p0 )kC 0 ≤ kH0 kC 2 · ρ = 2kH0 kC 2 M1  e1 > 1 depending on and k∂pjj g0 kC 0 ≤ kH0 kC 1+j for all j ≥ 2. Then for some C H0 , H1 , √ √ e1 M1 . k∂p g0 kC r−1 (Dρ ) ≤ max( )j−1 k∂pjj g0 kC 0 ≤ C I

j≥1

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CHAPTER 14

On the other hand, using kf kCIr ≤ kf kC r , we have e1 , kf0 kCIr (Dρ ) ≤ C √

e1 M1  ≥ Choose µ = C

√

e2 . k∂p f0 kC r−1 (Dρ ) ≤ C I

, we have

√ √ e1 M1  = µ, kg0 kCIr ≤ max{k∂θ g0 kC r−1 (Dρ ) , k∂p g0 kC r−1 (Dρ ) } ≤ C I

and kf0 kCIr (Dρ ) , k∂p f0 kC r−1 (Dρ ) ≤ I ensure

I

√

µ. By choosing  sufficiently small, we

¯ √ ρ 1 2E e1 M1  < min{ 1 , √ } = min{ , KT µ = KT C }. 4 2(r − 2)  4 2(r − 2) The conditions of Lemma 14.5 are satisfied with these parameters and we apply the lemma with j = 3. There exists a map Φ3 : Dρ/2 → Dρ , √ e2  kΦ3 − IdkCI2 (Dρ/2 ) ≤ max{2K 3 (T µ)2 , 2K 2 (T µ) } ≤ C e2 > 1 depending on T, C e1 , M1 . Moreover, for some C H ◦ Φ3 = l + g0 + [f0 ]ω0 + [f1 ]ω0 + [f2 ]ω0 + f3 , with

√ e3  32 , k[f1 ]ω0 kCI2 (Dρ/2 ) ≤ (KT µ) µ ≤ C √ e3 2 , k[f2 ]kCI2 (Dρ/2 ) ≤ (KT µ)2 µ ≤ C √ e3  52 , k[f3 ]kCI2 (Dρ/2 ) ≤ (KT µ)3 µ ≤ C

e3 depending on T, C e1 , M1 . Using l + g0 = H0 (p) + E − H0 (p0 ), define for some C Z = [f0 ]ω0 , Z1 = [f1 + f2 ]ω0 , and R = f3 , we obtain (H + E) ◦ Φ2 = H0 + Z + Z1 + R + E e2 , C e3 }. Finally, we define with the desired estimates and constant C1 = max{C Φ (θ, p, t) = Φ3 (θ, p, t, E). This is well defined since Φ3 is independent of E. We now use a smooth approximation technique to show the normal form Φ can be taken to be C ∞ . Using standard techniques, for every σ > 0, there are C = C(r) > 0 such that there is C ∞ functions H00 (p) and H10 (θ, p, t) satisfying kH0 − H00 kC 2 ≤ σ,

kH1 − H10 kC 2 ≤ σ,

kH0 kC r , kH1 kC r ≤ C.

Note that the estimates of the normal form depend on the C r norm of the smooth approximation. We apply the above procedure to H00 + H10 , and obtain

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DERIVATION OF THE SLOW MECHANICAL SYSTEM

a C ∞ coordinate change Φ0 such that (H00 + H10 + E) ◦ Φ0 = H0 + Z 0 + Z10 + R0 . ˜ kRkC 2 ≤ C ˜ 32 . We have kZ 0 − ZkC 2 ≤ kH1 − H10 kC 2 < σ and kZ10 kCI2 ≤ C, I We now compute (H0 + H1 + E) ◦ Φ0 = H0 + Z + Z10 + +R0 + (H0 − H00 ) ◦ Φ0 + (H1 − H10 ) ◦ Φ0 + (Z 0 − Z). √ −2 The last three terms are bounded in CI2 norms by C˜  σ. To ensure the remainder is small we can take σ = 4 . The normal form lemma stated here also applies to maximal resonances with long period, which then combined with the idea of Lochak ([53]), can be used to study single resonance. This is Proposition 3.6, which we prove here. Proof of Proposition 3.6. As in the proof of Theorem 14.1, we apply the Lemma √ 14.5 to l + g0 + f0 , where l + g0 = H0 + E and f0 = H1 . Choose ρ = K1 , then √ kg0 kCIr ≤ CK1 , kf0 kCIr ≤ , k∂p f0 kC r−1 ≤  I

where C depends on r, n, kH0 kC r . Choose √ µ = CK1 , √ by assumption T ≤ C1 /(K12 ), then KT µ ≤ (KCC1 )K1−1 . The condition of Lemma 14.5 is satisfied if K1 is sufficiently large. We apply Lemma 14.5 with j = 1, the conclusion is kΠθ (Φ − Id)kC r−1 ≤ K 2 (KCC1 )2 K1−2 , I √ kΠp (Φ − Id)kC r−1 ≤ K(KCC1 )K1−1 . I

and kf1 kC r−j ≤ (KCC1 )K1−1 . We can also apply the same smooth approximation to upgrade the coordinate change to C ∞ .

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14.2

AFFINE COORDINATE CHANGE, RESCALING, AND ENERGY REDUCTION

Definition of the slow system. Recall that an n-tuple of vectors k1 , · · · , kn ∈ Zn+1 defines an irreducible lattice, if the lattice Λ = SpanZ {k1 , · · · , kn } satisfies Span{k1 , · · · , kn } ∩ Zn+1 = Λ. Let B0 be the n × (n + 1) matrix whose rows are vectors k1T , · · · , knT from Zn+1 , and let kn+1 ∈ Zn+1 be such that  T  k1  ..  (14.10) B :=  .  ∈ SL(n + 1, Z). T kn+1

Such a kn+1 exists if and only if k1 , · · · , kn are irreducible. By possibly changing the signs of the rows of B, we ensure β := kn+1 · (ω0 , 1) > 0.

(14.11)

    ω 0 B 0 = . 1 β

(14.12)

In particular, we have

2 Letting A0 = ∂pp H0 (p0 ), we define

K(I) =


1 B0 A0 B0T I · I, 2

I ∈ Rn

(14.13)

and U (ϕ), ϕ ∈ Tn by the relation U (k1 · (θ, t), · · · , kn · (θ, t)) = −Z(θ, p0 , t).

(14.14)

Consider the corresponding autonomous system G (θ, p, t, E) = N + E. We show that G can be reduced to Hs (ϕ, I, τ ) =


√ 1 K(I) − U (ϕ) + P (ϕ, I, τ, ) , β

ϕ ∈ Tn , I ∈ Rn , τ ∈

√

T,

(14.15) with a coordinate change and an energy reduction. See Proposition 14.8.

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DERIVATION OF THE SLOW MECHANICAL SYSTEM

Linear and rescaling coordinate change. Our coordinate change is a combination of a linear and a rescaling coordinate change. Namely, we have        s θ p − p0 s f −1 ϕ T p (θ, p, t, E) = Φ1 (ϕ, p , s, p ) : =B , =B , t s E + H0 (p0 ) pf (ϕ, ps , s, pf ) = Φ2 (ϕ, I, τ, F ) = (ϕ,

√

√ I, τ / , F ).

We then have ΦL = Φ1 ◦ Φ2 :

(ϕ, I, τ, F ) ∈ Tn × Rn ×

by the formula     θ ϕ√ = B −1 , t τ/ 

√

T × R 7→ (θ, p, t, E)

   √ p − p0 I = B T √ . E + H0 (p0 ) F

(14.16)



(14.17)

Given any M1 > 0, there exists CB > 1 such that {k(I,

√


√ −1 F )k < CB M1 } ⊂ ΦL {k(p − p0 , E + H0 (p0 ))k <  M1 } √ ⊂ {k(I, F )k < CB M1 }. (14.18)

Define

1 Gs (ϕ, I, τ, F ) = √ G ◦ ΦL , 

then the Hamiltonian flows of Gs and G are conjugate via ΦL (with respect to the natural symplectic forms in their corresponding spaces). This can be seen from the equivalence of the following Hamiltonian systems: (G ,

√ (dϕ ∧ dI + dτ ∧ dF )) dθ ∧ dp + dt ∧ dE) ∼ (G ◦ ΦL ,   1 ∼ √ G ◦ ΦL , dϕ ∧ dI + dτ ∧ dF . 

We have the following lemma. Lemma 14.6. Assume that N satisfies (14.2) on DM1 √ . Then there exist CB > 1and C2 = C2 (H0 , C1 , M ) such that
√ √ √ Gs =  K(I) − U (ϕ) + (β +  l(I, F )) · F + P1 (ϕ, I, τ, F, ) , (14.19) where β = kn+1 · (ω(p0 ), 1), and klkC 2 , kP1 kC 2 ≤ C1 ,

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CHAPTER 14

with the norm taken on the set {k(I,

√

−1 F )k < CB M1 }.

(14.20)

Proof of Lemma 14.6. Due to (14.18), G ◦ ΦL is defined on the {k(I, −1 CB M1 }. Consider the expansion of N from Theorem 14.1

√

F )k
0,

such that for any 0 <  < 0 , there is a function √ n Hs : Tn × BM (0) × T → R 2 uniquely solving the equation Gs (ϕ, I, τ, −Hs ) = 0 on the set {k(I,

√

F )k ≤ M2 }. Moreover, Hs has the form (14.15), i.e.

Hs (ϕ, I, τ ) =


√ 1 K(I) − U (ϕ) + P (ϕ, I, τ, ) β

where kP kC 2 ≤ C3 . In particular, Hs → H s /β uniformly in C 2 (T2 × R2 × R).

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CHAPTER 14

Proof of Lemma 14.7. We can choose 0 such that for any 0 <  < 0 ∂ β (Gs ) > > 0 ∂F 2 √ on {k(I, F )k ≤ M2 }. Therefore, Hs exists by the implicit function theorem. Moreover, there exists a constant C 0 , depending on H0 and U , but independent of , such that kHs kC 2 ≤ C 0 . Let Q = Hs − β1 (K(I) − U (ϕ)) = Hs − H s , then kQkC 2 ≤ kH s kC 2 + C 0 . We know 0 = Gs (ϕ, I, τ, −Q − H s )
√ √ √ =  −βQ + l(I, Q + H s )(Q + H s ) + P1 (ϕ, I, τ, Q + H s ) √ √ =: (−βQ + P2 (ϕ, I, τ, Q + H s , )). Therefore,

√ Q=

 P2 (ϕ, I, τ, Q + H s , ). β

To solve this implicit equation notice that there exists C 00 > 0 depending only on C1 , H s such that kP2 kC 2 ≤ C 00 . Application of the Faa-di Bruno formula ((14.3)) shows that for some C > 0 depending only on n we have √  kQkC 2 = kP2 (ϕ, I, τ, Q + H s , )kC 2 β √ √ ≤ CC 00 kQk2C 2 ≤ CC 00 (kH s kC 2 + C 0 ), and the lemma follows. We have Gs (ϕ, I, τ, F ) = 0,

⇐⇒

Hs + F = 0.

(14.22)

The following Proposition follows from the standard energy reduction (see for example [8]). Proposition 14.8. Assume the conclusions of Theorem 14.1 hold on the set √ DM1 := {k(p − p0 , H + H0 (p0 )k < M1 } −2 for sufficently large M1 depending only on H0 . Then for M3 = CB M1 , 0 <  < 0 (H0 , C1 , M1 ), the following are equivalent:

1. The curve (θ, t, p, E)(t) is an orbit of N (θ, p, t) + E inside DM1 . 2. The curve (ϕ, τ, I, F )(t) = ΦL (θ, t, p, E)(t) is the time change of an orbit of Ns (ϕ, I, τ ) + F , with τ as the new time.

DERIVATION OF THE SLOW MECHANICAL SYSTEM

14.3

177

VARIATIONAL PROPERTIES OF THE COORDINATE CHANGES

We have made two reductions: The normal form H (θ, p, t) → N (θ, p, t) = H ◦ Φ , and the coordinate change with time change N (θ, p, t) + E → Hs (ϕ, I, τ ) + F. In this section, we discuss the effect of these reductions on the Lagrangian, barrier function, and the Mather, Aubry, and Ma˜ n´e sets. The main conclusion of this section is the following proposition, which follows directly from Propositions 14.10 and 14.11. Proposition 14.9. For M1 large enough depending only on H0 , k1 , · · · , kn , n, there exist C2 > 1 and 0 > 0 depending on H0 , k1 , · · · , kn , n, M1 , such that the following hold for any 0 <  < 0 . √ −2 1. For M 0 = CB M1 /2 (see (14.20)), and kc − p0 k ≤ M 0  and α = αH (c), let c¯ and α ¯ satisfy   √  c − p0  c¯ T =B −α + H0 (p0 ) − α ¯ then αHs (¯ c) = α ¯. 2. Let c, αN (c), c¯, αHs (¯ c) satisfy the relation in item 1, and suppose (ϕi , ti ) ∈ √ Tn × T, (ϕi , τi ) ∈ Tn ×  T, i = 1, 2 satisfies     θi ϕ√ i = B −1 mod Zn × Z ti τi /  then |hH ,c (θ1 , t1 ; θ2 , t2 ) −

√

hHs ,¯c (ϕ1 , τ1 ; ϕ2 , τ2 )| ≤ C.

3. Let c, αH (c), c¯, αHs (¯ c) be as before; then fH s (¯ fH (c)), M c) = ΦL ◦ Φ (M  

AeHs (¯ c) = ΦL ◦ Φ (AeH (c)),

eH s (¯ eH (c)). N c) = ΦL ◦ Φ (N   Recall that Φ can be extended to the whole phase space, and N = H ◦ Φ is considered as a function on Tn × Rn × T. Proposition 14.10. In the setup of 14.9 there exists C3 > 1 depending on H0 , k1 , · · · , kn , n, M1 such that we have the following relation. fH (c) = M fN (c), Φ AeH (c) = AeN (c), Φ N eH (c) = 1. αH (c) = αN (c), Φ M      e NN (c).

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CHAPTER 14

2. |AH ,c (θ1 , e t1 ; θ2 , e t2 ) − AN ,c (θ1 , e t1 ; θ2 , e t2 )| ≤ C3 . 3. |hH ,c (θ1 , t1 ; θ2 , t2 ) − hN ,c (θ1 , t1 ; θ2 , t2 )| ≤ C3 . Proof of Proposition 14.10. The symplectic invariance of the alpha function and e  is exact and the Mather, Aubry and Ma˜ n´e sets follows from the fact that Φ Hamiltonian isotopic to identity (see [10] and Proposition 2.7). In order to get the quantitative estimate of action, we need more detailed estimates. e  (θ, p, t, E) = (Θ, P, t, ΦE ), from Theorem 14.1 and using the Writing Φ rescaled norm (14.3), we have √ kΦ − IdkC 1 ≤ C1 

e  − IdkC 0 ≤ C1 , kΦ

e = ΦE − E. By exactness of Φ e  , we have there exists a function Denote E n n S : T × R × T × R → R such that e = dS(θ, p, t, E) = dS(θ, p, t) P dΘ + ΦE dt − (pdθ + Edt) = P dΘ − pdθ + Edt (14.23) In particular, given a curve (θ, p, t, E)(t), t ∈ [t˜1 , t˜2 ] with N + E = 0, we have e  (θ, p, t, E) (and hence H (Θ, P, t) + ΦE = 0), and we for (Θ, P, t, ΦE )(t) = Φ apply (14.23) to the tangent vector of the curve to get d ˙ + ΦE − (p · θ˙ + E) S(θ, p, t, E) = P · Θ dt ˙ − H (Θ, P, t) − (p · θ˙ − N (θ, p, t)) =P ·Θ ˙ t). ˙ t) − LN (θ, θ, = LH (Θ, Θ, 



As a result, Z

t˜2

t˜1



Z  ˙ ˙ LH (Θ, Θ, t) − c · Θ dt −

t˜2

t˜1



 ˙ t) − c · θ˙ dt LN (θ, θ,

 t˜2 e , e − θ) = S(θ, p, t, E) − c · (Θ

(14.24)



t˜1

e θe are lifts of Θ(t), θ(t) to the universal cover. Moreover, from where Θ, kΦ − IdkC 0 ≤ C1 , we get Θ−θ is a well defined vector function on Tn ×BM1 (p0 )n ×T. In particular, e − θe = Θ − θ. we have Θ We now estimate the C 0 -norm of S. Write S0 = p · (Θ − θ), using (14.23) we have e = (P − p)dΘ + dS0 − (Θ − θ)dp + Edt. e dS = (P − p)dΘ + pd(Θ − θ) + Edt e C 0 = kΦE −EkC 0 ≤ C1 , we have kdSkC 0 ≤ C1 , Since kΘ−θkC 0 , kP −pkC 0 , kEk

179

DERIVATION OF THE SLOW MECHANICAL SYSTEM

and kS − S(0)kC 0 ≤ C 0  for some C 0 depending on C1 . It follows that the integral in (14.24) is bounded by C 0 . Applying this estimate to a minimizer, we have |AN ,c (θ(t˜1 ), e t1 ; θ(t˜2 ), e t2 ) − AH ,c (Θ(t˜1 ), t˜1 ; Θ(t˜2 ), t˜2 ))| ≤ C 0 . Since kΦ − IdkC 0 ≤ C1 , we have kθ(t˜i ) − Θ(t˜i )k ≤ C1 , i = 1, 2. The estimate follows from the Lipschitz property of AH,c . Taking the limit, we obtain the estimate for the barrier hH,c . We now study the relation between N and Hs . Extend the √ definition of Hs √ n n to T ×R ×R so that the P term is supported on the set {k(I, F )k < 2M2 }. Proposition 14.11. Assume the conclusions of Theorem 14.1 hold on the set √ {k(p − p0 , H + H0 (p0 )k < M1 } −2 for sufficently large M1 depending only on H0 . Then for M3 = CB M1 , we have:

1. Let (θ, p, t, E) satisfies k(p − p0 , E + H0 (p0 ))k < M3 , N (θ, p, t) + E = 0, and (ϕ, I, τ, F ) = ΦL (θ, p, t, E). Let LN and LHs be the Lagrangians for N and Hs . Then for  s   v v v = ∂p N (θ, p, t), = B , 1 vf we have f



LN (θ, v, t) − p0 · v + H0 (p0 ) = v L

Hs

 1 vs √ ϕ, √ f , s  . v

2. Suppose (c, α), (¯ c, α ¯ ) ∈ Rn × R satisfies kc − p0 k ≤ M23   √  c − p0 ¯ c = BT , −α + H0 (p0 ) ¯ α

√

 and

then α = αN (c) if and only if α ¯ = αHs (¯ c). 3. Let c, αN (c), c¯, αHs (¯ c) satisfies the relation in item 2, and suppose (ϕi , ti ) ∈ √ Tn × T, (ϕi , τi ) ∈ Tn ×  T, i = 1, 2 satisfies     θi ϕ√ i = B −1 mod Zn × Z, ti τi /  then hN ,c (θ1 , t1 ; θ2 , t2 ) =

√

hHs ,¯c (ϕ1 , τ1 ; ϕ2 , τ2 ).

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CHAPTER 14

4. Let c, αN (c), c¯, αHs (¯ c) be as before, then fH s (¯ fN (c)), M c) = ΦL (M  

AeHs (¯ c) = ΦL (AeN (c)),

eH s (¯ eN (c)). N c) = ΦL (N  

−1 Proof of Proposition 14.11. The choice of M3 is to ensure that for M2 = CB M1 as in Lemma 14.6, √ ΦL ({k(p − p0 , E + H0 (p0 ))k < M3 }) ⊂ {k(I, F )k < M2 }.

Item 1. Recall that G = H + E, and LN − p0 · v + H0 (p0 ) = sup {(p − p0 ) · v − N (θ, p, t) + H0 (p0 )} p

= sup {(p − p0 , E + H0 (p0 )) · (v, 1) :

N (θ, p, t) + E = 0} ,

(14.25)

p,E

where the supremum is acheived at a unique point since H + E ≤ 0 is a strictly convex set. Let us denote LpN0 = LN − p0 · v + H0 (p0 ). Continuing from (14.25) and using (14.17), we get       p − p0 v LpN0 (θ, v, t) = sup (B T )−1 ·B : G (θ, p, t, E) = 0 E + H0 (p0 ) 1 √ √ s f s = sup (I, F ) · (v , v ) : G (ϕ, I, τ, F ) = 0  √ √ =  sup (I, F ) · (v s , v f ) : Hs (ϕ, I, τ ) + F = 0 ,  √ = v f sup (I, F ) · (v s /(v f ), 1) : Hs (ϕ, I, τ ) + F = 0 √ = v f LHs (ϕ, v s /(v f ), τ ). Item 2. We first derive a relation between the integrals of the Lagrangians. Let (θ, p, t, E)(t), t ∈ [t1 , t2 ] be a solution to N + E with √ N + E = 0 and k(p − p0 , E + H0 (p0 ))k < M1 . Let (ϕ, I, τ, F )(t) = Φ−1 L (θ, p, t, E)(t). Then from (14.17),      s dϕ vs v ϕ√ ˙ θ˙ =B := f (t), = √ f. τ˙ /  v 1 dτ v From item 1 we get Z t2 Z t2
√ ˙ t)dt = LpN0 (θ, θ, v f (t)LHs ϕ, v s /(v f ), τ dt t1 t1 Z τ2 Z τ2 √ dϕ dt dϕ = v f (τ )LHs (ϕ(τ ), (τ ), τ ) (τ )dτ =  LHs (ϕ, , τ )dτ. dτ dτ dτ τ1 τ1 e be a lift of θ(t) to the universal cover and ϕ(t), Let θ(t) e τe(t) be a lift of

DERIVATION OF THE SLOW MECHANICAL SYSTEM

181

(ϕ(t), τ (t)). Then 

   e ϕ(t) e √ θ(t) =B + const. τe(t)/  t

Given any (c, −α) ∈ Rn × R, we have Z

t2



 ˙ t) − c · θ˙ + α dt LN (θ, θ,

t1

Z

t2

= t1

√

  ˙ t)dt − (c − p0 , −α + H0 (p0 )) · (θ(t e 2 ) − θ(t e 1 ), t2 − t1 ) LpN0 (θ, θ, τ2

   e 2 ) − θ(t e 1) c − p0 θ(t ·B −α + H0 (p0 ) t2 − t1 τ Z 1τ2 √ √ √ =  LHs dτ − ( ¯ c, −¯ α) · (ϕ(τ e 2 ) − ϕ(τ e 1 ), (e τ2 − τe1 )/ ) τ  Z 1τ2  √ dϕ dϕ =  LHs − c¯ · +α ¯ (ϕ, , τ )dτ. dτ dτ τ1 =

Z



LHs dτ − (B T )−1



(14.26) We now prove item 2. Suppose α = αN (c), let (θ, p, t, E)(t), t ∈ [−∞, ∞) be an orbit in the Ma˜ √n´e set NN (c). Then there exists C∗ > 0 such √ that we have kp(t) − ck ≤ C ∗ , by Proposition 7.6. Since kc − p k ≤ M /2, with 0 3 √ M3 large enough, we have kp(t) − p0 k ≤ M3  for all t. Moreover, using the fact that the orbit is semi-static, we have Z

T

−C ≤

˙ (LN − c · v + α) (θ(t), θ(t), t)dt ≤ C.

0

From (14.26), we get Z

τ (T )

−C ≤ τ (0)


LHs − c¯ · v + α ¯ dτ ≤ C

for all T > 0. This implies α ¯ is Ma˜ n´e critical for c¯. As a result, we obtain that α is Ma˜ n´e critical for LN − c · v implies α ¯ is Ma˜ n´e critical for LHs − c¯ · v. Moreover, the converse is true by reversing the above computations. Item 3. Apply (14.26) to any one-sided minimizer of hN ,c (θ1 , t1 ; θ2 , t2 ), and use the localization of calibrated orbits. Item 4. (14.26) implies that an orbit (θ, p, t, E)(t) is semi-static for LN ,c implies (ϕ, I, τ, F ) = Φ−1 L (θ, p, t, E) is a reparametrization of a semi-static orbit of LHs ,¯c . The converse also holds. This implies the relation between Ma˜ n´e sets. The same applies to Aubry sets. Note that the Mather set is precisely the closure of the union of the support of all invariant measures contained in the Aubry set, and therefore is also invariant.

Chapter Fifteen Variational aspects of the slow mechanical system In this chapter we study the variational properties of the slow mechanical system H s (ϕ, I) = K(I) − U (ϕ), with min U = U (0) = 0. S The main Egoal of this section is to derive some properties of the “channel” E E>0 LF β (λh h), and information about the Aubry sets for c ∈ LF β (λh h). More precisely, we prove Proposition 5.1 and justify the picture in Figure 5.1. ⊥ • In Section 15.1, we show that each LF β (λE h h) is a segment parallel to h . • In Section 15.2, we provide a characterization of the segment and provide information about the Aubry sets. • In Section 15.3, we provide a condition for the “width” of the channel to be non-zero. • In Section 15.4, we discuss the limit of the set LF β (λE h h) as E → 0, which corresponds to the “bottom” of the channel.

We drop all supscripts “s” to simplify the notations. The results proved in this section are mostly contained in [64] in some form. Here we reformulate some of them for our purpose and also provide some different proofs.

15.1

RELATION BETWEEN THE MINIMAL GEODESICS AND THE AUBRY SETS

Assume that H(ϕ, I) satisfies the conditions [DR1h ]–[DR3h ] and [DR1c ]–[DR4c ]. Then for E 6= Ej , 1 ≤ j ≤ N − 1, there exists a unique shortest geodesic γhE for the metric gE in the homology h. For the bifurcation values E = Ej , there are two shortest geodesics γhE and γ¯hE . The function lE (h) denotes the length of the shortest gE -geodesic in homology h. By Lemma 15.4 the length function lE (h) is continuous and strictly increasing in E ≥ 0. Assume that the curves γhE are parametrized using the Maupertuis principle, namely, it is the projection of the associated Hamiltonian orbit. Let T (γhE ) be the period under this parametrization, and write λ(γhE ) = 1/(T (γhE )). ¯ ∈ H1 (T2 , Z) such that h, h ¯ form a basis of H1 (T2 , Z) We pick another vector h

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¯ ∗ in H 1 (T2 , R) we have hh, h ¯ ∗ i = 0. We denote h ¯∗ and for the dual basis h∗ , h by h⊥ to emphasize the latter fact. Items 1–4 of the next theorem will be proved in this section, and item 5 is proved in Proposition 15.9. Theorem 15.1. 1. For E = Ej , LF β (λ(γhE ) · h) = LF β (λ(¯ γhE ) · h). E E As a consequence, write λE h = λ(γh ), then the set LF β (λh h) is well defined E (the definition is independent of the choice of γh ). + 2. For each E > 0, there exist −∞ ≤ a− E (h) ≤ aE (h) ≤ ∞ such that − + ∗ ⊥ LF β (λE h h) = lE (h)h + [aE (h), aE (h)] h . + Moreover, the set function [a− E , aE ] is upper semi-continuous in E. E 3. For each c ∈ LF β (λh h), E 6= Ej , there is a unique c-minimal measure supported on γhE . E E 4. For each c ∈ LF β (λh j h), there are two c-minimal measures supported on γh j and c. 5. For E > 0, assume that the torus T2 is not completely foliated by shortest − closed gE -geodesics in the homology h, then a+ E (h)−aE (h) > 0 and the channel has non-zero width.

We have the following well-known lemma, see for example [47]. Lemma 15.2. Let h, h1 , h2 ∈ H1 (T2 , Z), then • (Positive homogeneous) lE (nh) = hlE (h) for all n ∈ N. • (Sublinear) lE (h1 + h2 ) ≤ lE (h1 ) + lE (h2 ). Assume that γ is a geodesic parametrized according to the Maupertuis principle. First, we note the following useful relation. p L(γ, γ) ˙ + E = 2(E + U (γ)) = gE (γ, γ), ˙ (15.1) where L denote the associated Lagrangian. According to [21], the minimal measures for L is in one-to-one correspondence with the minimal measures of gE (ϕ, v) (considered as a Lagrangian function). On the other hand, any minimal measure 12 gE with a rational rotation number is supported on a closed geodesic. The following lemma characterizes minimal measures supported on a closed geodesic. Lemma 15.3. 1. Assume that c ∈ H 1 (T2 , R) is such that αH (c) = E > 0. Then for any h ∈ H1 (T2 , Z), lE (h) − hc, hi ≥ 0.

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2. Let γ be a closed geodesic of gE , E > 0, with [γ] = h ∈ H1 (T2 , Z). Let µ be the probability measure uniformly distributed on the periodic orbit associated to γ. Then given c ∈ H 1 (T2 , R) with αH s (c) = E, µ is c − minimal if and only if

lE (h) − hc, hi = 0.

(15.2)

3. Let γ be a closed geodesic gE , E ≥ 0, with [γ] = h ∈ H1 (T2 , Z) and αH s (c) = E. Then γ ⊂ AH s (c) if and only if (15.2) holds. Proof. Let γ be a closed geodesic of gE , E > 0, with [γ] = h. Assume that with the Maupertuis parametrization, the periodic of γ is T . Let µ be the associated invariant measure, then ρ(µ) = h/T . Assume that αH s (c) = E, by definition, we have Z Ldµ + E ≥ β(h/T ) + αH s (c) ≥ hc, h/T i. By (15.1), we have Z Ldµ + E =

1 T

Z

T

(L + E)(dγ) = 0

1 T

Z

T

p

gE (dγ) = lE (γ)/T.

0

Combining the two expressions, we have lE (γ) − hc, hi ≥ 0. By choosing γ such that lE (γ) = lE (h), statement 1 follows. To prove statement 2, notice that if µ is c-minimal, then αH s (c) = E and the equality Z Ldµ + E = hc, h/T i holds. Equality (15.2) follows from the same calculation as statement 1. For E > 0, γ ⊂ AH s (c) if an only if γ is a minimal measure. Hence we only need to prove statement 3 for E = 0. In this case, γ can be parametrized as a homoclinic orbit. γ ⊂ AH s (c) if and only if Z ∞ (L − c · v + α(c))(dγ) = 0. −∞

Since Z

∞

Z

∞

(L − c · v + α(c))(dγ) = −∞

(L + E)(dγ) − hc, hi = lE (h) − hc, hi, −∞

the statement follows. Proof of Theorem 15.1, item 1–4. By Lemma 15.3, if there are two shortest geodesics γhE and γ¯hE for gE , for any c, the invariant measure supported on γhE is c-minimal if and only if the measure on γ¯hE is c-minimal. This implies statement 1. Statement 2 follows from the fact that LF β (λE h h) is a closed convex set, and

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(15.2). Statements 3 and 4 follow directly from Lemma 15.3. Item 5 is proved in Proposition 15.9. We also record the following consequence of Theorem 4.1. Lemma 15.4. The period TE := T (γhE ) and length lhE as functions of E are strictly monotone for E > 0. Proof. Let γhE be a minimal geodesic of gE , then for any E 0 < E we have Z q Z q lE 0 (h) ≤ gE 0 (γ˙ hE ) < gE (γ˙ gE ) = lE (h), therefore lE (h) is strictly monotone. To prove strict monotonicity of TE , we apply Theorem 15.1, item 2 to get LFβ (h/TE ) · h = lhE h∗ · h, where h∗ · h > 0. Since LFβ (h/TE ) are distinct for different E, TE 6= TE 0 for E 6= E 0 , we obtain strict monotonicity.

15.2

CHARACTERIZATION OF THE CHANNEL AND THE AUBRY SETS

In this section we provide a precise characterization of the set − + ∗ ⊥ LF β (λE h h) = lE (h) h + [aE (h), aE (h)] h .

For each E > 0, define ¯ d± E (h) = ± inf (lE (nh ± h) − lE (nh)). n→∞

¯ − lE (nh) ≤ lE ((n − 1)h ± h) ¯ + lE ((n − 1)h) from the Note that lE (nh ± h) sub-additivity and positive homogeneity of lE ; therefore the infimum coincides with the limit. We will omit dependence on h when it is not important. Lemma 15.5. The function d± E (h) is continuous in E > 0. Proof. From the sub-linearity of lE (h), we get ¯ ≤ lE (nh ± h) ¯ − lE (nh) ≤ lE (h), ¯ −lE (h) ¯ as a result, the family lE (nh± h)−l E (nh) as a function of E ∈ [E1 , E2 ] ⊂ (0, ∞)

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is equi-continuous and equi-bounded. Then there exists a subsequence such that ¯ − lE (nh) = d+ (h) lim lE (nh ± h) E

n→∞

uniformly. This implies d+ E (h) is continuous. The same argument works for d− E (h). Proposition 15.6. For each E > 0, we have ± d± E (h) = aE (h).

Proof. We first show − + + d− E (h) ≤ aE (h) ≤ aE (h) ≤ dE (h). ⊥ + Denoting c+ = lE (h)h∗ +a+ E h , by definition, lE (h)−hc , hi = 0. By Lemma 15.3, statement 1, for n ∈ N,

¯ − hc+ , nh + hi ¯ = lE (nh + h) ¯ − nlE (h) − hc+ , hi ¯ 0 ≤ lE (nh + h) ¯ − nlE (h) − a+ . = lE (nh + h) E

+ Taking infimum in n, we have d+ E − aE ≥ 0. Performing the same calculation ¯ replaced by nh − h, ¯ we obtain 0 ≤ lE (nh − h) ¯ − nlE (h) + a− , and with nh + h E − − hence aE − dE ≥ 0. + ⊥ We now prove the opposite direction. Taking any c ∈ lE (h)h∗ + [d− E , dE ]h , we first show that α(c) = E. ¯ then any invariant measure µ with rotation number ρ is Take ρ ∈ Qh + Qh, ¯ with m1 , m2 ∈ Z. Let T denote the period. supported on some [γ] = m1 h + m2 h By Lemma 15.7 below,

¯ ¯ β(ρ) + E = lE (m1 h + m2 h)/T ≥ hc, m1 h + m2 hi/T = hc, ρi. Since β is continuous, we have α(c) = suphc, ρi−β(ρ) ≤ E, where the supremum is taken over all rational ρ’s. Since the equality is achieved at ρ = h, we conclude that α(c) = E. By Lemma 15.3, statement 2, the measure supported on γhE is c-minimal, and hence c ∈ LF β (λE h h). ¯ form a basis in H1 (T 2 , Z) and the dual of h ¯ is perpendicular to Recall h, h h and denoted by h⊥ . + ⊥ Lemma 15.7. For any c ∈ lE (h) h∗ + [d− E , dE ] h and m1 , m2 ∈ Z, we have

¯ − hc, m1 h + m2 hi ¯ ≥ 0. lE (m1 h + m2 h) + ⊥ Moreover, if c ∈ lE (h)h∗ + (d− E , dE )h and m1 , m2 6= 0, there exists a > 0 such

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that ¯ − hc, m1 h + m2 hi ¯ > a > 0. lE (m1 h + m2 h) Proof. The inequality for m1 = 0 or m2 = 0 follows from positive homogeneity of lE . We now assume m1 , m0 . If m2 > 0, for a sufficiently large n ∈ N, we have ¯ − hc, m1 h + m2 hi ¯ lE (m1 h + m2 h) ¯ + lE ((nm2 − m1 )h) − hc, nm2 h + m2 hi ¯ = lE (m1 h + m2 h) ¯ − hc, m2 (nh + h) ¯ = m2 (lE (nh + h) ¯ − hc, nh + hi) ¯ ≥ lE (m2 (nh + h)) ¯ ¯ ≥ lE (nh + h) − hc, nh + hi. Since ¯ − hc, nh + hi ¯ = lE (nh + h) ¯ − nlE (h) − hc, hi, ¯ lE (nh + h) + ⊥ for c ∈ lE (h)h∗ + (d− E , dE )h , then there exists a > 0 such that

¯ − nlE (h) − hc, hi ¯ > a. lim lE (nh + h)

n→∞

For m2 < 0, we replace the term (nm2 − m1 )h with (−nm2 − m1 )h in the above calculation. The next statement follows from a more general result of Massart (Theorem 1.3 of [57]) for Lagrangian systems on the surface. Here we give a proof in the simpler T2 case. + ⊥ Proposition 15.8. For any E > 0 and c ∈ lE (h)h∗ + (d− E , dE )h , we have

AH s (c) = γhE if E is not a bifurcation value and AH s (c) = γhE ∪ γ¯hE if E is a bifurcation value. Proof. We first consider the case when E is not a bifurcation value. Since γhE is the unique closed shortest geodesic, if AH s (c) ⊇ γhE , it must contain an infinite orbit γ + . Moreover, as γhE supports the unique minimal measure, the orbit γ + must be biasymptotic to γhE . As a consequence, there exist Tn , Tn0 → ∞ such that γ + (−Tn ) − γ + (Tn0 ) → 0. By closing this orbit using a geodesic, we obtain a closed piece-wise geodesic curve γn . Moreover, since γ + has no self-intersection, we can arrange it such that γn also have no self-intersection. We have Z Z (L − c · v + α(c))(dγn ) = (L + E)(dγn ) − hc, [γn ]i = lE (γn ) − hc, [γn ]i.

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By the definition of the Aubry set, and taking the limit as n → ∞, we have lim lE (γn ) − hc, [γn ]i = 0.

n→∞

Since γn has no self-intersection, we have [γn ] is irreducible. However, this contradicts the strict inequality obtained in Lemma 15.7. We now consider the case when E is a bifurcation value, and there are two shortest geodesics γhE and γ¯hE . Assume by contradiction that AH s (c) ⊇ γhE ∪ γ¯hE . For mechanical systems on T2 , the Aubry set satisfies an ordering property. As a consequence, there must exist two infinite orbits γ1+ and γ2+ contained in the Aubry set, where γ1+ is forward asymptotic to γhE and backward asymptotic to γ¯hE , and γ2+ is forward asymptotic to γ¯hE and backward asymptotic to γhE . Then there exist Tn , Tn0 , Sn , Sn0 → ∞ such that γ1+ (Tn0 ) − γ2+ (−Sn ), γ2+ (Sn0 ) − γ1+ (−Tn ) → 0 + as n → ∞. The curves γ1,2 , γhE , γ¯hE are all disjoint on T2 . Similar to the previous case, we can construct a piecewise geodesic, non-self-intersecting closed curve γn with Z

lim

n→∞

(L − c · v + α(c))(dγn ) = 0.

This, however, leads to a contradiction for the same reason as the first case.

15.3

THE WIDTH OF THE CHANNEL

We show that under our assumptions, the “width” of the channel − ¯ ¯ d+ E (h) − dE (h) = inf (lE (nh + h) − lE (nh)) + inf (lE (nh − h) − lE (nh)) n∈N

n∈N

is non-zero. The following statement is a small modification of a theorem of Mather (see [64] and Theorem 1 of [56], for a more general result), we provide a proof using our language. Proposition 15.9. For E > 0, assume that the torus T2 is not completely foliated by shortest closed gE -geodesics in the homology h. Then − d+ E (h) − dE (h) > 0.

Remark 15.10. This is the last item of Theorem 15.1. Proof. Let M denote the union of all shortest closed gE -geodesics in the ho− mology h. We will show that M 6= T2 implies d+ E (h) − dE (h) > 0. Omit h

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dependence. For n ∈ N, denote ¯ − lE (nh)) + (lE (nh − h) ¯ − lE (nh)). dn = (lE (nh + h) Assume by contradiction that inf dn = lim dn > 0. Let γ0 be a shortest geodesic in homology h. We denote γ e0 its lift to the universal cover, and use “≤γ˜ ” to denote the order on γ e defined by the flow. Let ¯ and nh − h ¯ respectively, γ1 and γ2 be shortest curves in the homology nh + h and let T1 and T2 be their periods. γi depends on n but we will not write it down explicitly. Let γ ei , i = 0, 1, 2 denote a lift of γi to the universal cover. Using the standard curve shortening lemma in Riemmanian geometry, it’s easy to see that γ ei and γ ej may intersect at most once. Let a ∈ γ0 ∩ γ1 and lift it to the universal cover without changing its name. Let b ∈ γ0 ∩ γ2 , and we choose a lift in γ e0 by the largest element such that b ≤ a. We now choose the lifts γ ei of γi , i = 1, 2, by the relations γ e1 (0) = a and γ e2 (T2 ) = b. We have for 1 ≤ k ≤ 2n, γ e2 (T2 ) + kh > γ e1 (0) and ¯ + kh ≤γ˜ a + nh + h ¯=γ γ e2 (0) + kh = b − (nh − h) e1 (T1 ). 0 As a consequence, γ e2 + kh and γ e1 has a unique intersection. Let xk = (e γ2 + kh) ∩ γ e1 ,

x ¯k = (e γ2 + kh) ∩ (e γ1 − h).

We have xk is in increasing order on γ e1 and x ¯k is in decreasing order after projection to γ2 (see Figure 15.1) . Define γ ek∗ = (e γ2 + kh)|[¯ xk , xk ] ∗ γ e1 |[xk , xk+1 ], and let γk∗ be its projection to T2 . We have [γk ] = h and 2n X

lE (γk∗ ) = lE (γ1 ) + lE (γ2 ).

k=1

Using lE (γk ) ≥ lE (h) and lE (γ1 ) + lE (γ2 ) ≤ 2nlE (h) + dn , we obtain lE (h) ≤ lE (γk ) ≤ lE (h) + dn . Any connected component in the completement of M is diffeomorphic to an annulus. Pick one such annulus, and let b > 0 denote the distance between its boundaries. Since γ1 intersects each boundary once, there exists a point yn ∈ γ1 S such that d(yn , M) = b/2. Since γ1 ⊂ k γk∗ there exists some γk∗ containing yn . By taking a subsequence if necessary, we may assume yn → y∗ ∈ / M. Using the above discussion, we have lE (h) ≤

inf

y∗ ∈γ,[γ]=h

lE (γ) ≤ inf lE (h) + dn = lE (h). n

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Figure 15.1: Proof of Proposition 15.9; the L-shaped curves are γ˜k∗ ’s.

Since the collection of rectifiable curves satisfying y∗ ∈ γ, [γ] = h is compact, and that lE is lower semi-continuous over such curves, there exists a rectifiable curve γ∗ containing y∗ with lE (γ∗ ) = lE (h), hence γ∗ is a shortest curve. But y∗ ∈ / M, leading to a contradiction. Proposition 15.9 clearly applies to the slow system, as there are either one or two shortest geodesics.

15.4

THE CASE E = 0

We now extend the earlier discussions to the case E = 0. While the function ± ¯ a± E is not defined at E = 0, the function dE is well defined at E = 0. Recall h, h ¯ is perpendicular to h and denoted form a basis in H1 (T 2 , Z) and the dual of h by h⊥ . Proposition 15.11. The properties of the channel and the Aubry sets depend on the type of homology h. 1. Assume h is simple and critical. − a) d+ 0 (h) − d0 (h) > 0. + ∗ ⊥ b) l0 (h)h + [d− 0 (h), d0 (h)] h ⊂ LF β (0). − ∗ ⊥ 0 c) For c ∈ l0 (h) h + [d0 (h), d+ 0 (h)] h , we have γh ⊂ AH s (c); − + ∗ ⊥ For c ∈ l0 (h)h + (d0 (h), d0 (h)) h , we have γh0 = AH s (c).

2. Assume h is simple and non-critical. − a) d+ 0 (h) − d0 (h) > 0. + ∗ ⊥ b) l0 (h)h + [d− 0 (h), d0 (h)] h ⊂ LF β (0). − ∗ ⊥ 0 c) For c ∈ l0 (h)h + [d0 (h), d+ 0 (h)] h , we have γh ∪ {0} ⊂ AH s (c); − + ∗ ⊥ For c ∈ l0 (h)h + (d0 (h), d0 (h))h , we have γh0 ∪ {0} = AH s (c). d) The function d± E (h) is right-continuous at E = 0.

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3. Assume h is non-simple and h = n1 h1 + n2 h2 , with h1 , h2 simple. − ∗ ∗ ∗ ∗ ∗ a) d+ 0 (h) = d0 (h). Moreover, let c (h) = lE (h1 )h1 + lE (h2 )h2 , where (h1 , h2 ) is the dual basis to (h1 , h2 ), then ⊥ c∗ = lE (h)h∗ + d± 0 (h)h ,

where h⊥ is a unit vector perpendicular to h. b) γh01 ∪ γh02 = AH s (c∗ ). − c) d+ 0 (h1 ) − d0 (h1 ) > 0 with ∗ ∗ lE (h1 )h∗1 + d+ 0 (h1 )h2 = c .

Before proving Proposition 15.11, we first explain how the proof of Proposition 15.9 can be adapted to work even for E = 0. Lemma 15.12. Assume that there is a unique g0 −shortest geodesic in the homology h. Then − d+ 0 (h) − d0 (h) > 0. Proof. We will try to adapt the proof of Proposition 15.9. Let γ0 , γ1 , and γ2 be ¯ and nh − h, ¯ respectively. We choose shortest geodesics in homologies h, nh + h an arbitrary parametrization for γi on [0, T ]. Note that the parametrization is just continuous in general. The proof of Proposition 15.9 relies only on the property that lifted shortest geodesics intersect at most once. For E = 0, we will rely on a weaker property. Let γ ei be the lifts to the universal cover R2 . The degenerate point {0} lifts to the integer lattice Z2 . Since g0 is a Riemannian metric away from the integers, using the shortening argument, we have: If γi intersect γj at more than one point, then either the intersections occur only at integer points, or the two curve coincide on a segment with integer end points. Let a0 ∈ γ0 ∩ γ1 and let γ e0 and γ1 be lifts with γ e0 (0) = γ e1 (0) = a0 . If a ∈ / Z2 , then it is the only intersection between the two curves. If a0 ∈ Z2 , we define a00 to be the largest intersection between γ e0 |[0, T ) and γ e1 according to the order on γ e0 . a00 is necessarily an integer point, and since a00 ∈ γ e0 , there exists n1 < n e0 is minimizing, we such that a00 − a0 = n0 h. Moreover, using the fact that γ have l0 (e γ0 |[a0 , a00 ]) = l0 (e γ1 |[a0 , a00 ]). ¯ and γ ¯=γ We now apply a similar argument to γ e0 + h e1 . Let a1 = γ e0 (T ) + h e1 (T ) 0 and let a1 be the smallest intersection between γ e0 |(0, T ] and γ e1 . Then there exists n1 ∈ N, n0 + n1 < n, such that a1 − a01 = n1 h. Moreover, 0 ¯ l0 ((e γ0 + h)|[a γ1 |[a01 , a1 ]). 1 , a1 ]) = l0 (e

¯ =: Let ηe1 = γ e1 |[a00 , a01 ] and η1 be its projection. We have [η] = (n − n0 − n1 )h + h

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¯ and m1 h + h, l0 (η1 ) − m1 l0 (h) = l0 (γ1 ) − nl0 (h). The curve η1 has the property that it intersects γ0 only once. Applying the ¯ and same argument to γ2 , we obtain a curve η2 with [η2 ] = m2 h − h, l0 (η2 ) − m2 l0 (h) = l0 (γ2 ) − nl0 (h). To proceed as in the proof of Proposition 15.9, we show that if ηe1 and ηe2 are lifts of η1 and η2 with the property that ηe1 (0), ηe2 (T ) ∈ {e γ0 (t)},

¯ ηe1 (T ), ηe2 (0) ∈ {e γ0 (t) + h},

then ηe1 and ηe2 intersect only once. Indeed, there are no integer points between ¯ γ e0 and γ e0 + h. We have l0 (η1 ) − m1 l0 (h) + l0 (η2 ) − m2 l0 (h) = dn , where dn is as defined in Proposition 15.9. Assuming inf dn = 0, we proceed as in the proof of Proposition 15.9 and obtain curves [γk ] = h, positive distance away from γ0 , such that l0 (h) ≤ l0 (γk ) ≤ l0 (h) + dn . This leads to a contradiction. Proof of Proposition 15.11. Case 1, h is simple and critical. (a) This follows from Lemma 15.12. (b) We note that Lemma 15.7 depends only on positive homogeinity and sub-additivity of lE (h), and hence applies even when E = 0. We obtain for + ¯∗ c ∈ l0 (h)h∗ + [d− 0 (h), d0 (h)]h l0 (h0 ) − hc, h0 i ≥ 0, ∀h0 ∈ H1 (T2 , Z2 ). Since lE (h) is strictly increasing, we obtain lE (h0 ) − hc, h0 i > 0 for E > 0. By Lemma 15.3, there are no c-minimal measures with energy E > 0. As a consequence, α(c) = 0. Since {0} is a c-minimal measure with rotation number + ¯∗ 0, we conclude l0 (h)h∗ + [d− 0 (h), d0 (h)]h ⊂ LF β (0). (c) Since we proved α(c) = 0, the first conclusion follows from Lemma 15.3. For the second conclusion, we verify that the proof of Proposition 15.8 for nonbifurcation values applies to this case. + (d) The set function [d− E (h), dE (h)] is upper semi-continuous at E = 0 from the right, by definition. We will show that it is continuous. Assume by contradiction that + − + [lim inf d− E (h), lim sup dE (h)] ( [d0 (h), d0 (h)]. E→0+

E→0+

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+ ¯∗ Then there exists c ∈ l0 (h)h∗ + (d− 0 (h), d0 (h))h and + ⊥ c(E) ∈ / lE (h)h∗ + [d− E (h), dE (h)]h

such that c(E) → c. By part (c), the Aubry set AH s (c) supports a unique minimal measure. By Proposition 7.3, the Aubry set is upper semi-continuous in c. Hence any limit point of AH s (c(E)) as E → 0 is in AH s (c). This implies that AeH s (c(E)) approaches γhE as E → 0. Since γhE is the unique closed geodesic in a neighborhood of itself, we conclude that AeH s (c(E)) = γhE for sufficiently + ⊥ small E. But this is a contradiction with c(E) ∈ / lE (h)h∗ + [d− E (h), dE (h)]h . Case 2, h is simple and non-critical. (a) This follows from Lemma 15.12. (b) The proof is identical to that of Case 1. (c) For the first conclusion, we can directly verify that γh0 ⊂ AH s (c) and {0} ⊂ AH s (c). For the second conclusion, we note that proof of Proposition 15.8 for bifurcation values applies to this case. Case 3, h is non-simple with h = n1 h1 + n2 h2 . ¯ = m1 h1 + m2 h2 for some m1 , m2 ∈ Z. For sufficiently (a) Assume that h ¯ ∈ Nh1 + Nh2 . As a consequence, large n ∈ N, we have nh ± h ¯ − l0 (nh) l0 (nh ± h) = (nn1 ± m1 )l0 (h1 ) + (nn2 ± m2 )l0 (h2 ) − (nn1 l0 (h1 ) + nn2 l0 (h2 )) = ±m1 l0 (h1 ) ± m2 l0 (h2 ). − We obtain d+ 0 (h) − d0 (h) = 0 by definition. We check directly that

l0 (h) − hc∗ , hi = 0. + ⊥ ∗ ⊥ Since l0 (h)h∗ + d− 0 (h)h = l0 (h)h + d0 (h)h is the unique c with this property, the second claim follows. (b) We note that any connected component of the complement to γh01 ∪ 0 γh2 is contractible. If AH s (c) has other components, the only possibility is a contractible orbit bi-asymptotic to {0}. However, such an orbit can never be minimal, as the fixed point {0} has smaller action. − (c) The statement d+ 0 (h1 )−d0 (h1 ) > 0 follows from part 1(a). for the second claim, we compute

d+ 0 (h1 ) = inf l0 (nh1 + h2 ) − l0 (nh2 ) = l0 (h2 ) n

and the claim follows.

Appendix Notations We provide a list of notations for the reader’s convenience.

FORMULATION OF THE MAIN RESULT (θ, p, t) φtH φs,t H φH H = H0 + H1 D Z3∗ = Z3 \ {(0, 0, 1)}Z k = (k 1 , k 2 , k 0 ) Sk Γk Γk1 ,k2 Sr P K = {(k, Γk )} U = U(P) λ USR (k1 , Γk1 )

UDR (k1 , k2 )

A point in the phase space Tn × Rn × T$ . The flow of H(θ, p, t) on Tn × Rn × T$ , or the flow of H(θ, p) on Tn × Rn . The Hamiltonian diffeomorphism of H(θ, p, t) from time s to times t. The time-$ map of φtH . Equal to the Poincar´e map of φtH on {t = 0}. Nearly integrable system. A fixed constant controlling the norm and convexity of H0 . Set of integer vectors defining a resonance relation. Resonance vectors, contained in Z3∗ . Single-resonance surface in the action space given by k ∈ Z3∗ . Singe-resonance segment, a closed segment contained in Sk . Double resonance point in the action space given by k1 , k2 ∈ Z3∗ . The unit sphere of the C r functions. Diffusion path consisting of segments of single resonance segments. Collection of resonances and resonance segments making up a diffusion path. An open and dense set in S r defining the “nondegenerate perturbations” relative to a diffusion path. Set of H1 ∈ S r satisfying the quantitative non-resonance conditions [SR1λ ]–[SR3λ ] relative to the resonant segment (k1 , Γk1 ). Set of H1 ∈ S r satisfying the non-degeneracy conditions [DR1h ] − [DR3h ] and [DR1c ]–[DR4c ].

196 V = V(U, 0 ) Kst (k1 , Γk1 , λ)

Bσ Vσr , Vσ

APPENDIX A

A “cusp” set of perturbations, equal to {H1 : H1 ∈ U, 0 <  < 0 (H1 )}. Set of strong additional resonances relative to the resonance segment Γk1 , for an perturbation H1 ∈ λ USR (k1 , Γk1 ). Ball in Euclidean space Rn . Ball in the functional space C r . When supscript is not indicated, then stands for C r where r is from the main theorem.

WEAK KAM AND MATHER THEORY $ = $(H) T$ = R/($Z) H = H(D) L = LH LH,c AH,c (x, s, y, t) αH (c), αL (c) βH (ρ), βL (ρ) hH,c (x, s, y, t) hH,c (x, y) dH,c (x, s, y, t) Tcs,t u(x) ∂ + u(x) Gc,w , w = w(θ, t) e w) I(c, fH (c), AeH (c), N eH (c) M f0 (c), Ae0 (c), N e 0 (c) M H H H

e S(H, c) c ` c0 c a` c0

The period of a time periodic Hamiltonian, i.e. H(θ, p, t + $) = H(θ, p, t). Torus with period $. A family of Hamiltonians satisfying uniform conditions depending on the parameter D > 1. The Lagrangian of H. The “penalized” Lagrangian LH (θ, v, t) − c · v. The minimal action for the Lagrangian LH,c . Mather’s alpha function. Mather’s beta function. The time-dependent Peierl’s barrier function. The discrete time Peierl’s barrier function, equal to hH,c (x, 0, y, 0). Mather’s semi-distance Lax-Oleinik semi-group defined on Tn . The supergradient of a semi-concave function at x. The time-dependent pseudograph as a subset of Tn × Rn × R, or Tn × Rn × T$ . The maximum invariant set contained in the psudograph Gc,w . The continuous (Hamiltonian) Mather, Aubry, and Ma˜ ne set, defined on Tn × Rn × T$(H) . The discrete (Hamiltonian) Mather, Aubry, and Ma˜ ne set, defined on Tn × Rn , invariant under the map φH = $(H) φH . A static class of the Aubry set AeH (c). The forcing relation. The forcing equivalence relation.

197

APPENDIX

SINGLE RESONANCE [H1 ]k1 [H1 ]k1 ,k2 Φ N Zk1 (θs , p, t) R(θ, p, t) Kst (k1 , Γk1 , K) ΓSR k1 k · kCIr Tω B ∈ SL(3, Z) (θs , θf , ps , pf , t)

(Θs , P s )(θf , pf , t)

The average of H1 relative to the resonance k1 . The average of H1 relative to the double resoannce k1 , k2 . The averaging coordinate change at single resonance. Normal form under the coordinate change. Same notation is used at double resonance. The resonant component of H1 relative to the resonance k1 . The remainder in the single resonance normal form. Set of strong additional resonance k2 intersecting Γk1 with norm at most K, plus any resonances in the diffusion path that intersect Γk1 . √ The punctured resonance segment after removing O( ) neighborhoods of strong double resonances. The rescaled C r norm where √ the derivatives in the action variable are rescaled by . The period of the rational vector (ω, 1) ∈ R3 . An integer matrix defining the linear coordinate change relative to a double resonance k1 , k2 . The coordinate at single resonance after taking a linear coordinate change. The resonance becomes (1, 0, 0) · (ω, 1) = 0 in this coordinate. Normally hyperbolic invariant cylinder for the single resonant normal form, parametrized using the (θf , pf , t) variables.

DOUBLE RESONANCE Φ N ΦL K(I) U (ϕ) gE h γhE ηhE

The averaging coordinate change at double resonance. The normal form at double resonance. The linear coordinate change corresponding to the double resonance k1 , k2 . The kinetic energy of the slow mechanical system. The potential function of the slow mechanical system. The Jacobi metric at energy E of the slow mechanical system. A homology class in H 1 (T2 , Z). Shortest curves of the Jacobi metric in homology class h. The Hamiltonian periodic orbit corresponding to the geodesic γhE .

198 Φij loc , i, j ∈ {+, −} Φglob c¯h (E) ¯h Γ ¯e Γ h

¯ e,µ Γ h1 Φ∗L Φ∗L,Hs ΓDR k1 ,k2

APPENDIX A

Local maps near the saddle fixed point of H s . Global map along a homoclinic orbit η of H s . Curve of cohomologies chosen in the channel of h. Choice of cohomologies along the homology h. In the non-simple case, choice of cohomology curve above energy e. In the non-simple case, choice of cohomology along the adjacent simple homology h1 . The relation between cohomology class and alpha function after the coordinate change ΦL . The relation between the cohomology classes for the coordinate change ΦL . Depends on the alpha function of the system Hs . The choice of cohomology classes at a double resonance k1 , k2 .

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