Action-minimizing Methods in Hamiltonian Dynamics (MN-50): An Introduction to Aubry-Mather Theory [Pilot project, eBook available to selected US libraries only] 9781400866618

Citation preview

Action-minimizing Methods in Hamiltonian Dynamics

c 2015 by Princeton University Press Copyright Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved ISBN 978-0-691-16450-2 A list of titles in this series appears at the back of the book Library of Congress Control Number: 2014956149 British Library Cataloging-in-Publication Data is available This book has been composed in LATEX. (SJP) Printed on acid-free paper ∞ Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Action-minimizing Methods in Hamiltonian Dynamics

An Introduction to Aubry–Mather Theory

Alfonso Sorrentino Mathematical Notes 50

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

Contents

Preface

vii

1 Tonelli Lagrangians and Hamiltonians on Compact Manifolds 1.1 Lagrangian Point of View . . . . . . . . . . . . . . . . . . . . . . 1.2 Hamiltonian Point of View . . . . . . . . . . . . . . . . . . . . .

1 1 4

2 From KAM Theory to Aubry-Mather Theory 2.1 Action-Minimizing Properties of Measures and Orbits on KAM Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

3 Action-Minimizing Invariant Measures for Tonelli Lagrangians 3.1 Action-Minimizing Measures and Mather Sets . . . . . . . . . . . 3.2 Mather Measures and Rotation Vectors . . . . . . . . . . . . . . 3.3 Mather’s α- and β-Functions . . . . . . . . . . . . . . . . . . . . 3.4 The Symplectic Invariance of Mather Sets . . . . . . . . . . . . . 3.5 An Example: The Simple Pendulum (Part I) . . . . . . . . . . . 3.6 Holonomic Measures and Generic Properties of Tonelli Lagrangians

18 18 24 28 35 39 45

4 Action-Minimizing Curves for Tonelli Lagrangians 4.1 Global Action-Minimizing Curves: Aubry and Mañé Sets . 4.2 Some Topological and Symplectic Properties of the Aubry Mañé Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 An Example: The Simple Pendulum (Part II) . . . . . . . . 4.4 Mather’s Approach: Peierls’ Barrier . . . . . . . . . . . . .

48 48

. . . and . . . . . . . . .

8

66 68 71

5 The Hamilton-Jacobi Equation and Weak KAM Theory 76 5.1 Weak Solutions and Subsolutions of Hamilton-Jacobi and Fathi’s Weak KAM theory . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 Regularity of Critical Subsolutions . . . . . . . . . . . . . . . . . 85 5.3 Non-Wandering Points of the Mañé Set . . . . . . . . . . . . . . 87 Appendices A On the Existence of Invariant Lagrangian Graphs 89 A.1 Symplectic Geometry of the Phase Space . . . . . . . . . . . . . 89 A.2 Existence and Nonexistence of Invariant Lagrangian Graphs . . . 91

vi

CONTENTS

B Schwartzman Asymptotic Cycle and Dynamics 97 B.1 Schwartzman Asymptotic Cycle . . . . . . . . . . . . . . . . . . . 97 B.2 Dynamical Properties . . . . . . . . . . . . . . . . . . . . . . . . 99 Bibliography

107

Index

113

Preface In the 1950s and 1960s the celebrated Kolmogorov-Arnol0d-Moser theorem(s) (commonly abbreviated with KAM from the initials of the three main pioneers) finally settled the old question concerning the existence of quasi-periodic motions for nearly-integrable Hamiltonian systems, i.e., Hamiltonian systems that are slight perturbations of an integrable one. In the integrable case, in fact, the whole phase space is foliated by invariant Lagrangian submanifolds that are diffeomorphic to tori and on which the dynamics is conjugate to a rigid rotation. On the other hand, it is natural to ask what happens to such a foliation and to these stable motions once the system is perturbed. In 1954 Kolmogorov [48]—and later Arnol0d [3] and Moser [71, 70] in different contexts—proved that, in spite of the generic disappearence of the invariant tori filled by periodic orbits (already pointed out in the works of Henri Poincaré), for small perturbations of an integrable system it is still possible to find invariant Lagrangian tori corresponding to “strongly non-resonant” rotation vectors. This result, commonly referred to as KAM theorem, besides opening the way to a new understanding of the nature of Hamiltonian systems and their stable motions, contributed to raising new, interesting questions. For instance: what is the destiny of the stable motions (orbits on KAM tori) that are destroyed by the effect of the perturbation? Is it still possible to identify something reminiscent of their former presence? What can be said about a system that is not close to an integrable one? While all these questions are concerned with the investigation of stable motions of the perturbed system, another interesting issue soon took the stage: Does the breakdown of this stable picture open the way to orbits with unstable or chaotic behaviours? An answer to this latter question did not take long to arrive. In 1964 V. I. Arnol0d [4] constructed an example of a perturbed integrable system in which unstable orbits—resulting from the breaking of unperturbed KAM tori—coexist with the stable picture drawn by the KAM theorem. This striking, and somewhat unexpected, phenomenon, yet not completely understood, is nowadays called Arnol 0d diffusion. This new insight led to a change in perspective, and in order to make sense of the complex balance between stable and unstable motions that was becoming more evident, new approaches needed to be exploited. Among these, variational methods turned out to be particularly suitable and successful. Mostly inspired

viii

PREFACE

by the so-called principle of least action,1 a sort of widely accepted “thriftiness” of nature in all its actions, they seemed to provide the natural setting to overcome the local view given by the analytic methods and make progress towards a global understanding of the dynamics. Aubry-Mather theory represented undoubtedly a great leap forward in this direction. Developed independently by Serge Aubry [6, 7] and John Mather [57] in the 1980s, this novel approach to the study of the dynamics of twist diffeomorphisms of the annulus (which correspond to Poincaré maps of onedimensional Hamiltonian systems [71]) pointed out the existence of many actionminimizing sets, which in some sense generalize invariant rotational curves and which always exist, even after rotational curves are destroyed. Besides providing a detailed structure theory for these new sets, this powerful approach yielded to a better understanding of the destiny of invariant rotational curves and led to the construction of interesting chaotic orbits as a result of their destruction [59, 43, 63]. Motivated by these achievements, John Mather [62, 64]—and later Ricardo Mañé [53, 26, 27] and Albert Fathi [33, 34, 35, 36, 37] in different ways— developed a generalization of this theory to higher dimensional systems. Positive definite superlinear Lagrangians on compact manifolds, also called Tonelli Lagrangians (see Definition 1.1.1), were the appropriate setting to work in. Under these conditions, in fact, it is possible to prove the existence of interesting invariant (action-minimizing) sets, known as Mather, Aubry, and Mañé sets, which are obtained as minimizing solutions to variational problems. As a result, these objects present a much richer structure and rigidity than one might generally expect and, quite surprisingly, play a leading role in determining the global dynamics of the system. In particular, they can be considered as a sort of generalization of the KAM tori, but they continue to exist even after KAM tori’s disappearance or when it does not make sense to speak of them (for example when the system is “far” from any integrable one). These tools—while dealing with stable motions—were also quite promising in the construction of chaotic orbits, for instance, connecting orbits among the above-mentioned invariant sets [64, 10, 30]. Therefore they raised hopes for the possibility of proving the generic existence of Arnol0d diffusion in nearly integrable Hamiltonian systems [65, 47, 19, 25, 24]. However, in contrast with the case of twist diffeomorphisms, the situation turns out to be more complicated, due to a general lack of information on the topological structure of these action-minimizing sets. These sets, in fact, play a twofold role. Whereas on the one hand they may provide an obstruction to the existence of “diffusing or1 “Nature is thrifty in all its actions,” according to Pierre-Louis Moreau de Maupertuis (1744). A better-known special case of this principle is what is generally called Maupertius’ principle. Actually, as I learned from Leo Butler, König published a note arguing for priority to Leibniz in the Berlin Academy correspondences overseen by Maupertuis. A priority dispute brought in Euler and Voltaire, and ultimately a committee was convened by the King of Prussia. In 1913, the Berlin Academy reversed its previous decision and found that Leibniz had priority.

PREFACE

ix

bits,” on the other hand their topological structure plays a fundamental role in the variational methods that have been developed for the construction of orbits with prescribed behaviors. We will not enter further into the discussion of this problem, but we refer the interested readers to [10, 14, 30, 65, 68, 47, 19, 25, 24]. In addition to its fundamental impact on the modern study of classical dynamics, Mather’s theory has also contributed to pointing out interesting and sometimes unexpected links to other fields of research (both pure and applied), fostering a multidisciplinary interest in its ideas and in the techniques involved. These lecture notes aim at satisfying a twofold demand. On the one hand, they are addressed to researchers and students in the field of dynamical systems and intend to provide a comprehensive and pedagogic introduction to Mather’s theory and its subsequent developments. On the other hand, they are tailored to be used as an “interdisciplinary bridge,” which should help researchers and students coming from different fields and backgrounds, to get acquainted with the main ideas and techniques, and help orient them to the huge body of research literature available on the topic. Starting from the mathematical background on which Mather’s theory was born and has developed, these notes firstly focus on the main questions this theory aims to answer (the destiny of broken invariant KAM tori and the onset of chaos) and try to highlight the sense in which this theory can be viewed as a generalization of KAM theory (or a natural counterpart to it). This will be achieved by their guiding the reader through a detailed illustrative example, which, at the same time, will provide the pretext for introducing the main ideas and concepts of the general theory. Once the crucial ideas are clarified, the whole theory and its subsequent developments will be described in their full generality. We will consider only the autonomous case (i.e., no dependence on time in the Lagrangian and Hamiltonian). This choice has been made only to make the discussion easier and to avoid some technical issues that would be otherwise involved. However, all the theory that we are going to describe can be generalized, with some small modifications, to the non-autonomous timeperiodic case. During our discussion, in order to draw the most complete picture of the theory, we will point out and discuss such differences and the needed modifications. These lecture notes are organized as follows. They consist of five chapters: 1. Tonelli Lagrangian and Hamiltonians on compact manifolds. In this chapter we introduce the basic setting: Tonelli Lagrangians and Hamiltonians on a compact manifold. Besides discussing their main properties and some examples, this chapter provides the opportunity to recall some basic facts on Lagrangian and Hamiltonian dynamics (and on their mutual relation), which will be of fundamental importance in the discussion thereafter. 2. From KAM theory to Aubry-Mather theory. Before entering into the description of the general theory, in this chapter we discuss an illustra-

x

PREFACE

tive example, namely the properties of invariant probability measures and orbits on KAM tori (or more generally, on invariant Lagrangian graphs). This will prepare the ground for understanding the main ideas and techniques that will be developed in the following chapters, without several technicalities that might be confusing to a neophyte. 3. Action-minimizing invariant measures for Tonelli Lagrangians. In this chapter we discuss the notion of action-minimizing measures, recalling the needed measure-theoretical material. In particular, this will allow us to define a first family of invariant sets, the so-called Mather sets, and discuss their main dynamical and symplectic properties. As a by-product, we introduce the minimal average actions, sometimes called Mather’s αand β-functions. A thorough discussion of their properties (differentiability, strict convexity or lack thereof) will be provided and related to the dynamical and structural properties of the Mather sets. We also describe these concepts in a concrete physical example: the simple pendulum. 4. Global action-minimizing curves for Tonelli Lagrangians. In this chapter we discuss the notion of action-minimizing orbits. In particular, this will allow us to define other two families of invariant sets, the so-called Aubry and Mañé sets. We will discuss their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets will be carefully described. As a by-product, we will introduce the Mañé’s potential, Peierls’ barrier, and Mañé’s critical value. We will throughly discuss their properties; in particular, we will highlight how this critical value is related to the minimal average action and describe these new concepts in the case of the simple pendulum, completing the picture started in the preceding chapter. 5. Hamilton-Jacobi equation and weak KAM theory. Another interesting approach to the study of these invariant sets is provided by the so-called weak KAM theory, developed by Albert Fathi. In this chapter we briefly describe this approach, which could be considered as the functional analytic counterpart of the variational methods discussed in the previous chapters. The starting point is the relation between KAM tori (or more generally, invariant Lagrangian graphs) and classical solutions and subsolutions of the Hamilton-Jacobi equation. We will introduce the notion of weak (non-classical) solutions of the Hamilton-Jacobi equation and a special class of subsolutions (critical subsolutions). In particular, we will point out their relation to Aubry-Mather theory. These notes also include two appendices with some extra (complementary) material: A. On the existence of invariant Lagrangian graphs. In this appendix we recall some basic notions of symplectic geometry and illustrate

PREFACE

xi

what kind of information Aubry-Mather theory conveys about the study of the integrability of Hamiltonian systems, and, more generally, how this information relates to the existence or nonexistence of invariant Lagrangian graphs. B. Schwartzman asymptotic cycle and dynamics. In this appendix we introduce and discuss the notion of the Schwartzman asymptotic cycle of a flow, introduced by Sol Schwartzman in [74], as a first attempt to develop an algebraic topological approach to the study of dynamics. We will discuss its relation to the notion of rotation vector (or homology class) introduced in the previous chapters of these notes, and investigate its main properties. ACKNOWLEDGEMENTS A wholehearted acknowledgement is undoubtedly due to John Mather, Albert Fathi, Marie-Claude Arnaud, Patrick Bernard, Gonzalo Contreras, Vadim Kaloshin, Daniel Massart, and Gabriel Paternain, from whom I learned most of the material collected here and more. These notes originated from several lectures and graduate courses that the author had the privilege of teaching at: Centro de Ricerca Matemàtica (Spain, 2008), Università degli Studi di Napoli “Federico II” (Italy, 2009), University of Cambridge (UK, 2009 and 2012), Universitat Politècnica de Catalunya (Spain, 2010), Université de Neuchâtel (Switzerland, 2011), Scuola Normale Superiore di Pisa (Italy, 2014). I am very grateful to all these institutions for their invitations and for their warm hospitality, and to all attendees of these lectures for their active participation and their stimulating feedbacks. Finally, special thanks go to Gabriele Benedetti, Renato Bortolatto, Mauro Mariani, Marco Mazzucchelli, Will Merry, Pierre Pageault, Joana dos Santos, and Rodrigo Treviño for their valuable comments and suggestions on preliminary versions of these notes. Rome, June 2014

Chapter One Tonelli Lagrangians and Hamiltonians on Compact Manifolds

1.1

LAGRANGIAN POINT OF VIEW

In this section we want to introduce the basic setting that we will be considering hereafter. Let M be a compact and connected smooth manifold without boundary. Denote by TM its tangent bundle and by T∗ M the cotangent one. A point of TM will be denoted by (x, v), where x ∈ M and v ∈ Tx M , and a point of T∗ M by (x, p), where p ∈ T∗x M is a linear form on the vector space Tx M . Let us fix a Riemannian metric g on it and denote by d the induced metric on M ; let k · kx be the norm induced by g on Tx M ; we will use the same notation for the norm induced on T∗x M . We will consider functions L : TM −→ R of class C 2 , which are called Lagrangians. Associated with each Lagrangian is a flow on TM called the Euler-Lagrange flow, defined as follows. Let us consider the action functional AL from the space of continuous piecewise C 1 curves γ : [a, b] → M , with a ≤ b, defined by: Z b AL (γ) := L(γ(t), γ(t)) ˙ dt. a

Curves that extremize this functional among all curves with the same endpoints are solutions of the Euler-Lagrange equation: d ∂L ∂L (γ(t), γ(t)) ˙ = (γ(t), γ(t)) ˙ dt ∂v ∂x

∀ t ∈ [a, b] .

(1.1)

Observe that this equation is equivalent to ∂L ∂2L ∂2L (γ(t), γ(t))¨ ˙ γ (t) = (γ(t), γ(t)) ˙ − (γ(t), γ(t)) ˙ γ(t) ˙ ; 2 ∂v ∂x ∂v∂x therefore, if the second partial vertical derivative ∂ 2 L/∂v 2 (x, v) is nondegenerate at all points of TM , we can solve for γ¨ (t). The condition det

∂2L 6= 0 ∂v 2

is called the Legendre condition and allows one to define a vector field XL on TM such that the solutions of γ¨ (t) = XL (γ(t), γ(t)) ˙ are precisely the curves satisfying

2

CHAPTER 1

the Euler-Lagrange equation. This vector field XL is called the Euler-Lagrange vector field and its flow ΦL t is the Euler-Lagrange flow associated with L. It 1 turns out that ΦL is C even if L is only C 2 (see Remark 1.2.2). t Definition 1.1.1 (Tonelli Lagrangian). A function L : TM −→ R is called a Tonelli Lagrangian if: i) L ∈ C 2 (TM ); ii) L is strictly convex in each fiber, in the C 2 sense, i.e., the second partial vertical derivative ∂ 2 L/∂v 2 (x, v) is positive definite, as a quadratic form, for all (x, v); iii) L is superlinear in each fiber, i.e., lim

kvkx →+∞

L(x, v) = +∞. kvkx

This condition is equivalent to asking whether for each A ∈ R there exists B(A) ∈ R such that L(x, v) ≥ Akvk − B(A)

∀ (x, v) ∈ TM .

Observe that since the manifold is compact, condition iii) is independent of the choice of the Riemannian metric g. Remark 1.1.2. More generally, one can consider the case of a time-periodic Tonelli Lagrangian L : TM × T −→ R (also called non-autonomous case), as it was originally done by John Mather [62]. In fact, as it was pointed out by Jürgen Moser, this was the right setting for generalizing Aubry and Mather’s results for twist maps to higher dimensions; in fact, every twist map can be seen as the time-one map associated with the flow of a periodic Tonelli Lagrangian on the one-dimensional torus (see for instance [71]). In this case, a further condition on the Lagrangian is needed: iv) The Euler-Lagrange flow is complete, i.e., every maximal integral curve of the vector field XL has all R as its domain of definition. In the non-autonomous case, in fact, this condition is necessary in order to have that action-minimizing curves (or Tonelli minimizers; see section 4.1) satisfy the Euler-Lagrange equation. Without such an assumption Ball and Mizel [8] have constructed an example of Tonelli minimizers that are not C 1 and therefore are not solutions of the Euler-Lagrange flow. The role of the completeness hypothesis can be explained as follows. It is possible to prove, under the above conditions, that action-minimizing curves not only exist and are absolutely continuous, but also are C 1 on an open and dense full measure subset of the interval in which they are defined. It is possible to check that they satisfy the Euler-Lagrange equation on this set, while their velocity goes to infinity on the

TONELLI LAGRANGIANS AND HAMILTONIANS ON COMPACT MANIFOLDS

3

exceptional set on which they are not C 1 . A complete flow, therefore, implies that Tonelli minimizers are C 1 everywhere and that they are actual solutions of the Euler-Lagrange equation. A sufficient condition for the completeness of the Euler-Lagrange flow, for example, can be expressed in terms of a growth condition for ∂L/∂t:   ∂L ∂L (x, v, t) ≤ C 1 + (x, v, t) · v − L(x, v, t) ∀ (x, v, t) ∈ TM × T. − ∂t ∂v Examples of Tonelli Lagrangians. • Riemannian Lagrangians. Given a Riemannian metric g on M , the Riemannian Lagrangian on (TM, g) is given by the Kinetic energy: L(x, v) =

1 kvk2x . 2

Its Euler-Lagrange equation is the equation of the geodesics of g: ∇x˙ x˙ ≡ 0 (where ∇x˙ x˙ denotes the covariant derivative), and its Euler-Lagrange flow coincides with the geodesic flow. • Mechanical Lagrangians. These Lagrangians play a key role in the study of classical mechanics. They are given by the sum of the kinetic energy and a potential U : M −→ R: L(x, v) =

1 kvk2x + U (x) . 2

The associated Euler-Lagrange equation is given by: ∇x˙ x˙ = ∇U (x) , where ∇U is the gradient of U with respect to the Riemannian metric g, i.e., dx U · v = h∇U (x), vix ∀ (x, v) ∈ TM . • Mañé’s Lagrangians. This is a particular class of Tonelli Lagrangians, introduced1 by Ricardo Mañé in [51] (see also [39]). If X is a C k vector field on M , with k ≥ 2, one can embed its flow ϕX t into the Euler-Lagrange flow associated with a certain Lagrangian, namely LX (x, v) =

1 2 kv − X(x)kx . 2

1 It is interesting to notice that similar ideas had already appeared in embryonic stages in the literature; see, for example, Freidlin and Wentzell’s study of the exit time from a domain for a random perturbation of a vector field [44].

4

CHAPTER 1

It is quite easy to check that the integral curves of the vector field X are solutions to the Euler-Lagrange equation. In particular, the Euler-Lagrange X flow ΦL restricted to Graph(X) = {(x, X(x)), x ∈ M } (which is clearly t invariant) is conjugated to the flow of X on M , and the conjugation is given by π|Graph(X), where π : TM → M is the canonical projection. In other words, the following diagram commutes: L

Graph(X)

Φt X

/ Graph(X)

π

 M

π

ϕX t

 /M

X X ˙ xX (t)), that is, for every x ∈ M and every t ∈ R, ΦL t (x, X(x)) = (γx (t), γ (x). where γxX (t) = ϕX t

1.2

HAMILTONIAN POINT OF VIEW

In the study of classical dynamics, it turns often very useful to consider the associated Hamiltonian system, which is defined on the cotangent bundle T∗ M . Let us describe how to define this new system and what its relation is with the Lagrangian one. A standard tool in the study of convex functions is the so-called Fenchel transform, which allows one to transform functions on a vector space into functions on the dual space (see for instance [37, 73] for excellent introductions to the topic). Given a Lagrangian L, we can define the associated Hamiltonian as its Fenchel transform (or Legendre-Fenchel transform): H : T∗ M −→ (x, p) 7−→

R sup {hp, vix − L(x, v)} v∈Tx M

where h ·, · ix denotes the canonical pairing between the tangent and cotangent bundles. If L is a Tonelli Lagrangian, one can easily prove that H is finite everywhere (as a consequence of the superlinearity of L) and C 2 , superlinear and strictly convex in each fiber (in the C 2 sense). Such a Hamiltonian is called a Tonelli (or optical ) Hamiltonian. Definition 1.2.1 (Tonelli Hamiltonian). A function H : T∗ M −→ R is called a Tonelli (or optical) Hamiltonian if: i) H is of class C 2 ;

TONELLI LAGRANGIANS AND HAMILTONIANS ON COMPACT MANIFOLDS

5

ii) H is strictly convex in each fiber in the C 2 sense, i.e., the second partial vertical derivative ∂ 2 H/∂p2 (x, p) is positive definite, as a quadratic form, for any (x, p) ∈ T∗ M ; iii) H is superlinear in each fiber, i.e., H(x, p) = +∞ . kpkx →+∞ kpkx lim

Examples of Tonelli Hamiltonians Let us see what the Hamiltonians associated with the Tonelli Lagrangians that we have introduced in the previous examples are. • Riemannian Hamiltonians. If L(x, v) = 12 kvk2x is the Riemannian Lagrangian associated with a Riemannian metric g on M , the corresponding Hamiltonian will be 1 H(x, p) = kpk2x , 2 where k · k represents—in this last expression—the induced norm on the cotangent bundle T∗ M . • Mechanical Hamiltonians. If L(x, v) = 12 kvk2x + U (x) is a mechanical Lagrangian, the associated Hamiltonian is: H(x, p) =

1 kpk2x − U (x), 2

sometimes referred to as mechanical energy. • Mañé’s Hamiltonians. If X is a C k vector field on M , with k ≥ 2, 2 and LX (x, v) = kv − X(x)kx is the associated Mañé Lagrangian, one can check that the corresponding Hamiltonian is given by: H(x, p) =

1 kpk2x + hp, X(x)i . 2

Given a Hamiltonian, one can consider the associated Hamiltonian flow ΦH t on T∗ M . In local coordinates, this flow can be expressed in terms of the so-called Hamilton equations:  x(t) ˙ = ∂H ∂p (x(t), p(t)) (1.2) p(t) ˙ = − ∂H ∂x (x(t), p(t)) .   ∂H We will denote by XH (x, p) := ∂H (x, p), − (x, p) the Hamiltonian vec∂p ∂x tor field associated with H. This has a more intrinsic (geometric) definition in

6

CHAPTER 1

terms of the canonical symplectic structure ω on T∗ M (see appendix A). In fact, XH is the unique vector field that satisfies ∀(x, p) ∈ T∗ M.

ω (XH (x, p), ·) = dx H(·)

For this reason, it is sometime called the symplectic gradient of H. It is easy to check from both definitions that—only in the autonomous case—the Hamiltonian is a prime integral of the motion, i.e., it is constant along the solutions of these equations. Now, we would like to explain what the relation is between the EulerLagrange flow and the Hamiltonian one. It follows easily from the definition of Hamiltonian (and Fenchel transform) that for each (x, v) ∈ TM and (x, p) ∈ T∗ M the following inequality holds: hp, vix ≤ L(x, v) + H(x, p) .

(1.3)

This is called Fenchel inequality (or Legendre-Fenchel inequality) and plays a crucial role in the study of Lagrangian and Hamiltonian dynamics and in the variational methods that we are going to describe. In particular, equality holds if and only if p = ∂L/∂v(x, v). One can therefore introduce the following diffeomorphism between TM and T∗ M , known as the Legendre transform: L : TM

−→

(x, v) 7−→

∗ T   M ∂L (x, v) . x, ∂v

(1.4)

Moreover the following relation with the Hamiltonian holds:   ∂L − L(x, v) . (x, v), v H ◦ L(x, v) = ∂v x A crucial observation is that this diffeomorphism L represents a conjugation between the two flows, namely the Euler-Lagrange flow on TM and the Hamiltonian flow on T∗ M ; in other words, the following diagram commutes: TM L

ΦL t



T∗ M

/ TM 

ΦH t

L

/ T∗ M

Remark 1.2.2. Since L and the Hamiltonian flow ΦH are both C 1 , then it follows from the commutative diagram above that the Euler-Lagrange flow is also C 1 . Therefore one can equivalently study the Euler-Lagrange flow or the Hamiltonian flow, obtaining in both cases information on the dynamics of the system.

TONELLI LAGRANGIANS AND HAMILTONIANS ON COMPACT MANIFOLDS

7

Each of these equivalent approaches will provide different tools and advantages, which may be very useful for understanding the dynamical properties of the system. For instance, the tangent bundle is the natural setting for the classical calculus of variations and for Mather’s and Mañé’s approaches (chapters 3 and 4); on the other hand, the cotangent bundle is equipped with a canonical symplectic structure (see appendix A), which allows one to use several symplectic topological tools coming from the study of Lagrangian graphs, Hofer’s theory, Floer homology, among other subjects. Moreover, a particular fruitful approach in T∗ M is the so-called Hamilton-Jacobi method (or weak KAM theory), which is concerned with the study of solutions and subsolutions of Hamilton-Jacobi equations. In a certain sense, this approach represents the functional analytic counterpart of the above-mentioned variational approach (chapter 5). In the following chapters we will provide a complete description of these methods and their implications on the study of the dynamics of the system.

Chapter Two From KAM Theory to Aubry-Mather Theory

2.1

ACTION-MINIMIZING PROPERTIES OF MEASURES AND ORBITS ON KAM TORI

Before entering into the details of Mather’s work, we would like to discuss some properties of orbits and invariant probability measures that are supported on KAM tori. This will provide us with a better understanding of the ideas behind Mather’s theory and allow to see in what sense these action-minimizing sets—namely, what we will call Mather, Aubry and Mañé sets—represent a generalization of KAM tori. Let H : T∗ Td −→ R be a Tonelli Hamiltonian and L : TTd −→ R its associated withnelli Lagrangian and let us denote by ΦH and ΦL their respective flows. Observe that one can identify T∗ Td and TTd with Td × Rd . First of all, let us define what we mean by KAM torus. Definition 2.1.1 (KAM Torus). T ⊂ Td × Rd is a (maximal) KAM torus with rotation vector ρ if: i) T ⊂ Td × Rd is a C 1 Lagrangian graph, i.e., T = {(x, c + du) : x ∈ Td }, where c ∈ Rd and u : Td −→ R; ii) T is invariant under the Hamiltonian flow ΦH t generated by H; iii) the Hamiltonian flow on T is conjugated to a uniform rotation on Td ; i.e., t there exists a diffeomorphism ϕ : Td → T such that ϕ−1 ◦ ΦH t ◦ ϕ = Rρ , ∀t ∈ R, t d where Rρ : x → x + ρt (mod. Z ). In other words, the following diagram commutes: ΦH t

TO ϕ

Td

/T O ϕ

t Rρ

/ Td

Remark 2.1.2. (i ) First of all, let us remark that, with some minor technical modifications, most of the results that we will discuss in this section will continue to hold if we assume T to be only Lipschitz rather than C 1 . (ii ) If ρ is rationally independent, i.e., ρ · k 6= 0 for all k ∈ Zd \ {0}, then each orbit is dense on it (i.e., the motion is said to be minimal ). Consider the case in which ρ admits some resonance, i.e., there exists k ∈ Zd \ {0} such that ρ · k = 0. The set of resonances of ρ forms a module over Z. Let r denote its

9

FROM KAM THEORY TO AUBRY-MATHER THEORY

rank. Then, T is foliated by an r-dimensional family of (d − r)-tori, on each of which the motion is minimal. (iii ) If ρ is rationally independent then a classical result by Michel Herman implies that this torus is automatically Lagrangian (see [46, Proposition 3.2]). (iv ) Since T is Lagrangian and invariant under the Hamiltonian flow, the Hamiltonian H is constant on it, i.e., H(x, c + du) = Ec for some Ec ∈ R (this follows from the definition of the Hamiltonian flow). In particular, u is a classical solution of Hamilton-Jacobi equation H(x, c + dv) = k. It is easy to check that, for a fixed c, all solutions have the same energy. In fact, let u and v be two solutions. Then, the function u − v will have at least a critical point x0 and at x0 the two differentials must coincide: dx0 u = dx0 v. This implies that H(x0 , c + dx0 u) = H(x0 , c + dx0 v). Moreover, Ec is the least possible k ∈ R such that there can exist subsolutions of the equation H(x, c+dv) = k, namely v ∈ C 1 (Td ), satisfying H(x, c+dv) ≤ k for all x ∈ Td . The proof is essentially the same as above. In fact, since the function u − v has at least a critical point x0 and at x0 , the two differentials coincide, then Ec = H(x0 , c + dx0 u) = H(x0 , c + dx0 v) ≤ k. Let us start by studying the properties of invariant probability measures that are supported on T . Let µ∗ be an ergodic invariant probability measure supported on T . It will be more convenient at this point to work in the Lagrangian setting. Let us consider the correspondent invariant probability measure for the Euler-Lagrange flow ΦL , obtained using the Legendre transform: µ = L∗ µ∗ , where L∗ denotes the pullback. We would like to point out some properties of this measure and see how T can be characterized using measures satisfying such properties. Properties: R 1. First of all, observe that vdµ = ρ. To see this, let us consider the universal cover of Td , i.e., Rd , and denote by • Te := {(q, c+d˜ u(q)), q ∈ Rd } the lift of T to Rd , where u ˜ is a periodic extension of u; g H the suspension of the Hamiltonian flow ΦH to Rd × Rd ; • Φ t

t

• ϕ e the lift of ϕ, i.e., ϕ : Rd → Te such that the following diagram commutes: TeO

g H Φ t

ϕ ˜

ϕ ˜

Rd

/ Te O

et R ρ

/ Rd

10

CHAPTER 2

eρt : q → q + ρt. Moreover, ϕ where R e has the form ϕ(q) e = (ϕ eq (q), c + d˜ u(ϕ eq (q))) and ϕ eq satisfies ϕ eq (q + k) = ϕ eq (q) + k for each k ∈ Zd . Now, let (x0 , c + du(x0 )) be a point in the support of µ∗ and consider the ∗ d d corresponding orbit γx0 (t) := πΦH t (x0 , c + du(x0 )), where π : T T −→ T d is the canonical projection. Denote by γf (t) its lift to R . Then, using the x0 ergodic theorem (remember that we are assuming that µ∗ , and hence µ, is ergodic), we have that for a generic point (x0 , c + du(x0 )) in the support of µ∗ the following holds: Z Z 1 n ˙ γf x0 (n) − γf x0 (0) vdµ = lim γf x0 (t) dt = lim n→+∞ n 0 n→+∞ n TTd
ϕ eq ϕ e−1 γf x0 (0)) + nρ x0 (n) q (γf = lim = lim n→+∞ n→+∞ n n " #
−1 ϕ eq ϕ eq (γf [nρ] x0 (0)) + {nρ} = lim + n→+∞ n n = ρ, where [·] and {·} denote respectively the integer and fractional part of the vector (component by component). Remark 2.1.3. We will say that µ has rotation vector ρ. See also section 3.2 and appendix B. R 2. Let f : Td −→ R be a C 1 function. Then, df (x) · vdµ = 0. In fact, take (x0 , v0 ) ∈ supp µ and denote the corresponding orbit by (xt , vt ) = L ΦL t (x0 , v0 ). Since µ is invariant, Φt ∗ µ = µ, and therefore: Z df (x) · vdµ = TTd

= =

1 T

Z

1 T

Z

1 T Z

Z

T

Z dt TTd

0 T

Z dt

df (xt ) · vt dµ TTd Z T

0

dµ TTd

= TTd

df (x) · v dΦL t ∗µ

df (xt ) · vt dt 0

f (xT ) − f (x0 ) dµ T

T →+∞

−→ 0 ,

since f is bounded. A measure satisfying this condition is said to be closed (see also section 3.6). Observe that in the above proof we have not used anything other than the invariance of µ. See also Proposition 3.1.7. 3. Recall that Ec is the energy of T , i.e., H(x, c + du) = Ec . We want to show that this energy value is somehow related to the average value of L

11

FROM KAM THEORY TO AUBRY-MATHER THEORY

in the support of µ. Observe in fact that: Z Z Z   L(x, v) dµ = L(x, v) − (c + du) · v dµ + TTd

TTd

(c + du) · v dµ .

TTd

Using the property just pointed out in (2) and observing that, because of the Legendre-Fenchel (in)equality (1.3), along the orbits (xt , vt ) in the support of µ we have (c + du(xt )) · vt = L(xt , vt ) + H(x, c + du(xt )) = L(xt , vt ) + Ec , we can conclude that: Z Z Z L(x, v) dµ = = − Ec dµ + c · v dµ TTd TTd TTd Z = −Ec + c · v dµ = −Ec + c · ρ. TTd

  In particular, TTd L(x, v) − c · v dµ = −Ec . Observe that this new Lagrangian Lc (x, v) = L(x, v) − c · v is still Tonelli and it is easy to check that it has the same Euler-Lagrange flow as L. R

We can now show this first result. Proposition 2.1.4. If µ ˜ is another invariant probability measure of ΦL , then Z Z Lc (x, v) d˜ µ≥ Lc (x, v) dµ. (2.1) TTd

TTd

Therefore, µ minimizes the average value of Lc (or action of Lc ) in the set of L c all invariant probability measures of ΦL t (or Φt , since they are the same). Warning: In general, it does not minimize the action of L ! Proof. The proof is an easy application of the Legendre-Fenchel inequality (1.3). Observe that in the support of µ ˜, in contrast with what happens in the support of µ (see property (3) above), this is not an equality, i.e., (c+du(x))·v ≤ L(x, v) + H(x, c + du(x)) = L(x, v) + Ec . Then: Z Z   Lc (x, v) d˜ µ = L(x, v) − (c + du) · v d˜ µ TTd TTd Z Z ≥ − Ec d˜ µ = −Ec = Lc (x, v) dµ . TTd

TTd

 Remark 2.1.5. (i ) It follows from the above proposition that Z  Ec = − min Lc (x, v) d˜ µ: µ ˜ is a ΦL -invariant probability measure . TTd

12

CHAPTER 2

The minimizing action on the right-hand side is also denoted by α(c) (see the definition of Mather’s α-function in sections 3.1 and 3.3). (ii ) A measure µ that satisfies the inequality (2.1) for all invariant probability measures µ ˜ is called c-action-minimizing measure (or Mather’s measure with cohomology class c); see section 3.1. (iii ) It is easy to see from the proof of Proposition 2.1.4 and the LegendreFenchel inequality that if supp µ ˜ is not contained in L−1 (T ), then the inequality in (2.1) is strict. (iii ) In particular, using the previous remark and (ii ) in Remark 2.1.2, we obtain that [ L−1 (T ) = {supp µ : µ is a c-action-minimizing measure}. fc and called Mather set of The set on the right-hand side is often denoted by M cohomology class c. It will be defined in section 3.1. Let us now observe that although µ does not minimize the action of L among all invariant probability measures (we have pointed out in fact that it minimizes the action of a modified Lagrangian with the same Euler Lagrange flow as L), then it does minimize it if we add some extra constraints. Proposition 2.1.6. If µ ˜ is another invariant probability measure of ΦL with rotation vector ρ (in the sense of property (1) above), then Z Z L(x, v) d˜ µ≥ L(x, v) dµ. (2.2) TTd

TTd

Proof. The proof is the R same as before, using the Legendre-Fenchel inequality (1.3), the fact that v d˜ µ = ρ, and property (3) above. In fact: Z Z Z   L(x, v) d˜ µ = L(x, v) − (c + du) · v d˜ µ+ (c + du) · v d˜ µ TTd TTd TTd Z ≥ −Ec + c · v d˜ µ TTd Z = −Ec + c · ρ = L(x, v) dµ . TTd

 Remark 2.1.7. (i ) It follows from the above proposition that Z  −Ec +c·ρ = min L(x, v) d˜ µ: µ ˜ is a ΦL -inv. pb. meas. with rot. vect. ρ . TTd

The minimizing action on the right-hand side is also denoted by β(ρ) (see the definition of Mather’s β-function in sections 3.2 and 3.3).

FROM KAM THEORY TO AUBRY-MATHER THEORY

13

(ii ) A measure µ that satisfies the inequality (2.2) for all invariant probability measures µ ˜ with rotation vector ρ is called an action-minimizing measure (or Mather’s measure) with rotation vector ρ; see section 3.2. (iii ) It is easy to see from the proof of Proposition 2.1.6 and the LegendreFenchel inequality that if µ ˜ has rotation vector ρ, but its support is not contained in L−1 (T ), then the inequality in (2.2) is strict. (iv ) In particular, using the previous remark and (ii ) in Remark 2.1.2, we obtain that [ L−1(T ) = {supp µ : µ is an action-minim. measure with rotation vector ρ}. fρ and called the Mather The set on the right-hand side is often denoted by M set of homology class ρ. It will be defined in section 3.2. Summarizing, if µ is any invariant probability measure of ΦL supported on L (T ), then: −1

- µ minimizes the action of L(x, v) − c · v among all invariant probability measures of ΦL ; - µ minimizes the action of L(x, v) among all invariant probability measures of ΦL with rotation vector ρ. Let us now shift our attention to orbits on KAM tori and see what are the properties they enjoy. Let (x0 , c + dx0 u) ∈ T be a point on the KAM torus and consider its orbit under the Hamiltonian flow, i.e., γ(t) = πΦH t (x0 , c + dx0 u), where as usual π : T∗ Td −→ Td denotes the canonical projection along the fiber. Let us fix any time a < b and consider the corresponding Lagrangian action of this curve. Using the Legendre-Fenchel (in)equality we get: Z b Z b  L(γ(t), γ(t)) ˙ dt = (c + dγ(t) u)γ(t) ˙ − H(γ(t), c + dγ(t) u) dt a

a b

Z

c γ(t) ˙ dt + u(γ(b)) − u(γ(a)) − Ec (b − a) .

= a

Therefore, considering as above the action of the modified Lagrangian Lc (x, v), we get: Z b Lc (γ(t), γ(t)) ˙ dt = u(γ(b)) − u(γ(a)) − Ec (b − a) . (2.3) a

Let us now take any other absolutely continuous curve ξ : [a, b] −→ Td with the same endpoints as γ, i.e., ξ(a) = γ(a) and ξ(b) = γ(b). Proceeding as before and using Legendre-Fenchel inequality, we obtain: Z b Z b  ˙ ˙ − H(ξ(t), c + dξ(t) u) dt L(ξ(t), ξ(t)) dt ≥ (c + dξ(t) u)ξ(t) a

a

Z = a

b

˙ dt + u(ξ(b)) − u(ξ(a)) − Ec (b − a) . c ξ(t)

14

CHAPTER 2

Hence, using (2.3) and the fact that ξ(a) = γ(a) and ξ(b) = γ(b), we can conclude that: Z b Z b ˙ Lc (γ(t), γ(t)) ˙ dt ≤ Lc (ξ(t), ξ(t)) dt. (2.4) a

a

Therefore, for any time a < b, γ is the curve that minimizes the action of Lc over all absolutely continuous curves ξ : [a, b] −→ Td with ξ(a) = γ(a) and ξ(b) = γ(b). Actually something more is true. Let us consider a curve with the same endpoints, but a different time length, i.e., ξ : [a0 , b0 ] −→ Td , with a0 < b0 , such that ξ(a0 ) = γ(a) and ξ(b0 ) = γ(b). Proceeding as above, one obtains Z

b0

˙ Lc (ξ(t), ξ(t)) dt ≥ u(ξ(b0 )) − u(ξ(a0 )) − Ec (b0 − a0 )

a0

and consequently Z a

b



Z  Lc (γ(t), γ(t)) ˙ + Ec dt ≤

b0



 ˙ Lc (ξ(t), ξ(t)) + Ec dt .

a0

Hence, for any times a < b, γ minimizes the action of Lc + Ec among all absolutely continuous curves ξ that connect γ(a) to ξ(b) = γ(b) in any given time length (adding a constant does not change the Euler-Lagrange flow). We have just proven the following proposition. Proposition 2.1.8. For any given a < b, the projection γ of any orbit on T minimizes the action of Lc among all absolutely continuous curves that connect γ(a) to γ(b) in time b − a. Furthermore, γ minimizes the action of Lc + Ec among all absolutely continuous curves that connect γ(a) to γ(b) in any given time length. Remark 2.1.9. (i ) A curve γ : R −→ Td , such that for any a < b, γ|[a, b] minimizes the action of Lc among all absolutely continuous curves that connect γ(a) to γ(b) in time b − a, is called a c-global minimizer of L. (ii ) A curve γ : R −→ Td , such that for any a < b, γ|[a, b] minimizes the action of Lc among all absolutely continuous curves that connect γ(a) to γ(b) without any restriction on the time length, is called a c-time-free minimizer of L. (iii ) Proposition 2.1.8 can be restated by saying that the projection of each orbit on T is a c-global minimizer of L and a c-time free minimizer of L + Ec . Moreover, it follows easily from the proof and the Legendre-Fenchel inequality that if the curve ξ does not lie on T , then the inequality in (2.4) is strict. In particular, we obtain that: [ ˙ : γ is a c-global minimizer of L and t ∈ R}. L−1 (T ) = {(γ(t), γ(t))

15

FROM KAM THEORY TO AUBRY-MATHER THEORY

ec and called1 the Mañé set The set on the right-hand side is often denoted by N of cohomology class c. It will be defined in section 4.1. Actually these curves on T are more than c-global minimizers. As we will see, they satisfy a more restrictive condition. In order to introduce it, let us introduce what is called the Mañé potential. Let x1 , x2 ∈ Td and let us denote by hTc (x1 , x2 ) the minimimal action of Lc along curves that connect x1 to x2 in time T . We want to consider the infimum of these quantities for all positive times. Of course, without any “correction” this infimum might be −∞. Hence, let us consider the action of Lc + k for some k ∈ R and define:
φc,k (x1 , x2 ) := inf hTc (x1 , x2 ) + kT . (2.5) T >0

We want to see for which values of k this quantity is well defined, i.e., > −∞. Let γ : [0, T ] −→ Td be any absolutely continuous curve such that γ(0) = x1 and γ(T ) = x2 . Then, using Legendre-Fenchel inequality: T

Z

Z

Lc (γ(t), γ(t)) ˙ dt + kT

≥

0

T


c + dγ(t) u) dt + kT du(γ(t) · γ(t)−H(γ(t), ˙

0

= u(γ(T )) − u(γ(0)) + (k − Ec )T = u(x2 ) − u(x1 ) + (k − Ec )T .

(2.6)

Taking the infimum over all curves connecting x1 to x2 , we obtain: φc,k (x1 , x2 ) ≥ u(x2 ) − u(x1 ) + (k − Ec )T. Therefore, if k ≥ Ec , then φc,k (x1 , x2 ) > −∞ for all x1 , x2 ∈ Td . On the other hand, it is quite easy to show that for k < Ec , φc,k (x1 , x2 ) = −∞ for all x1 , x2 ∈ Td . Let us observe the following: • If k < Ec , then φc,k (x, x) = −∞ for all x ∈ Td . In fact, we know that the orbit γ(t) = πΦH t (x, c + du(x)) is recurrent, i.e., there exist Tn → +∞ such that δn := kγ(Tn ) − xk → 0 as n → ∞. Consider the curve ξ : [0, Tn + δn ] → Td obtained by joining the curve γ to the unit speed geodesic connecting γ(Tn ) to x. Then, proceeding as before and using the Legendre-Fenchel inequality and the continuity of u, we obtain: Z φc,k (x, x) ≤

Tn

Z

Tn +δn

Lc (γ, γ)dt ˙ + 0

˙ + k(Tn + δn ) Lc (ξ, ξ)dt

Tn n→+∞

≤ u(γ(Tn )) − u(x) + Aδn + (k − Ec )Tn + kδn −→ −∞, where A := maxkvk=1 |Lc (x, v)|. 1 Note that this definition of the Mañé set does not coincide with the original one given by Mather; however, since we are considering an autonomous Lagrangian, this definition turns out to be equivalent (see chapter 4).

16

CHAPTER 2

• It is also easy to check that for any k, φc,k satisfies a sort of triangular inequality: φc,k (x, y) ≤ φc,k (x, z) + φc,k (z, y)

∀ x, y, z ∈ Td .

For the proof of this, observe that any absolutely continuous curve connecting x to z and any absolutely continuous curve connecting z to y determine a curve from x to y. Passing to the infimum, one gets the result. Our claim follows easily from these two observations. In fact, let k < Ec and x1 , x2 ∈ Td . Then: φc,k (x1 , x2 ) ≤ φc,k (x1 , x1 ) + φc,k (x1 , x2 ) = −∞ . Summarizing, we have proven the following proposition. Proposition 2.1.10. Ec

= inf{k ∈ R : φc,k (x, y) > −∞ for all x, y ∈ Td } = sup{k ∈ R : φc,k (x, y) = −∞ for all x, y ∈ Td } = sup{k ∈ R : φc,k (x, x) = −∞ for all x ∈ Td } = sup{k ∈ R : φc,k (x, x) = −∞ for some x ∈ Td } .

Observe that the second and fourth equalities are simply consequences of the triangular inequality. Let us go back to our c-global minimizing curves. We have seen (Proposition 2.1.8) that if γ is the projection of an orbit on T , then it is a c-global minimizer of L and furthermore it is a c-time-free minimizer of L + Ec . The last property can be rewritten in terms of the Mañé potential: Z b (Lc (γ(t), γ(t)) ˙ + Ec ) = φc,Ec (γ(a), γ(b)) for any a < b. a

Therefore, these curves seems to be related to the Mañé potential corresponding to k = Ec (the least value for which it is defined). Observe that in this case we obtain from (2.6) that φc,Ec (x, y) ≥ u(y) − u(x) for all x, y ∈ Td , and consequently: φc,Ec (y, x) ≥ u(x) − u(y) = −(u(y) − u(x)) ≥ −φc,Ec (x, y) ,

(2.7)

which provides a lower bound on the action needed to “go back” from y to x. In particular, if ξ : [a, b] → Td is an absolutely continuous curve, then: Z b  ˙ + Ec dt ≥ φc,E (ξ(a), ξ(b)) ≥ −φc,E (ξ(b), ξ(a)) . Lc (ξ, ξ) c c a

Question: Do there exist curves for which these inequalities are equalities? Observe that if such a curve exists, then it must be necessarily a c-minimizer. We will show that the answer is affirmative and characterize such curves in our case.

17

FROM KAM THEORY TO AUBRY-MATHER THEORY

Proposition 2.1.11. Let γ be the projection of an orbit on T . Then, for each a < b we have: Z b (Lc (γ, γ) ˙ + Ec ) dt = φc,Ec (γ(a), γ(b)) = −φc,Ec (γ(b), γ(a)) . a

Remark 2.1.12. (i ) A curve γ : R −→ Td satisfying the conditions in Proposition 2.1.11 is called a c-regular global minimizer or a c-static curve of L. Observe that the adjective regular (coined by John Mather) has no relation to the smoothness of the curve, since this curve will be as smooth as all other solutions of the Euler-Lagrange flow (depending on the regularity of the Lagrangian). (ii ) Since a c-regular global minimizer is a c-global minimizer (see the comment above), it follows from (iii ) in Remark 2.1.9 that these are all the c-regular global minimizers of L, and the only ones. Therefore: [ ˙ : γ is a c- regular global minimizer of L and t ∈ R}. L−1 (T ) = {(γ(t), γ(t)) The set on the right-hand side is often denoted by Aec and called the Aubry set of cohomology class c. It will be defined in section 4.1. Proof. (Proposition 2.1.11) Let γ(t) := πΦH t (x, c + du(x)) and consider a < b. We already know from Proposition 2.1.8 that Z b (Lc (γ, γ) ˙ + Ec ) dt = φc,Ec (γ(a), γ(b)) . a

Since the orbit is recurrent, there exists Tn → +∞ such that δn := kγ(b + Tn ) − γ(a)k → 0. Consider the curve ξ : [b, b + Tn + δn ] → Td obtained by joining the curve γ|[b, b + Tn ] to the unit speed geodesic connecting γ(b + Tn ) to γ(a). Then, using the Legendre-Fenchel inequality and the continuity of u, we obtain: Z b+Tn Z b+Tn +δn ˙ + Ec (Tn + δn ) φc,Ec (γ(b), γ(a)) ≤ Lc (γ, γ)dt ˙ + Lc (ξ, ξ)dt b

b+Tn n→+∞

≤ u(γ(b + Tn )) − u(γ(b)) + (A + Ec )δn −→ u(γ(a)) − u(γ(b)), where A := maxkvk=1 |Lc (x, v)|. Therefore, using (2.6), we obtain φc,Ec (γ(b), γ(a)) ≤ u(γ(a)) − u(γ(b)) = −(u(γ(b)) − u(γ(a))) = −φc,Ec (γ(a), γ(b)), which, together with (2.7), allows us to deduce that φc,Ec (γ(b), γ(a)) = −φc,Ec (γ(a), γ(b)) and conclude the proof.



Chapter Three Action-Minimizing Invariant Measures for Tonelli Lagrangians

3.1

ACTION-MINIMIZING MEASURES AND MATHER SETS

In this section, inspired by what happens for invariant measures and orbits on KAM tori, we would like to attempt a similar approach for general Tonelli Lagrangians on compact manifolds, and prove the existence of some analogous compact invariant subsets for the Euler-Langrange (or Hamiltonian) flow. In many senses, these sets will resemble and generalize KAM tori, even when these do not exist or it does not make sense to speak of them. Let us start by studying invariant probability measures of the system and their action-minimizing properties. Then, we will use them to define a first family of invariant sets: the Mather sets (see Remark 2.1.5 (iii ) and Remark 2.1.7 (iii )). Let M(L) be the space of probability measuresR µ on TM that are invariant under the Euler-Lagrange flow of L and such that TM L dµ < ∞ (finite action). It is easy to see in the case of an autonomous Tonelli Lagrangian that this set is nonempty. In fact, recall that because of the conservation of the energy

 E(x, v) := H ◦ L(x, v) = ∂L (x, v), v − L(x, v) along the motions, each energy ∂v x level of E is compact (it follows from the superlinearity condition) and invariant under ΦL t . It is a well-known result by Kryloff and Bogoliouboff [49] that a flow on a compact metric space has at least an invariant probability measure. Proposition 3.1.1. Each nonempty energy level E(E) := {E(x, v) = E} contains at least one invariant probability measure of ΦL t . Proof. Let (x0 , v0 ) ∈ E(E) and consider the curve γ(t) := πΦL t (x0 , v0 ), where π : TM −→ M denotes the canonical projection. For each T > 0, we can consider the probability measure µT uniformly distributed on the piece of curve (γ(t), γ(t)) ˙ for t ∈ [0, T ], i.e., Z

1 f dµT = T

Z

T

f (γ(t), γ(t)) ˙ dt

∀ f ∈ C(E(E)).

0

The family {µT }T is precompact in the weak∗ topology; in fact, the set of probability measures on E(E) is contained in the closed unit ball in the space of signed Borel measures of E(E), which is weak∗ compact by the Banach-Alaouglu

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

19

theorem. Therefore, we can extract a converging subsequence {µTn }n such that µTn −→ µ in the weak∗ topology as Tn → ∞. We want to prove that µ is invariant, i.e., for each s ∈ R we have ΦL s ∗ µ = µ. Let f be any continuous function on E(E). Then : Z Z Z Z L L f dΦL µ − f dµ ≤ f dΦ µ − f dΦ µ T s∗ s∗ s∗ n Z Z L + f dΦs ∗ µTn − f dµTn Z Z + f dµTn − f dµ . (3.1) From the weak∗ convergence µTn −→ µ, we have that Z Z f dµTn − f dµ −→ 0 as Tn → +∞ and Z Z Z Z L L L f dΦL f dΦs ∗ µTn = f ◦ Φs dµ − f ◦ Φs dµTn −→ 0 as Tn → +∞. s ∗µ − Moreover, using the definition of µTn : Z Z Z Z Tn Tn +s 1 f dΦL f dµTn = ˙ dt − f (γ(t), γ(t)) ˙ dt f (γ(t), γ(t)) s ∗ µTn − Tn s 0 Z Z s 1 Tn +s = f (γ(t), γ(t)) ˙ dt − f (γ(t), γ(t)) ˙ dt Tn Tn 0 ≤

2s max |f | −→ 0 Tn E(E)

as Tn → +∞.

Therefore, it follows from (3.1) that for any f ∈ E(E): Z Z f dΦL = 0. µ − f dµ s∗ L Consequently, ΦL s ∗ µ = µ for any s ∈ R, i.e., µ is Φ -invariant.

With each µ ∈ M(L), we may associate its average action Z AL (µ) = L dµ . TM

Clearly, the measures constructed in Proposition 3.1.1 have finite action.



20

CHAPTER 3

Remark 3.1.2. One can show the existence of invariant probability measures with finite action also in the case of non-autonomous time-periodic Lagrangians. As was originally done by Mather in [62], one can apply Kryloff and Bogoliouboff’s result to a one-point compactification of TM and consider the extended Lagrangian system that leaves the point at infinity fixed. The main step consists in showing that the measure provided by this construction has no atomic part supported at ∞ (which is a fixed point for the extended system). Note: We will hereafter assume that M(L) is endowed with the vague topology, i.e., the weak∗ topology induced by the space C`0 of continuous functions f : TM −→ R having at most linear growth: sup (x,v)∈TM

|f (x, v)| < +∞ . 1 + kvk


∗ It is not difficult to check that M(L) ⊂ C`0 . Moreover it is a classical result that this space, with such a topology, is metrizable. See also section 3.6 for more details on this topology. Proposition 3.1.3. AL : M(L) −→ R is lower semicontinuous with the vague topology on M(L). R Proof. Let AL,K (µ) := min{L, K} dµ, for K ∈ R. Then, AL,K is clearly continuous, actually Lipschitz with constant K. In fact: |AL,K (µ) − AL,K (ν)| ≤ Kd(µ, ν). Moreover, since AL,K ↑ AL as K → ∞, it follows that AL is lower semicontinuous.  Remark 3.1.4. In general, this functional might not be necessarily continuous. This proposition has an important consequence. Corollary 3.1.5. There exists µ ∈ M(L), which minimizes AL over M(L). Observe in fact that M(L) is a nonempty, compact and convex set and that— it is a classical result—every lower semicontinuous function on a compact set attains its minimum. We will call a measure µ ∈ M(L), such that AL (µ) = minM(L) AL , an action-minimizing measure of L (compare with Remark 2.1.5 (ii )). Actually, one can find many other “interesting” measures besides those found by minimizing L. On the other hand, as we have already pointed out in Proposition 2.1.4 (and the following warning), in order to get information on a specific KAM torus and characterize it via the invariant probability measures supported on it, one needs necessarily to modify the Lagrangian. Think for instance of an integrable system for which we have a whole family of KAM tori. It would be unreasonable to expect to obtain all of them just by minimizing a single Lagrangian

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

21

action. One needs somehow to introduce some sort of “magnifying glass,” which, without modifying the dynamics of the system (i.e., the Euler Lagrange flow), allows one to reveal certain motions rather than others. What we found out in Proposition 2.1.4 was that this can be easily achieved by subtracting from our Lagrangian a linear function c · v, where c represents the cohomology class of the invariant Lagrangian torus we are interested in, or, in other words, the cohomology class of its graph (which is a closed 1-form) in T∗ Td . A similar idea can be implemented for a general Tonelli Lagrangian. Observe, in fact, that if η is a 1-form on M , we can interpret it as a function on the tangent bundle (linear on each fiber) ηˆ : TM −→ (x, v) 7−→

R hη(x), vix

and consider a new Tonelli Lagrangian Lη := L− ηˆ. The associated Hamiltonian will be given by Hη (x, p) = H(x, η(x) + p) (it is sufficient to write down the Legendre-Fenchel transform of L). Lemma 3.1.6. If η is closed, then L and Lη have the same Euler-Lagrange flow on TM . R  Proof. R Observe that, since η is closed, the variational equations δ L dt = 0 and δ (L − ηˆ) dt = 0 have the same extremals for the fixed endpoint problem, and consequently L and Lη have the same Euler-Lagrange flows. One can also check this directly, writing down the Euler-Lagrange equations associated with L and Lη and observe that, since η is closed (i.e., dη = 0), then   d ∂ d ∂ ηˆ(γ, γ) ˙ = (η(γ)) = (ˆ η (γ, γ)) ˙ . dt ∂v dt ∂x Let us show that they in fact are equal component by component. If we denote by ηj and γj the jth components of η and γ, then the ith component of the left-hand side of the above equation becomes d X ∂ηi (γ) · γ˙ j , ∂xj j=1

and on the other side: d X ∂ηj j=1 ∂η

∂xi

(γ) · γ˙ j .

∂ηi Since η is closed, ∂xji = ∂x for each i and j. Therefore, the two expressions are j the same. Therefore, the term coming from ηˆ does not make any contribution to the equations. 

22

CHAPTER 3

Although the extremals of these two variational problems are the same, this is not generally true for the orbits “minimizing the action” (we will give a precise definition of “minimizers” later in section 4.1). We have already given evidence of this in Proposition 2.1.4. What one can say is that these action-minimizing objects stay the same when we change the Lagrangian by an exact 1-form. R

Proposition 3.1.7. If µ ∈ M(L) and η = df is an exact 1-form, then b dµ = 0. df

The proof is essentially the same as in section 2.1 (Property 2), so we omit it. See also [62, lemma on page 176]. Let us point out that a measure (not necessarily invariant) that satisfies this condition is called a closed measure (see also section 3.6). Thus, for a fixed L, the minimizing measures will depend only on the de Rham cohomology class c = [η] ∈ H1 (M ; R). Therefore, instead of studying the action-minimizing properties of a single Lagrangian, one can consider a family of such “modified” Lagrangians, parameterized over H 1 (M ; R). Hereafter, for any given c ∈ H 1 (M ; R), we will denote by ηc a closed 1-form with that cohomology class. Definition 3.1.8. Let ηc be a closed 1-form of cohomology class c. Then, if µ ∈ M(L) minimizes ALηc over M(L), we will say that µ is a c-actionminimizing measure (or c-minimal measure, or Mather’s measure with cohomology c). Remark 3.1.9. Observe that the cohomology class of an action-minimizing invariant probability measure is something intrinsic neither in the measure itself nor in the dynamics, but depends on the specific choice of the Lagrangian L. Changing the Lagrangian L 7−→ L − η by a closed 1-form η, we will change all the cohomology classes of its action-minimizing measures by −[η] ∈ H1 (M ; R). Compare this with Remark 3.2.1 (ii ). One can consider the function on H1 (M ; R), which associates with each cohomology class c, minus the value of the corresponding minimal action of the modified Lagrangian Lηc (the minus sign is introduced as a convention that might become clearer later): α : H1 (M ; R) −→ c 7−→

R − min ALηc (µ) .

(3.2)

µ∈M(L)

This function α is well defined (it does not depend on the choice of the representatives of the cohomology classes) and it is easy to see that it is convex. It is generally known as Mather’s α-function. As was the case for KAM tori (see Remark 2.1.5), we will see in section 4.1 that the value α(c) is related to the energy level containing such c-action-minimizing measures (they coincide). See also [23]. Therefore, if we have an integrable Tonelli Hamiltonian H(x, p) = h(p), it is easy to deduce that α(c) = h(c). For this and several other reasons that

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

23

we will see later on, this function is sometime called effective Hamiltonian. It is interesting to observe that the value of this α-function coincides with what is called Mañé’s critical value, which will be introduced later in sections 4.1 and 5.1. We will denote by Mc (L) the subset of c-action-minimizing measures: Mc := Mc (L) = {µ ∈ M(L) : AL (µ) < +∞ and ALηc (µ) = −α(c)}. We can now define the first important family of invariant sets (see Remark 2.1.5 (iii )): the Mather sets. Definition 3.1.10. For a cohomology class c ∈ H1 (M ; R), we define the Mather set of cohomology class c as: [ fc := M supp µ ⊂ TM . (3.3) µ∈Mc

  fc ⊆ M is called the projected The projection on the base manifold Mc = π M Mather set (with cohomology class c). Remark 3.1.11. (i ) This set is clearly nonempty and invariant. Moreover, it is also closed. Observe that usually it is defined as the closure of the set on the right-hand side. This is indeed the original definition given by Mather in [62]. However, it is not difficult to check that this set is already closed. In fact, since the space of probability measures on TM is a separable metric space, one can take a countable dense set {µn }∞ n=1 of Mather’s measures and P∞ ˜ = n=1 21n µn . This is still an invariant probability consider the new measure µ measure and it is c-action-minimizing (it follows from the convexity of α), hence fc . But, from the definition of µ supp µ ˜⊆M ˜ one can clearly deduce that supp µ ˜= S f fc . Therefore, as the support ˜=M µ∈Mc supp µ ⊇ Mc . This implies that supp µ fc is closed (the support of a measure, of a single action-minimizing measure, M by definition, is closed). Moreover, this remark points out that there always fc . exists a Mather’s measure µc of full support, i.e., supp µc = M (ii ) As we have pointed out in Remark 2.1.5, if there is a KAM torus T of fc = L−1 (T ). In particular, the same proof continues cohomology class c, then M to hold if we replace T with any invariant Lagrangian graph Λ of cohomology class c that supports an invariant measure µ of full support (i.e., supp µ = Λ). In [62] Mather proved the celebrated graph theorem: fc be defined as in (3.3). Theorem 3.1.12 (Mather’s graph theorem). Let M fc is compact and invariant under the Euler-Lagrange flow, and π|M fc The set M −1 f f is an injective mapping of Mc into M , and its inverse π : Mc −→ Mc is Lipschitz. We will not prove this theorem here, but it will be deduced from a more general result in section 4.1, namely, the graph property of the Aubry set. Moreover, similarly to what we have seen for KAM tori (see Remark 2.1.5), in the

24

CHAPTER 3

autonomous case this set is contained in a well-defined energy level, which can be characterized in terms of the minimal action α(c). fc is contained in Theorem 3.1.13 (Dias Carneiro, [23]). The Mather set M the energy level {H ◦ L(x, v) = α(c)}. Also, this result will be deduced from a similar result for the Aubry and Mañé sets. See Proposition 4.1.23 in section 4.1. 3.2

MATHER MEASURES AND ROTATION VECTORS

Now, we would like to shift our attention to a related problem. As we have seen in section 2.1, instead of considering different minimizing problems over M(L), obtained by modifying the Lagrangian L, one can alternatively try to minimize the Lagrangian L by adding “constraints,” for instance, fixing the rotation vector of the measures. In order to generalize this to Tonelli Lagrangians on compact manifolds, we first need to define what we mean by rotation vector of an invariant measure (see appendix B for more details). R Let µ ∈ M(L). Thanks to the superlinearity of L, the integral TM ηˆdµ is well defined and finite for any closed 1-form η on M . Moreover, we have proven in Proposition 3.1.7 that if η is exact, then such an integral is zero, i.e., R η ˆ dµ = 0. Therefore, one can define a linear functional: TM H1 (M ; R) −→

R Z

c 7−→

ηˆdµ , TM

where η is any closed 1-form on M with cohomology class c. By duality, there exists ρ(µ) ∈ H1 (M ; R) such that Z ηˆ dµ = hc, ρ(µ)i ∀ c ∈ H1 (M ; R) TM

(the bracket on the right-hand side denotes the canonical pairing between cohomology and homology). We call ρ(µ) the rotation vector of µ (compare this with the definition given in section 2.1 Property (1)). This rotation vector is the same as Schwartzman’s asymptotic cycle of µ (see [74] and appendix B for more details). Remark 3.2.1. (i ) It is possible to provide a more “geometrical” interpretation of this. Suppose for the moment that µ is ergodic. Then, it is known that a generic orbit γ(t) := πΦL t (x, v), where π : TM −→ M denotes the canonical projection, will return infinitely many oftentimes close (as close as we like) to its initial point γ(0) = x. We can therefore consider a sequence of times Tn → +∞ such that d(γ(Tn ), x) → 0 as n → +∞, and consider the closed loops σn obtained by “closing” γ|[0, Tn ] with the shortest geodesic connecting γ(Tn ) to x. Denoting by [σn ] the homology class of this loop, one can verify

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

25

[74] that limn→∞ [σTnn ] = ρ(µ), independently of the chosen sequence {Tn }n . In other words, in the case of ergodic measures, the rotation vector tells us how many times on average a generic orbit winds around TM . If µ is not ergodic, ρ(µ) loses this neat geometric meaning, yet it may be interpreted as the average of the rotation vectors of its different ergodic components. (ii ) It is clear from the discussion above that the rotation vector of an invariant measure is an intrinsic property of the measure (and therefore of the dynamics in its support) and does not depend on the chosen Lagrangian. Therefore, differently from what happens with the cohomology class (see Remark 3.1.9), the rotation vector of an action-minimizing measure does not change when we modify our Lagrangian by adding a closed 1-form. A natural question is whether or not for a given Tonelli Lagrangian L there exist invariant probability measures for any given rotation vector. The answer turns out to be affirmative. Using the fact that the action functional AL : M(L) −→ R is lower semicontinuous, one can prove the following [62]: Proposition 3.2.2. (i) The map ρ : M(L) −→ H1 (M ; R) is continuous. (ii) The map ρ : M(L) −→ H1 (M ; R) is affine, i.e., for any µ, ν ∈ M(L) and a1 , a2 ≥ 0 with a1 + a2 = 1, ρ(a1 µ + a2 ν) = a1 ρ(µ) + a2 ρ(ν). (iii)For every h ∈ H1 (M ; R) there exists µ ∈ M(L) with AL (µ) < ∞ and ρ(µ) = h. In other words, the map ρ is surjective. Proof. (i) Let us fix a basis h1 , . . . , hb in H1 (M ; R), where b is the first Betti number of M , i.e., b = dim H1 (M ; R). If µn → µ and ηc is any closed 1-form on M of cohomolgy class c, then Z Z n→+∞ hc, ρ(µn )i = ηc · vdµn −→ ηc · vdµ = hc, ρ(µ)i, which is equivalent to saying that: n→+∞

hc, ρ(µn ) − ρ(µ)i −→ 0 ∀ c ∈ Rb ' H 1 (M ; R) . Therefore, ρ(µn ) − ρ(µ)−→0 as n → +∞. (ii) The fact that the map ρ is affine is a trivial consequence of the definition of the rotation vector. (iii) Let h ∈ H1 (M ; R) be a homology class and let us consider the abelian c of M , i.e., the covering space of M whose fundamental group covering space M is given by the kernel of the Hurewicz homomorphism h : π1 (M ) −→ H1 (M ; R) and whose group of Deck transformations is the image of h. Let T1 , . . . , Tk , . . . be a sequence of Deck transformations such that Tk −→ h ∈ H1 (M ; R) k

as k → +∞.

c and let x Fix x ˆ0 ∈ M ˆk := Tk x ˆ0 . We want to apply Tonelli’s theorem (see c c that minimizes the Theorem 4.1.1) in M and find a curve γˆk : [0, k] −→ M

26

CHAPTER 3

Lagrangian action among all curves connecting x ˆ0 to x ˆk in time k (it is an orbit; hence it is smooth). Let now γk be the projection of γˆk on M and denote by µk the probability measure evenly distributed along (γk , γ˙ k ). Similarly to what we have seen in Proposition 3.1.1, one can consider an accumulation point µ of these measures with respect to the weak∗ topology: It is easy to see that this measure has finite action and it is invariant. As far as its rotation vector is concerned, observe that if each η is a closed 1-form on M , then: Z Tk ηˆ(x) · v dµk = h[η], i. k TM Hence, taking the limit as k → +∞, it follows that µ has rotation vector h.  As already pointed out in section 2.1, among all probability measures with a prescribed rotation vector, a peculiar role—from a dynamical systems point of view—will be played by those minimizing the average action. Following Mather, let us consider the minimal value of the average action AL over the probability measures with rotation vector h. Observe that this minimum is actually achieved because of the lower semicontinuity of AL and the compactness of ρ−1 (h) (ρ is continuous and L superlinear). Let us define β : H1 (M ; R) −→ h 7−→

R min µ∈M(L): ρ(µ)=h

AL (µ) .

(3.4)

This function β is what is generally known as Mather’s β-function and it is easy to check that it is convex. As we have noted in Remark 2.1.7, if there is a KAM torus of cohomology class c and rotation vector ρ, then β(ρ) = −Ec + c · ρ. Therefore, if we have an integrable Tonelli Hamiltonian H(x, p) and the associated Lagrangian L(x, v) = `(v), it is easy to deduce that β(h) = `(h). For this and several other reasons we will see later, this function is sometime called the effective Lagrangian. Moreover, this function is also related to the notion of stable norm for a metric d (see for instance [54]). We can now define what we mean by action-minimizing measure with a given rotation vector (compare this with Remark 2.1.7(ii )). Definition 3.2.3. A measure µ ∈ M(L) realizing the minimum in (3.4), i.e., such that AL (µ) = β(ρ(µ)), is called an action-minimizing (or minimal or Mather’s) measure with rotation vector ρ(µ). Remark 3.2.4. We will see in section 4.1 that, differently from what happens with invariant probability measures, it will not be always possible to find actionminimizing orbits for any given rotation vector (and not even to define a rotation vector for each action-minimizing orbit). This is one of the main differences with the twist map case. For higher dimensions, in fact, there is a well-known counterexample by Hedlund [45], consisting of a Riemannian metric on a threedimensional torus, for which minimal geodesics exist only in three directions.

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

27

We will denote by Mh (L) the subset of action-minimizing measures with rotation vector h: Mh := Mh (L) = {µ ∈ M(L) : AL (µ) < +∞, ρ(µ) = h and AL (µ) = β(h)}. This allows us to define another important family of invariant sets (see also Remark 2.1.7 (iii )). Definition 3.2.5. For a homology class (or rotation vector) h ∈ H1 (M ; R), we define the Mather set corresponding to a rotation vector h as [ fh := M supp µ ⊂ TM, (3.5) µ∈Mh

  fh ⊆ M . and the projected one as Mh = π M Remark 3.2.6. (i ) Similarly to what we have pointed out in Remark 3.1.11, this set is also nonempty and invariant. Moreover, it is also closed. Also, in this case, it is not necessary to consider the closure of this set—as is usually done in the literature—since it is already closed. Just take a countable dense set {µn }∞ Mather’s measures with rotation vector h and consider the new n=1 ofP ∞ 1 measure µ ˜ = n=1 2n µn . This is still an invariant probability measure and its rotation vector is h. Moreover, it follows from the convexity of β that this measure is action-minimizing among S all measures with rotation vector h. Hence, fh . Clearly supp µ fh . This implies that supp µ ˜ ⊆ M ˜ = µ∈Mh supp µ ⊇ M fh . Therefore, M fh is closed. Moreover, this shows that there always supp µ ˜=M fh . exists a Mather’s measure µh of full support, i.e., supp µh = M (ii ) As we have pointed out in Remark 2.1.7, if there is a KAM torus T of fh = L−1 (T ). In particular, the same proof continues rotation vector h, then M to hold if we replace T with any invariant Lagrangian graph Λ that supports an invariant measure µ of rotation vector ρ and of full support (i.e., supp µ = Λ). Also for the Mather set corresponding to a rotation vector, we can prove a result similar to Theorem 3.1.12. fh be defined as in (3.5). M fh is compact and invariTheorem 3.2.7. Let M h f is an injective mapping of M fh ant under the Euler-Lagrange flow, and π|M fh is Lipschitz. into M , and its inverse π −1 : Mh −→ M fh is not explicitly menRemark 3.2.8. Although the graph property for M tioned in [62], it is easy to deduce it from Theorem 3.1.12 using the facts that fh can be seen as the support of a single action-minimizing measure (Remark M fc , for some suitable c ∈ H1 (M ; R) 3.2.6) and that this set is included in some M (Proposition 3.3.4).

28 3.3

CHAPTER 3

MATHER’S α- AND β-FUNCTIONS

The above discussion leads to two equivalent formulations for the minimality of a measure µ: • there exists a homology class h ∈ H1 (M ; R), namely its rotation vector ρ(µ), such that µ minimizes AL among all measures in M(L) with rotation vector h; i.e., AL (µ) = β(h). • There exists a cohomology class c ∈ H1 (M ; R) such that µ minimizes ALηc among all probability measures in M(L); i.e., ALηc (µ) = −α(c). What is the S relation betweenStwo these different approaches? Are they equivalent, i.e., h∈H1 (M ;R) Mh = c∈H1 (M ;R) Mc ? In order to comprehend the relation between these two families of actionminimizing measures, we need to understand better the properties of the functions α : H1 (M ; R) −→ R and β : H1 (M ; R) −→ R which we introduced above. Let us start with the following trivial remark. Remark 3.3.1. As we have previously pointed out, if we have an integrable Tonelli Hamiltonian H(x, p) = h(p) and the associated Lagrangian L(x, v) = `(v), then α(c) = h(c) and β(h) = `(h). In this case, the cotangent bundle T∗ Td is foliated by invariant tori Tc∗ := Td × {c} and the tangent bundle TTd by invariant tori Te h := Td × {h}. In particular, we have proved that fc = L−1 (Tc ) = Te h = M fh , M where h and c are such that h = ∇h(c) = ∇α(c) and c = ∇`(h) = ∇β(h). We would like to prove that this relation that links Mather sets of a certain cohomology class to Mather sets with a given rotation vector goes beyond the specificity of this situation. Of course, one main difficulty is that in general the effective Hamiltonian α and the effective Lagrangian β, although being convex and superlinear (see Proposition 3.3.2), are not necessarily differentiable. Before stating and proving the main relation between these two functions, let us recall some definitions and results from classical convex analysis (see [73]). Given a convex function ϕ : V −→ R ∪ {+∞} on a finite dimensional vector space V , one can consider a dual (or conjugate) function defined on the dual
space V ∗ via the so-called Fenchel transform: ϕ∗ (p) := supv∈V p · v − ϕ(v) . Proposition 3.3.2. α and β are convex conjugate, i.e., α∗ = β and β ∗ = α. In particular, it follows that α and β have superlinear growth.

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

29


∗ ∗ Proof. First of all, recall that (H1(M ; R)) ' H1 (M ; R) and H1(M ; R) ' H1 (M ; R). Let us compute β ∗ : β ∗ (c)

=

max h∈H1 (M ;R)

= −

(hc, hi − β(h))

min h∈H1 (M ;R)

= −

min h∈H1 (M ;R)

= − = −

(β(h) − hc, hi)   min AL (µ) − hc, hi µ∈Mh (L)

min

min (AL (µ) − hc, ρ(µ)i)

min

min

h∈H1 (M ;R) µ∈Mh (L) h∈H1 (M ;R) µ∈Mh (L)

ALηc (µ)

= − min ALηc (µ) µ∈M(L)

= α(c) . Similarly, one can check that α∗ = β: α∗ (h)

= = = = =

max

c∈H1 (M ;R)

max

c∈H1 (M ;R)

(hc, hi − α(c))   hc, hi + min ALηc (µ) µ∈M(L)

min (AL (µ) + hc, h − ρ(µ)i)

max

c∈H1 (M ;R) µ∈M(L)

min

max

µ∈M(L) c∈H1 (M ;R)

min

(AL (µ) + hc, h − ρ(µ)i)

min

max

ρ∈H1 (M ;R) µ∈Mρ (L) c∈H1 (M ;R)

(AL (µ) + hc, h − ρi) ,

where in the second-last line we could exchange the order of the max and the min using a general result by Rockafellar that requires concavity in one variable, convexity in the other one, and some compactness assumptions (see [73, Section 36]). Observe now that if h 6= ρ, then maxc∈H1 (M ;R) (AL (µ) + hc, h − ρi) = +∞. Therefore: α∗ (h)

= ... = =

min

min

min

max

ρ∈H1 (M ;R) µ∈Mρ (L) c∈H1 (M ;R)

µ∈Mh (L)

(AL (µ) + hc, h − ρi)

AL (µ) = β(h).

The second statement of this proposition follows from a general property of convex conjugation. Let ϕ : V → R be a convex function on a finite dimensional vector space V and let ϕ∗ : V ∗ −→ R ∪ {+∞} be its convex conjugate. Then (see [73]): ϕ∗ is finite everywhere if and only if ϕ has superlinear growth, i.e., ϕ(x)  kxk −→ +∞ as kxk → +∞. The next proposition will allow us to clarify the relation (and duality) between the two minimizing procedures described above. To state it, recall that,

30

CHAPTER 3

like any convex function on a finite-dimensional space, β admits a subderivative at each point h ∈ H1 (M ; R), i.e., we can find c ∈ H1 (M ; R) such that ∀h0 ∈ H1 (M ; R),

β(h0 ) − β(h) ≥ hc, h0 − hi.

(3.6)

As it is usually done, we will denote by ∂β(h) the set of c ∈ H1 (M ; R) that are subderivatives of β at h, i.e., the set of c’s that satisfy the above inequality. Similarly, we will denote by ∂α(c) the set of subderivatives of α at c. Fenchel’s duality implies an easier characterization of subdifferentials. Proposition 3.3.3. c ∈ ∂β(h) if and only if hc, hi = α(c) + β(h). Similarly h ∈ ∂α(c) if and only if hc, hi = α(c) + β(h). In particular, c ∈ ∂β(h) if and only if h ∈ ∂α(c). Proof. We will prove only the first statement. The second one is analogous and the third one is a trivial consequence. [⇐=] Using the fact that β(h0 ) ≥ hc, h0 i − α(c) for each h0 ∈ H1 (M ; R), it follows that for each h0 ∈ H1 (M ; R): β(h0 ) − β(h) ≥ hc, h0 i − α(c) − β(h) = hc, h0 i − (α(c) + β(h)) = hc, h0 i − hc, hi = hc, h0 − hi. [=⇒] It follows from the Legendre-Fenchel inequality that α(c) + β(h0 ) ≥ hc, h0 i for each h0 ∈ H1 (M ; R). Therefore we only need to prove the reverse inequality. In fact, using (3.6) one can deduce that for each h0 ∈ H1 (M ; R): α(c) + β(h) ≤ α(c) + β(h0 ) − hc, h0 − hi = α(c) + β(h0 ) − hc, h0 i + hc, hi   = α(c) + β(h0 ) − hc, h0 i + hc, hi . Therefore, taking the minimum over h0 on the left-hand side we obtain:   α(c) + β(h) ≤ 0 min α(c) + β(h0 ) − hc, h0 i + hc, hi = hc, hi, h ∈H1 (M ;R)

where
in the last equality we used the fact that minh0 ∈H1 (M ;R) α(c) + β(h0 ) − hc, h0 i = 0, as it follows easily from the Legendre-Fenchel duality.  We can now prove that what observed in Remark 3.3.1 continues to hold in the general case. Proposition 3.3.4. Let µ ∈ M(L) be an invariant probability measure. Then: (i) AL (µ) = β(ρ(µ)) if and only if there exists c ∈ H1 (M ; R) such that µ minimizes ALηc (i.e., ALηc (µ) = −α(c)). (ii) If µ satisfies AL (µ) = β(ρ(µ)) and c ∈ H1 (M ; R), then µ minimizes ALηc if and only if c ∈ ∂β(ρ(µ)) (or equivalently hc, hi = α(c) + β(ρ(µ)).

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

31

Proof. We will prove both statements at the same time. Assume AL (µ0 ) = β(ρ(µ0 )). Let c ∈ ∂β(ρ(µ0 )); by Fenchel’s duality this is equivalent to α(c) = hc, ρ(µ0 )i − β(ρ(µ0 )) = hc, ρ(µ0 )i − AL (µ0 ) = −ALηc (µ0 ). Therefore, ALηc (µ0 ) = minµ∈M(L) ALηc (µ). Assume conversely that ALηc (µ0 ) = minµ∈M(L) ALηc (µ), for some given cohomology class c. Then, it follows that α(c) = −ALηc (µ0 ), which can be written as hc, ρ(µ0 )i = α(c) + AL (µ0 ). It now suffices to use the Fenchel inequality hc, ρ(µ0 )i ≤ α(c) + β(ρ(µ0 )), and the inequality β(ρ(µ0 )) ≤ AL (µ0 ), given by the definition of β, to obtain hc, ρ(µ0 )i = α(c) + β(ρ(µ0 )). In particular, we have AL (µ0 ) = β(ρ(µ0 )).  Remark 3.3.5. (i ) It follows from the above proposition that both minimizing procedures lead to the same sets of invariant probability measures: [ [ Mh = Mc . h∈H1 (M ;R)

c∈H1 (M ;R)

In other words, minimizing over the set of invariant measures with a fixed rotation vector or minimizing, globally, the modified Lagrangian (corresponding to a certain cohomology class) are dual problems, such as the ones that often appears in linear programming and optimization. In some (rough) sense, modifying the Lagrangian by a closed 1-form is analogous to the method of Lagrange multipliers for searching for constrained minima or maxima of a function. (ii ) In particular, we have the following inclusions between Mather sets: c ∈ ∂β(h)

⇐⇒

h ∈ ∂α(c)

⇐⇒

fh ⊆ M fc . M

Moreover, for any c ∈ H1 (M ; R): fc = M

[

fh . M

h∈∂α(c)

(iii ) The minimum of the α-function is sometime called Mañé’s strict critical value. Observe that if α(c0 ) = min α(c), then 0 ∈ ∂α(c0 ) and β(0) = −α(c0 ).

32

CHAPTER 3

Therefore, the measures with zero homology are contained in the least possible f0 ⊆ M fc . This inclusion might be strict, energy level containing Mather sets: M 0 unless α is differentiable at c0 ; in fact, there may be other action-minimizing measures with nonzero rotation vectors corresponding to the other subderivatives of α at c0 . (iv ) Note that measures of trivial homology are not necessarily supported on orbits with trivial homology or fixed points. For instance, one can consider the following example (see also [29, Section 5.2]). Let M = T2 be equipped with the flat metric and consider a vector field X with norm 1 and such that X has two closed orbits γ1 and γ2 in opposite homology classes and any other orbit asymptotically approaches γ1 in forward time and γ2 in backward time; for example, one can consider X(x1 , x2 ) = (cos(2πx1 ), sin(2πx1 )), where (x1 , x2 ) ∈ T2 = R2 /Z2 . As we have described in section 1.1, we can embed this vector field into the Euler-Lagrange vector field given by the Tonelli Lagrangian LX (x, v) = 12 kv − X(x)k2 . Let us now consider the probability measure µγ1 and µγ2 , uniformly distributed respectively on (γ1 , γ˙ 1 ) and (γ2 , γ˙ 2 ). Since these two curves have opposite homologies, ρ(µγ1 ) = −ρ(µγ2 ) =: h0 6= 0. Moreover, it is easy to see that ALX (µγ1 ) = ALX (µγ2 ) = 0, since the Lagrangian vanishes on Graph(X). Using the facts that LX ≥ 0 (in particular that it is strictly positive outside of Graph(X)) and that there are no other invariant ergodic probability measures contained in Graph(X), we can conclude that M0 = γ1 ∪ γ2 and α(0) = 0. Moreover, µ0 := 12 µγ1 + 12 µγ2 has zero homology and its support is f0 . Therefore (see Proposition 3.3.4 (i)), µ0 is action-minimizing contained in M f0 ⊆ M f0 ; in particular, M f0 = M f0 . This also with rotation vector 0 and M implies that β(0) = 0 and α(0) = min α(c) = 0. Observe that α is not differentiable at 0. In fact, reasoning as we have done fth0 = before for the zero homology class, it is easy to see that for all t ∈ [−1, 1], M f0 . It is sufficient to consider the convex combination µλ = λµγ + (1 − λ)µγ M 1 2 for any λ ∈ [0, 1]. Therefore, ∂α(0) = {th0 , t ∈ [−1, 1]} and β(th0 ) = 0 for all t ∈ [−1, 1]. As we have just seen in item (iv) of Remark 3.3.5, it may happen that the Mather sets corresponding to different homology (or cohomology) classes coincide or are included one into the other. This is something that cannot happen in the integrable case: in this situation, in fact, these sets form a foliation and are disjoint. The problem in the above-mentioned example seems to be related to the lack of strict convexity of β and α. See also the discussion on the simple pendulum in section 3.5: in this case the Mather sets, corresponding to a nontrivial interval of cohomology classes about 0, coincide. In light of this, let us try to understand better what happens when α and β are not strictly convex, i.e., when we are in the presence of “flat” pieces. Let us first fix some notation. If V is a real vector space and v0 , v1 ∈ V , we will denote by σ(v0 , v1 ) the segment joining v0 to v1 , that is, σ(v0 , v1 ) := {tv0 + (1 − t)v1 : t ∈ [0, 1]}. We will say that a function f : V −→ R is affine on σ(v0 , v1 ), if there exists v ∗ ∈ V ∗ (the dual of V ), such that f (v) = f (v0 ) +

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

33

hv ∗ , v − v0 i for each v ∈ σ(v0 , v1 ). Moreover, we will denote by Int(σ(v0 , v1 )) the interior of σ(v0 , v1 ), i.e., Int(σ(v0 , v1 )) := {tv0 + (1 − t)v1 : t ∈ (0, 1)}. Proposition 3.3.6. (i) Let h0 , h1 ∈ H1 (M ; R); β is affine on σ(h0 , h1 ) if fh ⊇ M fh0 ∪ M fh1 . and only if for any h ∈ Int(σ(h0 , h1 )) we have M 1 (ii) Let c0 , c1 ∈ H (M ; R); α is constant on σ(c0 , c1 ) if and only if for any fc ⊆ M fc ∩ M fc . c ∈ Int(σ(c0 , c1 )) we have M 0

1

Remark 3.3.7. The inclusions in Proposition 3.3.6 may not be true at the endpoints of σ. For instance, Remark 3.3.5 (iv ) provides an example in which the inclusion in Proposition 3.3.6 (i) is not true at the endpoints of σ(−h0 , h0 ). Proof. (i) [=⇒] Assume that β is affine on σ(h0 , h1 ), i.e., there exists c ∈ H1 (M ; R) such that β(h) = β(h0 ) + hc, h − h0 i for all h ∈ σ(h0 , h1 ). For i = 0, 1, let µi be an action-minimizing measure with rotation vector hi . If t ∈ (0, 1), let us consider µt := tµ1 + (1 − t)µ0 . Clearly, ρ(µt ) = th1 + (1 − t)h0 ∈ Int(σ(h0 , h1 )); therefore β(ρ(µt )) = β(h0 ) + thc, h1 − h0 i. Then: Z Z L(x, v)dµt = L(x, v)d(tµ1 + (1 − t)µ0 ) TM TM Z Z = t L(x, v)dµ1 + (1 − t) L(x, v)dµ0 TM

TM

= tβ(h1 ) + (1 − t)β(h0 ) = t(β(h0 ) + hc, h1 − h0 i) + (1 − t)β(h0 ) = β(h0 ) + thc, h1 − h0 i = β(ρ(µt )). Therefore µt is action-minimizing with rotation vector th1 + (1 − t)h0 . Since this is true for any µi with rotation vector hi (i = 0, 1), it follows that: fh0 ∪ M fh1 ⊆ M fth1 +(1−t)h0 . M fh ⊇ M fh0 ∪ M fh1 for all h ∈ Int(σ(h0 , h1 )), using the observation [⇐=] Since M fh0 and M fh1 must in Remark 3.3.5 (ii ) and Proposition 3.3.4, we obtain that M 1 fc for some c0 ∈ H (M ; R). In particular, c0 ∈ ∂β(h0 ) ∩ be contained in M 0 ∂β(h1 ). Let us show that c0 ∈ ∂β(h) for all h ∈ Int(σ(h0 , h1 )). In fact, let h = th1 + (1 − t)h0 for some t ∈ (0, 1). Then, using the convexity of β: α(c0 ) + β(h)

= ≤ ≤ =

α(c0 ) + β(th1 + (1 − t)h0 )

t α(c0 ) + β(h1 ) + (1 − t) α(c0 ) + β(h0 ) thc0 , h1 i + (1 − t)hc0 , h0 i hc0 , th1 + (1 − t)h0 i = hc0 , hi.

On the other hand, the reverse inequality is always true (Legendre-Fenchel inequality).

34

CHAPTER 3

Now using this fact, it follows that if h ∈ σ(h0 , h1 ) then: α(c0 ) + β(h) = hc0 , hi

and

α(c0 ) + β(h0 ) = hc0 , h0 i;

therefore subtracting the second equality from the first one, we obtain what we wanted: β(h) − β(h0 ) = hc0 , h − h0 i ∀ h ∈ σ(h0 , h1 ) . (ii) [⇐=] It is a trivial consequence of Theorem 3.1.13. [=⇒] Let t ∈ (0, 1) and consider ct := tc1 + (1 − t)c0 . We want to show that for each h ∈ ∂α(ct ) we have hc1 − c0 , hi = 0. First observe that since ct ∈ Int(σ(c0 , c1 )), there exists δ > 0 such that ct + s(c1 − c0 ) ∈ σ(c0 , c1 ) for all s ∈ (−δ, δ). Then, using the facts that h ∈ ∂α(ct ) and that α is constant on σ(c0 , c1 ), we obtain: 0 = α(ct + s(c1 − c0 )) − α(ct ) ≥ shc1 − c0 , hi

∀ s ∈ (−δ, δ).

But this can be true only if hc1 − c0 , hi = 0. Now let us prove that, for i = 0, 1, ∂α(ci ) ⊇ ∂α(ct ) for all t ∈ (0, 1). If h ∈ ∂α(ct ), then using the facts that hc1 , hi = hc0 , hi and that α is constant on σ(c0 , c1 ), we get: α(ci ) + β(h)

= = = = =

α(ct ) + β(h) = hct , hi htc1 + (1 − t)c0 , hi thc1 , hi + (1 − t)hc0 , hi thci , hi + (1 − t)hci , hi hci , hi.

This and Remark 3.3.5 (ii ) immediately allow us to conclude that fc ⊆ M fc M i

for i = 0, 1

⇐⇒

fc ⊆ M fc ∩ M fc . M 0 1 

Remark 3.3.8. It follows from the previous remarks and Proposition 3.3.6 that, in general, the action-minimizing measures (and consequently the Mather fc and M fh ) are not necessarily ergodic. Recall that an invariant probsets M ability measure is said to be ergodic if all invariant Borel sets have measure 0 or 1. These measures play a special role in the study of the dynamics of the system; therefore, one could ask what the ergodic action-minimizing measures are. It is a well-known result from ergodic theory that the ergodic measures of a flow correspond to the extremal points of the set of invariant probability measures (see for instance [49]), where by “extremal point” of a convex set we mean an element that cannot be obtained as a nontrivial convex combination of other elements of the set. Since β has superlinear growth, its epigraph {(h, t) ∈ H1 (M ; R) × R : t ≥ β(h)} has infinitely many extremal points. Let

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

35

(h, β(h)) denote one of these extremal points. Then, there exists at least one ergodic action-minimizing measure with rotation vector h. It is in fact sufficient to consider any extremal point of the set {µ ∈ Mh (L) : AL (µ) = β(h)}: this measure will be an extremal point of M(L) and hence ergodic. Moreover, as we have already recalled in Remark 3.2.1, for such an ergodic measure µ, Birkhoff’s ergodic theorem implies that for µ-almost every initial datum, the corresponding trajectory has rotation vector h. 3.4

THE SYMPLECTIC INVARIANCE OF MATHER SETS

In this section we would like to discuss some symplectic aspects of the theory that we have started to develop. In particular, as a first step, we would like to understand how the Mather sets behave under the action of symplectomorphisms (the author learned part of this material from [16] and [13]). In order to do this, we need to move to the Hamiltonian setting, rather than the Lagrangian one, and consider the Hamiltonian H : T∗ M −→ R, defined by Fenchel duality (see section 1.2). Recall that the associated Hamiltonian flow ΦH is conjugate to the Euler-Lagrange flow of L; therefore, from a dynamical systems point of view, the two systems are equivalent. We can define the associated Mather sets in the cotangent bundle as fc (L)), M∗c (H) := LL (M where LL : TM −→ T∗ M is the Legendre transform associated with L (see (1.4)). These sets are nonempty, compact, and invariant for the Hamiltonian flow and they continue to satisfy the graph property (see Theorems 3.1.12 and 3.2.7). The main advantage of shifting our point of view is that the cotangent Pd bundle T∗ M can be naturally equipped with an exact symplectic form ω = i=1 dxi ∧ Pd dpi = −dλ, where λ = i=1 pi dxi is what is called the tautological form or Liouville form. See appendix A for a more intrinsic definition of λ. Let Ψ : T∗ M −→ T∗ M be a diffeomorphism. We will say that Ψ is a symplectomorphism if it preserves the symplectic form ω, i.e., Ψ∗ ω = ω. An easy class of examples is provided by translations in the fibers. Let η be a closed 1-form and consider τη : T∗ M −→ T∗ M , (x, p) 7−→ (x, p + η(x)). Then, τ is a symplectomorphism. Observe that since ω = −dλ, one has that d(Ψ∗ λ−λ) = 0; in other words, the 1-form Ψ∗ λ−λ is closed. We will call the cohomology class of Ψ the cohomology class of Ψ∗ λ − λ and denote it by [Ψ] ∈ H1 (T∗ M ; R) ' H1 (M ; R), where the latter identification follows from the fact that T∗ M can be retracted to M along the fibers (hereafter we will always identify these two spaces). Obviously, going back to the previous example, [τη ] = [η]. In particular, Ψ is said to be exact if and only if [Ψ] = 0. Lemma 3.4.1. Any symplectomorphism Ψ : T∗ M −→ T∗ M can be written as Ψ = Φ ◦ τη , where Φ is an exact symplectomorphism and [η] = [Ψ].

36

CHAPTER 3

Proof. Let η be any closed 1-form on M such that [η] = [Ψ], and define τη as above. Moreover, we define Φ = Ψ ◦ (τη )−1 = Ψ ◦ τ−η . In order to conclude the proof of the lemma, we need to check that Φ is exact. In fact, ∗ ∗ ∗ Φ∗ λ = (Ψ ◦ τ−η )∗ λ = τ−η (Ψ∗ λ) = τ−η (λ + Θ) = λ + τ−η Θ − η, ∗ where Θ := Ψ∗ λ − λ. Since τ−η is isotopic to the identity, then τ−η Θ and η ∗ ∗ are cohomologous ([τ−η Θ] = [η] = [Ψ]); therefore, [Φ λ − λ] = [Θ − η] = 0 and consequently it is exact. 

Therefore, if we want to understand the interplay between Mather sets and the action of symplectomorphisms, it will be sufficient to analyze the behavior of these two kinds of symplectomorphisms: translations in the fibers and exact symplectomorphisms. Proposition 3.4.2. Let H : T∗ M −→ R be a Tonelli Hamiltonian and η a closed 1-form on M . Then:   M∗c (H ◦ τη ) = τ−η M∗c+[η] (H) ∀ c ∈ H1 (M ; R). Remark 3.4.3. If H is a Tonelli Hamiltonian, then also H ◦ τη is a Tonelli Hamiltonian (we are just composing it with a vertical translation in the fibers). Proof. Let us start by observing that the Lagrangian associated with H ◦τη is Lη := L − ηˆ and that the associated Legendre transform LLη = τ−η ◦ L (just derive Lη with respect to v). Therefore, using the definition of Mather sets for a given cohomology class, we get:     fc (L − ηˆ) = LL M fc+[η] (L) M∗c (H ◦ τ ) := LLη M η     fc+[η] (L) = τ−η M∗ = (τ−η ◦ LL ) M (H) . c+[η]  Let us see now what happens with exact symplectomorphisms (see also [77]). Proposition 3.4.4. Let H : T∗ M −→ R be a Tonelli Hamiltonian and Φ : T∗ M −→ T∗ M an exact symplectomorphism such that H ◦ Φ is still of Tonelli type. Then: M∗0 (H ◦ Φ) = Φ−1 (M∗0 (H)). Moreover, if Φ is isotopic to the identity, then M∗c (H ◦ Φ) = Φ−1 (M∗c (H))

∀ c ∈ H1 (M ; R).

Remark 3.4.5. Observe that even if H is Tonelli, H ◦ Φ is not necessarily of Tonelli type. Moreover, without assuming Φ isotopic to the identity, the latter statement might be false (see for instance [69]).

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

37

This proposition can be easily deduced from the following lemma (see also [77]). Lemma 3.4.6. Let H : T∗ M −→ R be a Tonelli Hamiltonian and Φ : T M −→ T∗ M an exact symplectomorphism such that H 0 := H ◦ Φ is still of Tonelli type. Then: (i) µ is an invariant probability measure of H if and only if Φ∗ µ is an invariant probability measure of H 0 = H ◦ Φ; (ii) for any µ, an invariant probability measure of H, the following holds:   Z  Z  ∂H 0 ∂H 0 (x, p) − H(x, p) dµ = p (x, p) − H (x, p) dΦ∗ µ. (3.7) p ∂p ∂p ∗

Remark 3.4.7. The identity in (3.7) represents the equality between the associated Lagrangian actions (in the Hamiltonian formalism). Therefore, since the respective actions coincide on all measures and the invariant measures are in 1-1 correspondence, there must be a correspondence between the minimizing ones: µ is action-minimizing for H if and only if Φ∗ µ is action-minimizing for H ◦ Φ. From this, Proposition 3.4.4 follows easily. Proof. (Lemma 3.4.6) (i) The first part of the statement follows from the classical fact that Φ tranforms the associated Hamiltonian vector fields in the following way: XH (Φ(x, p)) = DΦ(x, p)XH◦Φ (x, p)

∀ (x, p) ∈ T∗ M.

(3.8)

(ii) If we denote by λ(x, p) the Liouville form pdx, then:  Z  Z   ∂H p (x, p) − H(x, p) dµ = λ(x, p)[XH (x, p)] − H(x, p) dµ . ∂p Therefore, using the fact that | det DΦ| = 1, the relation in (3.8), and the fact that Φ∗ λ − λ = df (since Φ is an exact symplectomorphism), we obtain:  Z  Z   ∂H p (x, p) − H(x, p) dµ = λ(x, p)[XH (x, p)] − H(x, p) dµ ∂p Z   = Φ∗ λ(x0 , p0 )[DΦ−1 (x0 , p0 )XH (Φ(x0 , p0 ))] − H(Φ(x0 , p0 )) dΦ∗ µ Z  
= λ(x0 , p0 ) + df (x0 , p0 ) [XH◦Φ (x0 , p0 )] − H(Φ(x0 , p0 )) dΦ∗ µ Z   = λ(x0 , p0 )[XH 0 (x0 , p0 )] − H 0 (x0 , p0 ) dΦ∗ µ Z + df (x0 , p0 )[XH 0 (x0 , p0 )] dΦ∗ µ Z   ∂H 0 = p0 0 (x0 , p0 ) − H(x0 , p0 ) dΦ∗ µ. ∂p

38

CHAPTER 3

R In the last equality we used the fact that df (x0 , p0 )[XH 0 (x0 , p0 )] dΦ∗ µ = 0, as it follows easily from the invariance of Φ∗ µ. For the sake of simplifying the notation, let us denote ν := dΦ∗ µ and assume that ν is ergodic (otherwise consider each ergodic component). Using the ergodic theorem and the compactness of the suppport of ν, we obtain that for a generic point (x0 , y0 ) in the support of ν: Z Z 0 1 N H0 0 0 0 0 df (Φt (x0 , p0 ))[XH 0 (ΦH df (x , p )[XH 0 (x , p )] dν = lim t (x0 , p0 ))] dt N →+∞ N 0 0

=

f (ΦH N (x0 , p0 )) − f (x0 , p0 ) = 0. N →+∞ N lim

 Finally, we can deduce the main result of this section. (See also [69], where the case in which Ψ is not isotopic to the identity is discussed.) Theorem 3.4.8. Let H : T∗ M −→ R be a Tonelli Hamiltonian and Ψ : T M −→ T∗ M a symplectomorphism of class [Ψ], isotopic to the identity such that H ◦ Ψ is of Tonelli type. Then,   M∗c (H ◦ Ψ) = Ψ−1 M∗c+[Ψ] (H) ∀ c ∈ H1 (M ; R). ∗

Proof. Suppose that Ψ = Φ ◦ τη , with [η] = [Ψ]. Then, using Propositions 3.4.2 and 3.4.4, we get for all c ∈ H1 (M ; R):   M∗c (H ◦ Ψ) = M∗c (H ◦ Φ ◦ τη ) = τ−η M∗c+[η] (H ◦ Φ)      = τη−1 Φ−1 M∗c+[η] (H) = (Φ ◦ τη )−1 M∗c+[η] (H)   = Ψ−1 M∗c+[Ψ] (H) .  Corollary 3.4.9. Let H : T∗ M −→ R be a Tonelli Hamiltonian and Ψ : T M −→ T∗ M a symplectomorphisms of class [Ψ] and isotopic to the identity, such that H ◦ Ψ is of Tonelli type. Then: ∗

(i) (ii)

αH◦Ψ (c) = αH (c + [Ψ]) βH◦Ψ (h) = βH (h) − h[Ψ], hi

∀ c ∈ H1 (M ; R) ∀ h ∈ H1 (M ; R).

Remark 3.4.10. For a more symplectic interpretation (and more properties) of Mather’s α-function in terms of Hofer and Viterbo’s distance in the group of Hamiltonian diffeomorphisms, see [79].

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

39

Proof. (i) From Theorem 3.4.8 it follows that if (x, p) ∈ M∗c (H ◦ Ψ), then Ψ(x, p) ∈ M∗c+[Ψ] (H). Therefore, using Theorem 3.1.13, for any (x, p) ∈ M∗c (H◦ Ψ) we have: αH◦Ψ (c) = (H ◦ Ψ)(x, p) = H(Ψ(x, p)) = αH (c + [Ψ]). (ii) Observe that if h ∈ ∂αH◦Ψ (c), then h ∈ ∂αH (c + [Ψ]). In fact, for every c0 ∈ H1 (M ; R): αH (c0 + [Ψ]) − αH (c0 + [Ψ])

= αH◦Ψ (c0 ) − αH◦Ψ (c) ≥ hc0 − c, hi = h(c0 + [Ψ]) − (c − [Ψ]), hi.

Let now h ∈ ∂αH◦Ψ (c). Using the fact that α and β are conjugates of each other, we get: βH◦Ψ (h)

= hc, hi − αH◦Ψ (c) = hc, hi − αH (c + [Ψ]) = hc + [Ψ], hi − αH (c + [Ψ]) − h[Ψ], hi = βH (h) − h[Ψ], hi. 

We can summarize everything in the following commutative diagram. ΦH◦Ψ t



/ T∗ M O

Ψ

T∗O M H◦Ψ

$

z

αH◦Ψ

H1 (M ; R) 3.5

H

:Rd

c7→M∗ c (H◦Ψ)

ΦH t

c7→M∗ c (H) αH

c 7→c+[Ψ]

/ H1 (M ; R)

AN EXAMPLE: THE SIMPLE PENDULUM (PART I)

In this section we would like to describe the Mather sets, the α-function, and the β-function in a specific example: the simple pendulum. This system can be described in terms of the Lagrangian: L : TT −→ (x, v) 7−→

R
1 2 |v| + 1 − cos(2πx) . 2

It is easy to check that the Euler-Lagrange equation provides exactly the equation of the pendulum (see figure 3.1 for its phase-space portrait):  v = x˙ v˙ = 2π sin(2πx) ⇐⇒ x ¨ − 2π sin(2πx) = 0.

40

CHAPTER 3

E>0 ν E=0

−2 < E < 0 0

E = −2 1/2

x

1

E=0

E>0

Figure 3.1: The phase space of the simple pendulum.

The associated Hamiltonian (or energy) H : T∗ T −→ R is given by H(x, p) := − (1 − cos(2πx)). Observe that in this case the Legendre transform (x, p) = LL (x, v) = (x, v); therefore we can easily identify the tangent and cotangent bundles. In the following we will consider TT ' T∗ T ' T × R and identify H1 (M ; R) ' H1 (M ; R) ' R. First of all, let us study what are the invariant probability measures of this system. 1 2 2 |p|

• Observe that (0, 0) and ( 12 , 0) are fixed points for the system (respectively unstable and stable). Therefore, the Dirac measures concentrated on each of them are invariant probability measures. Hence, we have found two first invariant measures, δ(0,0) and δ( 12 ,0) , both with zero rotation vector: ρ(δ(0,0) ) = ρ(δ( 12 ,0) ) = 0. As far as their energy is concerned (i.e., the energy levels in which they are contained), it is easy to check that E(δ(0,0) ) = H(0, 0) = 0 and E(δ( 21 ,0) ) = H( 12 , 0) = −2. Observe that these two energy levels cannot contain any other invariant probability measure. • If E > 0, then the energy level {H(x, v) = E} consists of two homotopically nontrivial periodic orbits (rotation motions): p ± PE := {(x, v) : v = ± 2[(1 + E) − cos(2πx)], ∀ x ∈ T}. The probability measures evenly distributed along these orbits—which we will denote µ± E —are invariant probability measures of the system. If we

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

41

denote by Z T (E) := 0

1

1 p

2[(1 + E) − cos(2πx)]

dx

(3.9)

the period of such orbits, then it is easy to check that (see Remark 3.2.1) ±1 ρ(µ± E ) = T (E) . Observe that the function T : (0, +∞) −→ (0, +∞), which associates with a positive energy E the period of the correspond± ing periodic orbits PE , is continuous and strictly decreasing. Moreover, T (E) → ∞ as E → 0 (it is easy to see this by observing that motions on the separatrices take “infinite” time to connect 0 to 1 ≡ 0 mod.1). Therefore, ρ(µ± E ) → 0 as E → 0. • If −2 < E < 0, then the energy level {H(x, v) = E} consists of one contractible periodic orbit (libration motion): PE := {(x, v) : v 2 = 2(1 + E) − 2 cos(2πx),

x ∈ [xE , 1 − xE ]},

1 2π

where xE := arccos(1+E). The probability measure evenly distributed along this orbit—which we will denote by µE —is an invariant probability measure of the system. Moreover, since this orbit is contractible, its rotation vector is zero: ρ(µE ) = 0. The measures above are the only ergodic invariant probability measures of the system. Other invariant measures can be easily obtained as convex combinations of them. Now we want to understand which of these are action-minimizing for some cohomology class. Remark 3.5.1. (i ) Let us start by remarking that for −2 < E < 0 the support of the measure µE is not a graph over T; therefore it cannot be actionminimizing for any cohomology class, since otherwise it would violate Mather’s graph theorem (Theorems 3.1.12 and 3.2.7). Therefore all action-minimizing measures will be contained in energy levels corresponding to energy greater than zero. It follows from Theorem 3.1.13 that α(c) ≥ 0 for all c ∈ R. (ii ) Another interesting property of the α-function (in this specific case) is that it is an even function: α(c) = α(−c) for all c ∈ R. This is a consequence of the particular symmetry of the system, i.e., L(x, v) = L(x, −v). In fact, let us denote τ : T × R −→ T × R, (x, v) 7−→ (x, −v) and observe that if µ is an invariant probability measure, then also τ ∗ µ is an invariant probability measure. Moreover, τ ∗ M(L) = M(L), where M(L) denotes the set of all invariant probability measures of R R L. It is now sufficient to observe that, for each µ ∈ M(L), (L − c · v) dµ = (L + c · v)dτ ∗ µ, and hence conclude that Z Z α(c) = − inf (L − c · v) dµ = − inf (L + c · v)dτ ∗ µ = α(−c) . M(L)

M(L)

(iii ) It follows from the above symmetry and the convexity of α that minR α(c) = α(0).

42

CHAPTER 3

Let us now start by studying the 0-action-minimizing measures, i.e., invariant probability measures that minimize the action of L without any “correction.” Since L(x, v) ≥ 0 for each (x, v) ∈ T × R, then AL (µ) ≥ 0 for all µ ∈ M(L). In particular, AL (δ(0,0) ) = 0; therefore δ(0,0) is a 0-action-minimizing measure and α(0) = 0. Since there are not other invariant probability measures supported in the energy level {H(x, v) = 0} (i.e., on the separatrices), then we can conclude that: f0 = {(0, 0)} . M Moreover, since α0 (0) = 0 (see Remark 3.5.1 (iii )), it follows from Remark 3.3.5 that: f0 = M f0 = {(0, 0)} . M On the other hand, this could be also deduced from the fact that the only other measures with rotation vector 0 cannot be action-minimizing since they do not satisfy the graph theorem (Remark 3.5.1). Now let us investigate what happens for other cohomology classes. A naïve observation is that since the α-function is superlinear and continuous, all energy levels for E > 0 must contain some Mather set; in other words, they will be achieved for some c. + Let E > 0 and consider the periodic orbit PE and the invariant probability + measure µE evenly distributed on it. The graph of this orbit can be seen as the p + graph of a closed 1-form ηE := 2[(1 + E) − cos(2πx)] dx, whose cohomology class is Z 1p + c+ (E) := [ηE ]= 2[(1 + E) − cos(2πx)] dx, (3.10) 0

which can be interpreted as the (signed) area between the curve and the positive x-semiaxis. This value is clearly continuous and strictly increasing with respect to E (for E > 0) and, as E → 0: Z 1p 4 + 2[1 − cos(2πx)] dx = . c (E) −→ π 0 Therefore, it defines an invertible function c+ : (0, +∞) −→ ( π4 , +∞). + We want to prove that µ+ E is c (E)-action-minimizing. The proof will be an imitation of what was already seen for KAM tori in section 2.1 (see Proposition 2.1.4). + Let us consider the Lagrangian Lη+ (x, v) := L(x, v) − ηE (x) · v. Then, using E

the Legendre-Fenchel inequality (1.3) (in the support of µ+ E , because of our + choice of ηE , this is indeed an equality): Z Z
+ Lη+ (x, v)dµ+ = L(x, v) − ηE (x) · v dµ+ E E E Z + = −H(x, ηE (x))dµ+ E = −E.

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

43

Now, let ν be any other invariant probability measure and apply again the same procedure as above (warning: this time the Legendre-Fenchel inequality is not an equality anymore!): Z Z
+ Lη+ (x, v)dν = L(x, v) − ηE (x) · v dν E Z + ≥ −H(x, ηE (x))dν = −E . + Therefore, we can conclude that µ+ E is c (E)-action-minimizing. Since it already projects over the whole T, it follows from the graph theorem that it is unique: p fc+ (E) = P + = {(x, v) : v = 2[(1 + E) − cos(2πx)], ∀ x ∈ T}. M E

Furthermore, since ρ(µ+ E) =

1 T (E) ,

1 f T (E) fc+ (E) = P + . M =M E

− Similarly, one can consider the periodic orbit PE and the invariant probabil− ity measure µE evenly distributed on p it. The graph of this orbit can be seen as − + the graph of a closed 1-form ηE := − 2[(1 + E) − cos(2πx)] dx = −ηE , whose cohomolgy class is c− (E) = −c+ (E). Then (see also Remark 3.5.1 (ii )): p fc− (E) = P − = {(x, v) : v = − 2[(1 + E) − cos(2πx)], ∀ x ∈ T}, M E

and

1 fc− (E) = P − . f− T (E) =M M E

Note that this completes the study of the Mather sets for any given rotation vector, since ρ(µ± E) = ±

1 T (E)

E→+∞

−→ ±∞

and

ρ(µ± E) = ±

1 E→0+ −→ 0 . T (E)

What remains to study is what happens for nonzero cohomology classes in [− π4 , π4 ]. The situation turns out to be quite easy. Observe that α(c± (E)) = E. Therefore, from the continuity of α it follows that (take the limit as E → 0): α(± π4 ) = 0. Moreover, since α is convex and min α(c) = α(0) = 0, α(c) ≡ 0 on [− π4 , π4 ]. Therefore, the corresponding Mather sets will lie in the zero energy level. From the above discussion, it follows that in this energy level there is a unique invariant probability measure, namely δ(0,0) , and consequently: fc = {(0, 0)} M

for all −

4 4 ≤c≤ . π π

Let us summarize what we have found so far. Recall that in (3.9) and (3.10) we have introduced these two functions: T : (0, +∞) −→ (0, +∞) and

44

CHAPTER 3

c+ : (0, +∞) −→ ( π4 , +∞), representing respectively the period and the “cohomology” (area below the curve) of the “upper” periodic orbit of energy E. These functions (for which we have an explicit formula in terms of E) are continuous and strictly monotone (respectively, decreasing and increasing). Therefore, we can define their inverses which provide the energy of the periodic orbit with period T (for all positive periods) or the energy of the periodic orbit with cohomology class c (for |c| > π4 ). We will denote them by E(T ) and E(c) (observe that this last quantity is exactly −α(c)). Then:  if − π4 ≤ c ≤ π4  {(0, 0)} + fc = P if c > π4 M  E(c) − PE(−c) if c < − π4 and

 0)}   {(0, + h P f 1 M = E( h )   P− 1

E(− h )

if h = 0 if h > 0 if h < 0 .

We can provide an expression for these functions in terms of the quantities introduced above (see figure 3.2 for a sketch of their graphs):  0 if − π4 ≤ c ≤ π4 α(c) = E(|c|) if |c| > π4 and  β(h) =

0 1 1 c(E( |h| ))|h| − E( |h| )

if h = 0 if h = 6 0.

Observe that the α-function is C 1 . In fact, the only problem might be at c = ± π4 , but also there it is differentiable, with derivative 0. If it were not

α(c)

−4/π

4/π

β(h)

c

4/π

4/π 0

h

Figure 3.2: Sketch of the graphs of the α- and β-functions of the simple pendulum.

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

45

differentiable, then there would exist a subderivative h 6= 0 and consequently fh ⊆ M f± 4 , which is absurd since the set on the right-hand side consists of M π a single point. However, α is not strictly convex, since there is a flat piece on which it is zero. As far as β is concerned, it is strictly convex (as a consequence of α being C 1 ), but it is differentiable everywhere except at the origin. At the origin, in fact, there is a corner and the set of subderivatives (i.e., the slopes of tangent lines) is given by ∂β(0) = [− π4 , π4 ] (this is related to the fact that α has a flat on this interval). 3.6

HOLONOMIC MEASURES AND GENERIC PROPERTIES OF TONELLI LAGRANGIANS

In this section we would like to stress that using the above approach the minimizing measures are obtained through a variational principle over the set of invariant probability measures. Because of the request for “invariance,” this set clearly depends on the Lagrangian that one is considering. Moreover, it is somehow unnatural for variational problems to ask a priori for invariance. What generally happens, in fact, is that invariance is obtained as a byproduct of the minimization process carried out. An alternative approach, slightly different in this respect, was due to Ricardo Mañé [51] (see also [52]). This deals with the bigger set of holonomic measures (or closed measures; see Remark 3.6.1) and has proved to be extremely advantageous in dealing with different Lagrangians at the same time. In this section we want to sketch the basic ideas behind it. Let C`0 be the set of continuous functions f : TM → R growing (fiberwise) at most linearly, i.e., kf k` :=

sup (x,v)∈TM

f (x, v) < +∞ , 1 + kvk

and let MR`M be the set of probability measures on the Borel σ-algebra of TM such that TM kvk dµ < ∞ , endowed with the unique metrizable topology given by: Z Z µn −→ µ ⇐⇒ f (x, v) dµn −→ f (x, v) dµ ∀ f ∈ C`0 . TM

TM

Let (C`0 )∗ be the dual of C`0 . Then M`M can be naturally embedded in (C`0 )∗ and its topology coincides with that induced by the weak∗ topology on (C`0 )∗ . The space of probability measures that we will be considering is a closed subset of M`M (endowed with the induced topology), which is defined as follows. If γ : [0, T ] → M is a closed absolutely continuous curve, let µγ be such that Z Z 1 T f (γ(t), γ(t)) ˙ dt ∀ f ∈ C`0 . f (x, v) dµγ = T 0 TM

46

CHAPTER 3

R Observe that µγ ∈ M`M because if γ is absolutely continuous then |γ(t)| ˙ dt < ` +∞. Let C(M ) be the set of such µγ ’s and C(M ) its closure in MM . This set is convex and it is called the set of holonomic measures on M . One can check that the following properties are satisfied (see [52]): i) M(L) ⊆ C(M ) ⊆ M`M . In particular, for every Tonelli Lagrangian L on TM , all probability measures µ thatR are invariant with respect to the Euler-Lagrange flow and such that TM L dµ < +∞, are contained in C(M ). ii) With any given probability µ ∈ C(M ), one can associate a rotation vector ρ(µ) ∈ H1 (M ; R). This map extends continuously to a map ρ : C(M ) −→ H1 (M ; R), and this extension is surjective. n o iii) For each C ∈ R the set µ ∈ C(M ) : AL (µ) ≤ C is compact. iv) If a measure µ ∈ C(M ) satisfies n o AL (µ) = min AL (ν) : ν ∈ C(M ) , then µ ∈ M(L) (and in particular it is invariant). Observe that the existence of probabilities attaining the minimum follows from iii). In view of these properties, it is clear that the corresponding minimizing problem, although on a bigger space of measures, will lead to the same results as before and to the same definition of Mather sets. Remark 3.6.1. One can define the set of closed measures on TM as:   Z ` 1 K(TM ) := µ ∈ MM such that df (x) · v dµ = 0 ∀ f ∈ C (M ) . It is easy to verify that holonomic measures satisfy Proposition 3.1.7 and therefore C(M ) ⊆ K(TM ). It is definitely less trivial to prove that indeed these two sets coincide: C(M ) ⊆ K(TM ). Although this was originally noticed by John Mather, as far as I know it has never been published by the author. A proof of this result may be found in [15]. As we have already pointed out, this different approach is more suitable for working with different Lagrangians, for instance if one wants to study properties of a family of Lagrangians or wants to consider perturbations of a given system, as usually done in perturbation theory. Using these ideas, Ricardo Mañé [52] showed that one can prove much stronger results if, instead of considering all Lagrangians, one considers generic Lagrangians. Let us clarify what we mean by “generic.”

ACTION-MINIMIZING INVARIANT MEASURES FOR TONELLI LAGRANGIANS

47

Definition 3.6.2. A property P is said to be generic (in the sense of Mañé) for a Lagrangian L if there exists a residual set (i.e., dense Gδ set) SL ⊆ C 2 (M ), such that if U ∈ SL then L + U satisfies the property P. In [52] Ricardo Mañé proved the following result. Theorem 3.6.3 (Mañé). For a fixed c ∈ H1 (M ; R), having a unique c-actionminimizing measure is a generic property in the sense of Mañé. In other words, for any Tonelli Lagrangian L, there exists a residual subset SL ⊆ C 2 (M ) such that, for each U ∈ SL , the Lagrangian L + U has a unique c-action-minimizing measure. This result has been recently improved by Patrick Bernard and Gonzalo Contreras [17]: Theorem 3.6.4 (Bernard-Contreras). The following property is generic in the sense of Mañé: for all c ∈ H1 (M ; R), there are at most 1 + dim H1 (M ; R) ergodic c-action-minimizing measures.

Chapter Four Action-Minimizing Curves for Tonelli Lagrangians In the previous chapter we have described the construction and the main properties of Mather sets. One of the main limitations of these sets is that, being the support of invariant probability measures, they are recurrent under the flow (Poincaré recurrence theorem), i.e., each orbit after a sufficiently long time (and therefore infinitely many often times) will return arbitrarily close to its initial point. This property excludes many interesting invariant sets that are somehow “invisible” to such a construction; for instance, think about the stable and unstable manifolds of some hyperbolic invariant set, or about heteroclinic and homoclinic orbits between invariant sets. In this chapter we will construct other (possibly) “larger” compact invariant sets and discuss their significance for the dynamics: the Aubry sets and the Mañé sets. 4.1

GLOBAL ACTION-MINIMIZING CURVES: AUBRY AND MAÑÉ SETS

The key idea is the same as the one we have already explored in section 2.1: instead of considering action-minimizing invariant probability measures, one can look at action-minimizing curves for some modified Lagrangian. We showed in section 2.1 (see Remarks 2.1.9 and 2.1.12) that orbits on KAM tori could be characterized in terms of this property. In this section we will imitate that construction in the general case of a Tonelli Lagrangian. In the light of Lemma 3.1.6 and the discussion in section 3.1, let us fix a cohomology class c ∈ H1 (M ; R) and choose a smooth 1-form η on M that represents c. As we have already pointed out in section 1.1, there is a close relation between solutions of the Euler-Lagrange flow and extremals of the action functional ALη for the fixed endpoint problem (which are the same as the extremals of AL ). In general, these extremals are not minima (they are local minima only if the time length is very short [37, Section 3.6]). One could wonder if such minima exist, namely if for any given endpoints x, y ∈ M and any given positive time T , there exists a minimizing curve connecting x to y in time T . From what has been already said, this curve will correspond to an orbit for the Euler-Lagrange flow. Under our hypothesis on the Lagrangian, the answer to this question turns out to be affirmative. This is a classical result known as the Tonelli theorem in the calculus of variations.

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

49

Theorem 4.1.1 (Tonelli theorem, [62]). Let M be a compact manifold and L a Tonelli Lagrangian on TM . For all a < b ∈ R and x, y ∈ M , there exists, in the set of absolutely continuous curves γ : [a, b] −→ M such that γ(a) = x Rb and γ(b) = y, a curve that minimizes the action ALη (γ) = a Lη (γ(t), γ(t)) ˙ dt. Rb Remark 4.1.2. (i ) A curve minimizing ALη (γ) = a Lη (γ(t), γ(t)) ˙ dt subject to the fixed endpoint condition γ(a) = x and γ(b) = y is called a c-Tonelli minimizer. Recall that such minimizers do only depend on c and not on the chosen representative η. In fact, adding an exact 1-form df to L will contribute a constant term f (y)−f (x) that does not play any role in selecting the minimizers (see also Proposition 3.1.7). (ii ) As Mañé pointed out in [53], in order for these minimizers to exist, it is not necessary to assume the compactness of M : the superlinear growth condition with respect to some complete Riemannian metric on M is enough. (iii ) A Tonelli minimizer which is C 1 is in fact C r (if the Lagrangian L is C r ) and satisfies the Euler-Lagrange equation; this follows from the usual elementary argument in the calculus of variations, together with Caratheodory’s remark on differentiability. In the autonomous case, Tonelli minimizers will be always C 1 . In the non-autonomous time-periodic case (the Tonelli theorem holds also in this case [62]), as already noted in Remark 1.1.2, one needs to require that the Euler-Lagrange flow be also complete. We will sketch here the proof of the Tonelli theorem. We refer the reader to [62, Appendix 1] for more details. A different proof of this theorem can be found in [15]. Proof. (Tonelli theorem) The proof of this theorem follows from the following result (by C ac we denote the set of absolutely continuous functions). Claim. Let K ∈ R. The set SK := {γ ∈ C ac ([a, b], M ) : ALη (γ) ≤ K} is compact in the C 0 topology. First, let us see how to obtain the Tonelli theorem from this claim. Let k0 := inf{ALη (γ) : γ ∈ C ac ([a, b], M )}. Observe that k0 > −∞ since Lη is bounded from below. Therefore, for any K > k0 , SK 6= ∅ and it is compact because of the above claim. Moreover, SK ⊆ SK 0 if K ≤ K 0 . Hence: \ SK 6= ∅ K>k0

and any element in this intersection is a c-Tonelli minimizer. The proof of the above claim consists of several steps (see [62, Appendix 1] for the missing details). - The first step is the observation that the family of curves in Sk is absolutely equicontinuous, i.e., for every ε > 0 there existP δ > 0 such that if a ≤ n a < b ≤ a < b ≤ . . . ≤ a < b ≤ b and 0 0 1 1 n n i=0 (bi − ai ) < δ, then Pn i=0 d(γ(ai ), γ(bi )) < ε. For this, one needs to use the superlinearity of L.

50

CHAPTER 4

- Now, one can apply the Ascoli-Arzelà theorem to deduce that every sequence {γn }n in SK has a convergent subsequence with respect to the C 0 topology. Moreover, it follows easily from the definition of absolutely equicontinuity that the limit of any convergent subsequence must be also absolutely equicontinuous. - The last, and more involved, step is to show that if γ is the limit of a sequence {γn }n in SK , then γ ∈ Sk , namely ALη (γ) ≤ K. See [62, pages 199–201]. These three steps conclude the proof of the claim.



Remark 4.1.3. Observe that the lemma used in the proof of the Tonelli theorem is a sort of semicontinuity result for the Lagrangian action (compare this with Proposition 3.1.3). In fact, it implies that if {γn }n is a sequence in C ac ([a, b], M ), which converges to γ in the C 0 topology, then γ ∈ C ac ([a, b], M ) and ALη (γ) ≤ lim inf n→+∞ ALη (γn ). In the following we will be interested in particular Tonelli minimizers that are defined for all times and whose action is minimal with respect to any given time length. We will see that these curves present a very rich structure. Definition 4.1.4 (c-minimizers). An absolutely continuous curve γ : R −→ M is a c-(global)-minimizer for L if, for any given a < b ∈ R, ALη (γ [a, b]) = min ALη (σ), where the minimum is taken over all σ : [a, b] → M such that σ(a) = γ(a) and σ(b) = γ(b). We have already seen in section 2.1 that one can give an apparently stronger notion of minimizer, asking that the minimum be realized among all curves connecting the two endpoints, independently of their time lengths. Definition 4.1.5 (c-time-free minimizers). An absolutely continuous curve γ : R −→ M is a c-time-free minimizer for L if, for any given a < b ∈ R, ALη (γ [a, b]) = min ALη (σ), where the miminimum is taken over all σ : [a0 , b0 ] → M such that σ(a0 ) = γ(a) and σ(b0 ) = γ(b). Remark 4.1.6. (i ) We have proven in section 2.1 that orbits on KAM tori satisfy this stronger condition, modulo adding a constant to the Lagrangian (see Remark 2.1.9). In fact, it is quite easy to see that this condition is “sensitive” to the addition of constants to the Lagrangian (although this is something totally irrelevant for its being or not being a c-minimizer). For example, suppose that RT R T0 L (γ, γ)dt ˙ < Lη (σ, σ)dt ˙ with T 0 < T and let k be a constant such that η 0 0 ! Z T0 Z T 1 Lη (σ, σ)dt ˙ − Lη (γ, γ)dt ˙ . k> 0 T −T 0 0

51

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

Then, adding k to the Lagrangian, we even reverse the inequality: Z T Z T Lη (γ, γ)dt ˙ + kT = Lη (γ, γ)dt ˙ + k(T − T 0 ) + kT 0 ALη +k (γ) = 0

0 T

Z ≥

Z

T0

Lη (γ, γ)dt ˙ + 0

Z Lη (σ, σ)dt ˙ −

0

T

Lη (γ, γ)dt ˙ + kT 0

0

T0

Z

Lη (σ, σ)dt ˙ + kT 0 = ALη +k (σ).

= 0

(ii ) Obviously a c-time-free minimizer is also a c-minimizer. Fathi [36] proved that in the autonomous case, modulo adding a well-specified (and unique) right constant, these two notions of minimizers indeed coincide: if γ is a c-minimizer for L, then, γ is a c-time-free minimizer for L + α(c), where α is the α-function associated with L. We proved this in the special setting of section 2.1 (see Proposition 2.1.8). We will discuss the general case in chapter 5. (iii ) The equivalence between these two notions of minimizers is not true anymore when we consider time-periodic Tonelli Lagrangians (see [41]). Tonelli Lagrangians for which this equivalence result holds are called regular. Patrick Bernard [10] showed that under suitable assumptions on the Mather set it is possible to prove that the Lagrangian is regular. For instance, if the Mather set fc is the union of 1-periodic orbits, then Lη is regular. This problem turned out M to be strictly related to the convergence of the so-called Lax-Oleinik semigroup (see [37] and section 5.1 for its definition). Now we will study the existence and the properties of c-minimizers and ctime-free minimizers. There are two equivalent approaches that one can pursue: one is essentially due to Ricardo Mañé [53, 26] (see also [27]), while the other has been developed by John Mather [64]. In the following, in order to keep the analogy with the illustrative example discussed in section 2.1, we prefer following the first of these two approaches. We will discuss Mather’s approach in section 4.4. Given any x, y ∈ M and T>0, let us denote by CT (x, y) the set of absolutely continuous curves γ : [0, T ] −→ M such that γ(0) = x and γ(T ) = y. The Tonelli theorem implies that there exists γmin ∈ CT (x, y) realizing the minimum, i.e., ALη (γmin ) = minγ∈CT (x,y) ALη (γ). Our goal here is to study the existence of c-time-free minimizers for L. For this, let us fix k ∈ R and consider the following quantity: φη,k (x, y) = inf min ALη +k (γ) ∈ R ∪ {−∞} . T >0 γ∈CT (x,y)

This quantity is commonly called Mañé potential (compare this with (2.5) in section 2.1). First of all we would like to understand when it is finite and what its properties are. Let us introduce what is called the Mañé critical value. Definition 4.1.7 (Mañé critical value). c(Lη )

:= =

sup{k ∈ R : ∃ a closed curve γ s.t. ALη +k (γ) < 0} inf{k ∈ R : ∀ closed curves γ s.t. ALη +k (γ) ≥ 0}.

52

CHAPTER 4

Remark 4.1.8. (i ) It is easy to check that c(Lη ) < ∞. In fact, since L is superlinear, there exists a sufficiently large k such that L + k ≥ 0 everywhere. (ii ) Moreover c(Lη ) only depends on c = [η] and not on the chosen representative. In fact, it is sufficient to notice that the integral of exact 1-forms along closed curves is zero. We will see in the following that this “critical value” is something that we have already encountered before: c(Lη ) = α(c), where α is Mather’s α-function associated with L. In chapter 5 we will also point out its relation to viscosity solutions and subsolutions of the Hamilton-Jacobi equation and to the critical value introduced by Lions, Papanicolaou, and Varadhan in [50]. Proposition 4.1.9 (See also [26, 27]). (1) (2) (3) (4) (5)

∀k∈R: If k < c(Lη ) : If k ≥ c(Lη ) : If k ≥ c(Lη ) : If k ≥ c(Lη ) : If k ≥ c(Lη ) : If k > c(Lη ) :

φη,k (x, y) ≤ φη,k (x, z) + φη,k (z, y) φη,k (x, y) ≡ −∞ φη,k (x, y) ∈ R φη,k : M × M −→ R is Lipschitz. φη,k (x, x) ≡ 0 φη,k (x, y) + φη,k (y, x) ≥ 0 φη,k (x, y) + φη,k (y, x) > 0

∀ x, y, z ∈ M. ∀ x, y ∈ M. ∀ x, y ∈ M. ∀ x ∈ M. ∀ x, y ∈ M. ∀ x 6= y ∈ M.

Proof. (1) First of all observe that this inequality makes sense also if φη,k (x, y) = −∞ for some x, y ∈ M . Let γ1 ∈ CT (x, z) and γ2 ∈ CT 0 (z, y) and consider the new curve obtained by joining them: γ1 ∗ γ2 ∈ CT +T 0 (x, y). Since the action is linear, it follows that ALη (γ1 ∗ γ2 ) = ALη (γ1 ) + ALη (γ2 ), and therefore φη,k (x, y) ≤ ALη (γ1 ) + ALη (γ2 ). It is now sufficient to take the infimum over all possible (γ1 , T ) and (γ2 , T 0 ) to conclude that φη,k (x, y) ≤ φη,k (x, z) + φη,k (z, y). (2) We will first prove that if k < c(Lη ), there exists x0 ∈ M such that φη,k (x0 , x0 ) = −∞. In fact, from the definition of c(Lη ), we know that there exists a γ : [0, T ] −→ M closed curve with ALη (γ) < 0. Let us denote by γ n the n-time iteration of γ, i.e., γ n := γ ∗ . . . ∗ γ (n times). Then, n→∞

φη,k (γ(0), γ(0)) ≤ ALη (γ n ) = nALη (γ) −→ −∞ . Choose x0 := γ(0) (or any other point on γ). The first claim will now follow from (1). In fact, if x, y ∈ M , then: φη,k (x, y) ≤ φη,k (x, x0 ) + φη,k (x0 , x0 ) + φη,k (x0 , y) = −∞ . As for the second claim, if k ≥ c(Lη ) it follows from the definition of c(Lη ) that all closed curves have positive action and therefore φη,k (x, x) ≥ 0 for all x ∈ M .

53

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

But, if there existed x, y ∈ M such that φη,k (x, y) = −∞, then applying (1), we would get a contradiction: φη,k (x, x) ≤ φη,k (x, y) + φη,k (y, x) = −∞ . Therefore, φη,k (x, y) > −∞ for all x, y ∈ M if k ≥ c(L). (3) Let k ≥ c(Lη ) and let Q := maxx∈M,kvk=1 L(x, v). For any x, y ∈ M let us consider the unit speed geodesic connecting x to y, γx,y : [0, d(x, y)] −→ M . Then: φη,k (x, y) ≤ ALη (γx,y ) ≤ (Q + k)d(x, y) . Using (1) we can conclude: φη,k (x2 , y2 ) − φη,k (x1 , y1 )

= φη,k (x2 , x1 ) + φη,k (x1 , y1 ) + φη,k (y1 , y2 ) −φη,k (x1 , y1 ) ≤ (Q + k)[d(x1 , x2 ) + d(y1 , y2 )].

(4) This follows immediately from (3). (5) The first part is a consequence of (1) and (4). Let us prove the second part. Assume that k > c(Lη ) and suppose by contradiction that there exist x 6= y ∈ M such that φη,k (x, y) + φη,k (y, x) = 0. Consider γn ∈ CTn (x, y) and σn ∈ CSn (y, x) such that: lim ALη (γn ) = φη,k (x, y)

and

n→+∞

lim ALη (σn ) = φη,k (y, x).

n→+∞

Let us prove that T := lim inf n→+∞ Tn > 0 (it might be +∞). Suppose by contradiction that lim inf n→+∞ Tn = 0 and select a subsequence {γnk } such that Tnk ) → 0. Using the superlinearity of L, we know that for each A > 0 there exists a B(A) such that Lη ≥ Akvk − B(A); then: Z φη,k (x, y)

= ≥

Tnk

lim

nk →∞

Lη (γnk , γ˙ nk )dt + kTnk

0

" Z lim A

nk →∞

Tnk

# kγ˙ nk kdt + Tnk (k − B(A))

0

= A d(x, y). From the arbitrariness of A it follows that φη,k (x, y) = +∞, which is a contradiction. Therefore, lim inf n→+∞ Tn > 0. Analogously one can prove that S := lim inf n→+∞ Sn > 0 (it might be +∞). Choose now subsequences {γnk } and {σmk } such that Tnk → T > 0 and Smk → S > 0. Using the facts that k > c(Lη ) and x 6= y ∈ M are such that φη,k (x, y) + φη,k (y, x) = 0, we obtain a contradiction to (4) (or to the definition

54

CHAPTER 4

of c(Lη )): φη,c(Lη ) (x, x) ≤ ≤

lim ALη +c(Lη ) (γnk ∗ σmk )

k→+∞

lim ALη +k (γnk ∗ σmk ) + lim (c(Lη ) − k)(Tnk + Snk ) k→+∞

k→+∞

= φη,k (x, y) + φη,k (y, x) + lim (c(Lη ) − k)(Tnk + Snk ) k→+∞

=

(c(Lη ) − k)(T + S) < 0

(or − ∞). 

Remark 4.1.10. It follows from (2) that c(Lη ) can be equivalently defined as: c(Lη )

:= =

inf{k ∈ R : ∃ x, y ∈ M s.t. φη,k (x, y) > −∞} sup{k ∈ R : ∃ x, y ∈ M s.t. φη,k (x, y) = −∞}.

In terms of the Mañé potential, the condition of being a c-time-free minimizer for L + k can be rewritten as: Z b ∀a c(Lη ). For all x, y ∈ M with x 6= y, there exists T > 0 and γ ∈ CT (x, y) such that ALη +k (γ) = φη,k (x, y). Proof. Let us define for T > 0 the function f (T ) := minγ∈CT (x,y) ALη +k (γ). This function is clearly continuous (for all k ≥ c(Lη )), and the following properties hold. • f (T ) → +∞ as t → 0+ (this is true for all k ≥ c(Lη )). In fact, let γT be the corresponding Tonelli minimizer connecting x to y in time T . Using the superlinearity of L, for each A > 0 there exists B = B(A) such that L(x, v) ≥ Akvk − B for all (x, v). Then: f (T )

=

min γ∈CT (x,y)

Z ≥ A

ALη +k (γ) = ALη +k (γT )

T

kγ˙ T k dt + (k − B)T 0 T →0+

≥ A d(x, y) + (k − B)T −→ A d(x, y) . T →0+

Since A > 0 is arbitrary and x 6= y, we can conclude that f (T ) −→ +∞.

55

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

• f (T ) → +∞ as t → +∞ (this is true only for k > c(Lη )). In fact: f (T )

=

min γ∈CT (x,y)

=

min γ∈CT (x,y)

ALη +k (γ) ALη +c(Lη ) (γ) + (k − c(Lη ))T

≥ φη,c(Lη ) (x, y) + (k − c(Lη ))T

T →+∞

−→ +∞. 

Remark 4.1.12. Hence if k > c(Lη ), c-time-free minimizers are not so special, since there are time-free minimizers for L + k connecting any two given points, furthermore in finite time. In light of this (and for other a fortiori reasons), one should probably be more interested in studying the “critical” case k = c(Lη ), i.e., c-time-free minimizers for L+c(Lη ), that is, for the least possible value of k for which they can exist. Definition 4.1.13 (c semi-static curves). We say that γ : R −→ M is a c semi-static curve for L if: Z b Lη (γ(t), γ(t)) ˙ dt + c(Lη )(b − a) = φη,c(Lη ) (γ(a), γ(b)) ∀ a < b. a

Remark 4.1.14. (i ) If γ is c semi-static for L, then it is a c-time-free minimizer for L + c(Lη ) and consequently a c-global minimizer for L. Therefore, it corresponds to a solution of the Euler-Lagrange flow of L. (ii ) We will prove in the following that for autonomous Tonelli Lagrangians, the converse is true: each c-global minimizer of L is indeed a c semi-static curve of L + c(Lη ) = L + α(c), where α denotes Mather’s α-function associated with L (see (3.2) in section 3.1). (iii ) In section 2.1 we proved that orbits on a KAM torus of cohomology class c are c-time-free minimizers for L + Ec , where Ec denotes the energy of the torus. It follows from Proposition 2.1.10 and the definition of the Mañé critical value (see also Remark 4.1.10) that in this case Ec = c(Lc ). Therefore, we can conclude that orbits on KAM tori are c semi-static. In particular, we can restate Remark 2.1.9 (iii ) by saying that if we have a KAM torus T of cohomology class c, then [ L−1 (T ) = {(γ(t), γ(t)) ˙ : γ is c semi-static for L and t ∈ R}, where L denotes the Legendre transform associated with L. Inspired by the last remark, we can define the following set. Definition 4.1.15 (Mañé set). The Mañé set (with cohomology class c) is: [ ec = ˙ : γ is a c semi-static curve and t ∈ R} N {(γ(t), γ(t)) [ = {(γ(t), γ(t)) ˙ : γ is a c-global minimizer and t ∈ R} . (4.1)

56

CHAPTER 4

Remark 4.1.16. The second equality in definition follows from Remark 4.1.14 and will be proven later. Observe that so far we have proven neither that such semi-static curves exist nor that this set is nonempty. We will prove these later, deducing them—among other properties—from analogous results for another family of sets that we are about to define: the Aubry sets. However, if such a set is nonempty, it is clearly invariant (it is the union of orbits) and also closed (the proof follows the same line as that of the Tonelli theorem). Let us start by recalling what happened in the case of orbits on a KAM torus. We saw in section 2.1 not only that these orbits were c-global minimizers (or c semi-static), but also that they satisfied a stronger property, stated in Proposition 2.1.11. Roughly speaking, the action of Lc + Ec on a piece of curve between two endpoints x and y was not only the minimal needed to connect x to y, but it was also equal to minus the minimal action needed to connect y back to x. We called a curve satisfying such a condition a regular minimizer. Recall, in fact, that, as it follows easily from Proposition 4.1.9 (5), for each x, y ∈ M we have φη,c(Lη ) (x, y) ≥ −φη,c(Lη ) (y, x). Let us define such curves in the general case. Definition 4.1.17 (c static curves). We say that γ : R −→ M is a c static curve for L (or a “c-regular minimizer”) if: Z b Lη (γ(t), γ(t)) ˙ dt + c(Lη )(b − a) = −φη,c(Lη ) (γ(b), γ(a)) ∀ a < b. a

Remark 4.1.18. (i ) If γ is c static for L, then it is a c semi-static (and therefore, as already observed before, it corresponds to a solution of the EulerLagrange flow of L). This is just a consequence of the fact that −φη,c(Lη ) (γ(b), γ(a)) ≤ φη,c(Lη ) (γ(a), γ(b)), and therefore, if γ is c static, then: Z

b

Lη (γ(t), γ(t)) ˙ dt + c(Lη )(b − a) ≤ φη,c(Lη ) (γ(a), γ(b))

∀ a < b.

a

Since φη,c(Lη ) (γ(a), γ(b)) was defined as the minimum over all connecting curves, equality must hold. (ii ) Observe that the adjective regular in the alternative appelation (coined by John Mather) has no relation to the smoothness of the curve, since, like all solutions of the Euler-Lagrange flow, this curve will be as smooth as the Lagrangian. (iii ) In section 2.1 (Proposition 2.1.11 and Remark 2.1.12) we proved that orbits on a KAM torus of cohomology class c were c static curves. Recall in fact that, as remarked before, Ec , which in that case denoted the energy of the torus, coincides with c(Lc ) (use Proposition 2.1.10 and the definition of the Mañé critical value or Remark 4.1.10). In particular, we can restate Remark 2.1.12 (ii ) by saying that if we have a KAM torus T of cohomology class c, then [ ˙ : γ is c static for L and t ∈ R}, L−1 (T ) = {(γ(t), γ(t))

57

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

where L denotes the Legendre transform given by L. (iv ) The terms semi-static and static are probably inspired by the fact that in the case of mechanical Lagrangians (see section 1.1), 0 static curves correspond to “some” fixed points of the flow (namely the minima of the potential), while 0 semi-static curves may possibly include also hetero/homoclinic connections among these fixed points (see section 4.3 on the pendulum). Inspired by the last remark, we define the following set. Definition 4.1.19 (Aubry set). The Aubry set (with cohomology class c) is: [ ˙ : γ is a c static curve and t ∈ R} . (4.2) Aec = {(γ(t), γ(t))   The projection on the base manifold Ac = π Aec ⊆ M is called the projected Aubry set (with cohomology class c). Remark 4.1.20. Observe that so far we have not proven that this set is nonempty! We will do it now. However, if such a set is nonempty, it is clearly invariant (it is union of orbits) and also closed (the proof follows the same approach as that of the Tonelli theorem). Moreover, Aec is clearly contained in ec (Remark 4.1.18 (i )). N We summarize in this diagram what we are going to prove in the remainder of this section. Theorem 4.1.21. (1), (2), (3), (4), (5), and (6) in the following diagram are true:1 fc M O

(1)

fc )−1 π (π|M (5)

 Mc

AecO

⊆ π

⊆

(2)

⊆

ec N

(3)

⊆

(4) Eec := {E(x, v) = α(c) = c(Lη )}

⊆

ec )−1 (π|A (6)

 Ac

⊆

TM π

⊆

 M

Remark 4.1.22. (i ) It follows from (1), (2) and the existence of c-actionminimizing measures (Corollary 3.1.5) that the Aubry and Mañé sets are nonempty. Therefore, there exist semi-static and static curves. (ii ) Inclusion (2), as already observed, follows obviously from the fact that static curves are also semi-static (Remark 4.1.18 (i )). (iii ) The above inclusions (1) and (2) may not be strict (see Remark 4.3.3 in section 4.3). ec are closed, it follow from (3) that they are compact. (iv ) Since Aec and N 1 This (unintentional?) typographical “coincidence” honoring Ricardo Mañé was first pointed out by Albert Fathi.

58

CHAPTER 4

(v ) The proof of (3) provides a proof of Carneiro’s theorem stated in section 3.1 (Theorem 3.1.13). (vi ) Properties (5) and (6) are what are generally called Mather’s graph theorem(s). Namely, the Mather set and the Aubry set are contained in a Lipschitz graph over M . This is probably the most important property of these sets and it has many dynamical consequences. In some sense, this is why they can be thought of as generalizations of KAM tori (or Lagrangian graphs). (vii ) The graph property does not hold in general for the Mañé set (see section 4.3 on the pendulum). Let us start by proving that c semi-static curves have energy c(Lη ). As we remarked above, the original version of this theorem (with c(Lη ) replaced by α(c)) is due to Carneio (see Theorem 3.1.13). The proof presented here follows an idea of Ricardo Mañé [53] (see also [26, Theorem XI]). ec ⊆ Eec := {E(x, v) = c(Lη )}, i.e., c Proposition 4.1.23 (Property (3)). N semi-static curves have energy equal to c(Lη ). Proof. Let γ : R −→ M be a c semi-static curve, i.e., for each T > 0 we have ALη +c(Lη ) (γ|[0, T ]) = φη,c(Lη ) (γ(0), γ(T )). Let us fix T > 0 and λ > 0, and consider a time-reparameterization of γ, given by γλ : [0, T /λ] −→ M , t 7→ γ(λt). Observe that the endpoints are not changed: γλ (0) = γ(0) and γλ (T /λ) = γ(T ), but only the time length is. Let us consider the action of these curves as a function of λ > 0: Z T /λ A(λ) := ALη +c(Lη ) (γλ ) = Lη (γλ (t), γ˙ λ (t)) dt + c(Lη )T /λ 0

Z

T /λ

Lη (γ(λt), λγ(λt)) ˙ dt + c(Lη )T /λ.

= 0

Since γ is c semi-static (and therefore a c-time-free minimizer for Lη + c(Lη )), then A has a minimum at λ = 1. Therefore: 0 = A0 (1)

= −T [Lη (γ(T ), γ(T ˙ )) + c(Lη )T ]  Z T   ∂L ∂L + (γ(t), γ(t)) ˙ γ(t)t ˙ + (γ(t), γ(t)) ˙ γ(t) ˙ + t¨ γ (t) dt. ∂x ∂v 0
d Integrating by parts (observe that dt L = ∂L ˙ + ∂L ¨ t) and recalling the def∂x γ ∂v γ inition of energy E(x, v) = ∂L ∂v (x, v)v − L(x, v) and the fact that is preserved

59

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

along the orbit (E(γ(t), γ(t)) ˙ = E(γ(0), γ(0)) ˙ for all t ∈ R), we obtain: 0

= A0 (1) = . . . = T = −T [Lη (γ(T ), γ(T ˙ )) + c(Lη )] + Lη (γ(t), γ(t)) ˙ 0  Z T ∂L − −L(γ(t), γ(t)) ˙ + (γ(t), γ(t)) ˙ γ(t) ˙ dt ∂v 0 Z T E(γ(t), γ(t)) ˙ dt = −T c(Lη ) + 0

=

[E(γ(0), γ(0)) ˙ − c(Lη )] T.

Hence, E(γ(0), γ(0)) ˙ = c(Lη ).



Let us now prove that the Mañé critical value coincides with Mather’s αfunction. We will follow the proof given in [26, Theorem II]. Proposition 4.1.24 (Property (4)). c(Lη ) = α(c), where c = [η]. Proof. Suppose that µ is an invariant ergodic probability measure. If we fix a generic point (x, v) in the support of µ, it follows from the ergodic theorem that there exists a sequence of times Tn → +∞, such that ΦL Tn (x, v) → (x, v) as Tn → +∞ and Z Tn Z 1 Lη dµ = lim Lη (ΦL t (x, v)) dt. n→+∞ Tn 0 For the sake of simplifying the notation, let (xt , vt ) := ΦL t (x, v). Let B := max{|Lη (x, v)| : kvk ≤ 1} and for each n denote by σn : [0, d(x, xTn )] −→ M the geodesic joining x to xTn and by γn : [0, Tn ] −→ M the projection of the orbit (i.e., γn (t) = xt ). We have: lim

n→∞

1 AL +k (γn ∗ σn ) Tn η 1 1 ALη +k (γn ) + Tn Tn

= lim

n→∞



Z

!

d(x,xTn )

Lη (σn (t), σ˙ n (t)) + k dt 0

1 1 ≤ lim ALη +k (γn ) + (B + k)diam(M ) n→∞ Tn Tn = ALη (µ) + k .



Therefore, if k < −ALη (µ) then Φk (x, x) = −∞. Hence, k ≤ c(Lη ). It follows that: c(Lη ) ≥ sup{−ALη (µ), µ ∈ Merg (L)} ≥ − inf{ALη (µ), µ ∈ M(L)} = α(c) .

60

CHAPTER 4

Now we want to prove the reversed inequality. Let k < c(Lη ) and x, y ∈ M . Since Φk (x, y) = −∞, there exists a sequence of absolutely continuous curves γn : [0, Tn ] −→ M such that γn (0) = x, γn (Tn ) = y (i.e., γn ∈ CTn (x, y)) and lim ALη +k (γn ) = −∞.

n→∞

Moreover, since L is bounded from below, we have that Tn → +∞. Let now yn be a Tonelli minimizer in CTn (x, y). Then, consider the invariant probability measure evenly distributed along (γn , γ˙ n ). The family of these measures is precompact and we can extract a subsequence converging (in the weak∗ topology) to an invariant probability measure µ. In particular, ALη (µ) + k = lim

n→∞

1 AL +k (γn ). Tn η

But since Tn > 0 and limn→∞ ALη +k (γn ) = Φk (x, y) = −∞, then ALη (µ) + k ≤ 0. Therefore, for any k < c(Lη ) we can find an invariant probability measure µ such that k ≤ −ALη (µ). Therefore: c(Lη ) ≤ sup{−ALη (µ), µ ∈ Merg (L)} ≤ − inf{ALη (µ), µ ∈ M(L)} = α(c).  We will provide two other alternative proofs of the above proposition. The first one uses the fact that the Mather set is included in the Mañé set.2 The second one is shorter but it requires the use of holonomic measures (see section 3.6). See also [26, Theorem II] for another proof. ec is compact (it follows from (3)) and invariAlternative Proof I. Since N ant under the Euler-Lagrange flow ΦL t , then there exists an invariant ergodic probability measure µ supported in it (Kryloff and Bogoliouboff [49]; compare this also with Proposition 3.1.1). If we fix a generic point (x, v) in the support of µ, it follows from the ergodic theorem that there exists a sequence of times Tn → +∞ such that ΦL Tn (x, v) → (x, v) as Tn → +∞, and Z

1 n→+∞ Tn

Z

Lη dµ = lim 2 Observe

Tn

Lη (ΦL t (x, v)) dt.

0

that in our proof of this fact—Proposition 4.1.26—we use the fact that α(c) = c(Lη ). In order to be precise, then, one should provide an alternative proof of this inclusion that is independent of this equality; this is possible, but we are not going to present it in these notes.

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

61

Then, using the definition of α(c) and the fact that orbits in the support of this measure are semi-static, we obtain: Z Tn Z 1 Lη (ΦL −α(c) ≤ Lη dµ = lim t (x, v)) dt n→+∞ Tn 0 Z Tn 1 Lη (ΦL = lim t (x, v)) + c(Lη ) dt − c(Lη ) n→+∞ Tn 0 φη,c(Lη ) (x, π(ΦL Tn (x, v))) = lim − c(Lη ) = −c(Lη ), n→+∞ Tn where in the last equality we used the fact that φη,c(Lη ) is bounded on M × M (being Lipschitz on a compact manifold). Therefore α(c) ≥ c(Lη ). For the reversed inequality, we will use the following version of the ergodic theorem (see for instance [52, Lemma 2.1] for a proof). Lemma 4.1.25. Let (X, d) be a complete metric space and (X, B, ν) a probability space. Let f be an ergodic measure-preserving map and F : X −→ R a ν-integrable function. Then, for ν-almost every x ∈ X the following property holds: N −1 Z X F (f j (x)) − N F dν < ε. ∀ ε > 0 ∃ N > 0 : d(f N (x), x) < ε and j=0 Let us now see how to use this lemma for our purposes. Let µ be a c-actionminimizing ergodic measure, i.e., Z (Lη (x, v) + α(c)) dµ = 0. Applying the above lemma with F = Lη + α(c) and X = TM , we obtain that there exists a µ-full measure set A such that if (x, v) ∈ A, then there exists a sequence Tn → +∞ such that Z Tn
n→+∞ n→+∞ d((x, v), ΦL (x, v)) −→ 0 and L(ΦL t (x, v)) + α(c) dt −→ 0 . Tn 0

Then: L φη,α(c) (x, π(ΦL 1 (x, v))) + φη,α(c) (π(Φ1 (x, v)), x) L L = lim φη,α(c) (x, π(ΦL 1 (x, v))) + φη,α(c) (π(Φ1 (x, v)), π(ΦTn (x, v))) n→∞ Z Tn
≤ lim Lη (ΦL t (x, v)) + α(c) dt = 0. n→∞




0

It follows from the second property in Proposition 4.1.9 (5) that α(c) ≤ c(Lη ), and this concludes the proof. 

62

CHAPTER 4

Alternative Proof II. Let γ be any closed curve and let µγ be the probability measure evenly distributed on it (see section 3.6). Let k ≥ c(Lη ). It follows from the definition of c(Lη ) that ALη +k (µγ ) ≥ 0 and therefore ALη (µγ ) ≥ −k. It follows from the definition of holonomic measure then that, for any µ holonomic probability measure, we have ALη (µ) ≥ −k. Taking the infimum over all holonomic measures and using property iv) mentioned in section 3.6, we can conclude that −α(c) ≥ −k. Since this holds for all k ≥ c(Lη ), we obtain: α(c) ≤ c(Lη ). To prove the reversed inequality, observe that if k < c(Lη ), then there exists a closed curve γ, such that ALη +k (γ) < 0. Therefore, if µγ is the associated holonomic measure, we obtain: −α(c) ≤ ALη (γ) < −k . Since this holds for all k < c(Lη ), we conclude that α(c) ≥ c(Lη ).  The proof of Property (1) is essentially similar to the (first) proof of Proposition 4.1.24. We will prove this more general result, due to Ricardo Mañé [53] (see also [26, Theorem IV]). Proposition 4.1.26 (Property (1)). µ ∈ M(L) is c-action-minimizing if fc ⊆ Aec . and only if supp µ ⊆ Aec . In particular, M Proof. Since Aec is closed, it is sufficent to prove the results only for the ergodic measures. [⇐=] Let µ ∈ M(L) be ergodic and suppose that supp µ ⊆ Aec . Applying the ergodic theorem, we know that for a µ-generic point (x, v) in the support of µ, there exists a sequence of times Tn → +∞ such that ΦL Tn (x, v) → (x, v) as Tn → +∞, and Z

1 n→+∞ Tn

Z

Lη dµ = lim

Tn

Lη (ΦL t (x, v)) dt.

0

Then, using the fact that orbits in the support of this measure are semi-static and that α(c) = c(Lη ), we obtain: Z Z Lη + α(c) dµ = Lη + c(Lη ) dµ Z Tn 1 = lim Lη (ΦL t (x, v)) + c(Lη ) dt n→+∞ Tn 0 −φη,c(Lη ) (π(ΦL Tn (x, v)), x) = lim = 0, n→+∞ Tn where in the last equality we used the fact that φη,c(Lη ) Ris bounded on M × M (being Lipschitz on a compact manifold).R Therefore, Lη dµ ≤ −α(c), and from the definition of α(c) it follows that Lη dµ = −α(c), i.e., µ is c-actionminimizing.

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

63

[=⇒] We will use the above-mentioned version of the ergodic theorem (see Lemma 4.1.25). Let µ be a c-action minimizing ergodic measure, i.e., Z (Lη (x, v) + α(c)) dµ = 0. Applying the above lemma with F = Lη + α(c) and X = TM , we obtain that there exists a µ-full measure set A such that if (x, v) ∈ A, then there exists a sequence Tn → +∞ such that Z Tn
n→+∞ n→+∞ L d((x, v), ΦTn (x, v)) −→ 0 and L(ΦL t (x, v)) + α(c) dt −→ 0 . 0

Then, let a > 0 and recall that α(c) = c(Lη ): L φη,c(Lη ) (x, π(ΦL a (x, v))) + φη,c(Lη ) (π(Φa (x, v)), x)


L L = lim φη,c(Lη ) (x, π(ΦL a (x, v))) + φη,c(Lη ) (π(Φa (x, v)), π(ΦTn (x, v))) n→∞ Z Tn
≤ lim Lη (ΦL t (x, v)) + c(Lη ) dt n→∞

0

Z ≤ lim

n→∞

Tn


Lη (ΦL t (x, v)) + α(c) dt = 0.

0

Recalling property (5) in Proposition 4.1.9, we can conclude that for any a > 0: L φη,c(Lη ) (x, π(ΦL a (x, v))) + φη,c(Lη ) (π(Φa (x, v)), x) = 0,

and therefore the orbit through the point (x, v) is c static. Since the points for which this reasoning can be applied are dense in the support of µ and Aec is closed, we prove the claim: supp µ ⊆ Aec .  Remark 4.1.27. Looking at the proof of Proposition 4.1.26, it is quite easy ec , then to see that we actually proved that: if µ ∈ M(L) is such that supp µ ⊆ N µ is c-action-minimizing (check that the same proof still works in this case). In fact, one can prove this stronger version: Proposition 4.1.28. µ ∈ M(L) is c-action-minimizing if and only if supp µ ⊆ ec . N We will deduce such a proposition from the fact (to be proven later in chapter 5, Proposition 5.3.2) that the “non-wandering points” of the Mañé’ set are contained in the Aubry set (see also [26, Theorem V.c]). It is also quite easy to check that orbits in the Mañé (or Aubry) set are asymptotic to the Mather set. Proposition 4.1.29. If γ : R −→ M is a c semi-static curve, then lim inf d(γ(t), Mc ) = 0. t→±∞

64

CHAPTER 4

Proof. Let T ≥ 1 and consider the probability measure µT evenly distributed along the piece of curve {(γ(t), γ(t)) ˙ : t ∈ [0, T ]} (for a definition, the reader may check the proof of Proposition 3.1.1). The Lagrangian actions of these measures are equi-bounded (we use here the fact that the orbit is semistatic): ALη (µT )

= ≤

Z φη,c(Lη ) (γ(0), γ(T )) 1 T Lη (γ(t), γ(t)) ˙ dt = − c(Lη ) T 0 T max φη,c(Lη ) (x, y) − c(Lη ) < ∞ .

M ×M

Therefore, this family of measures is pre-compact with respect to the weak∗ topology. Let us consider any converging subsequence µTk → µ, with Tk → +∞. Then, µ is invariant (see, again, the proof of Proposition 3.1.1) and: Z

Z Lη dµ =

lim

k→∞

φη,c(Lη ) (γ(0), γ(T )) − c(Lη ) k→∞ T

Lη dµk = lim

= −c(Lη ) = −α(c). Therefore, µ is c-action-minimizing.



Finally, we prove the most important result of this theory: the graph property of the Mather and Aubry sets, respectively, Properties (5) and (6) in the diagram (see also Theorem 3.1.12). Because of the inclusion proven in Proposition 4.1.26, it is sufficient to prove it for the Aubry set. Theorem 4.1.30 (Mather’s graph theorem, [62]). (Property (6)) π|Aec is an injective mapping of Aec into M , and its inverse (π|Aec )−1 : Ac −→ Aec is Lipschitz. The proof of this theorem will be based on the following “crossing” lemma, proven by Mather in [62] (to which we refer the reader for a complete proof). Lemma 4.1.31 (Mather’s crossing lemma, [62]). Let K > 0. There exist ε, δ, ϑ > 0, C > 0 such that if α, β : [−ε, ε] −→ M are solutions of the Euler˙ Lagrange equation with k(α(0), α(0))k ˙ ≤ K, k(β(0), β(0))k ≤ K and   ˙ d(α(0), β(0)) ≤ δ and d (α(0), α(0)), ˙ (β(0), β(0)) > Cd(α(0), β(0)), then there exist C 1 curves a, b : [−ε, ε] −→ M with endpoints a(−ε) = α(−ε), a(ε) = β(ε) and b(−ε) = β(−ε), b(ε) = α(ε) such that:  2 ˙ ALη (α) + ALη (β) − ALη (a) − ALη (b) ≥ ϑ d (α(0), α(0)), ˙ (β(0), β(0)) > 0.

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

α(−ε)

a

65

β(ε)

α β β(−ε)

b

α(ε)

Proof. (Theorem 4.1.30) We will first use Lemma 4.1.31 to prove the Lipschitz property, and the rest will be just a consequence of this property. Let K := maxAec k(x, v)k (this is finite since Aec is compact) and let ε, δ, ϑ, C be as in Lemma 4.1.31. Then, we will prove that: if (x1 , v1 ), (x2 , v2 ) ∈ Aec are such that d(x1 , x2 ) ≤ δ, then d ((x1 , v1 ), (x2 , v2 )) ≤ Cd(x1 , x2 ). Suppose by contradiction that d ((x1 , v1 ), (x2 , v2 )) > Cd(x1 , x2 ) and consider the flow lines through these points, namely α(t) := ΦL t (x1 , v1 ) and β(t) := ΦL t (x2 , v2 ). They satisfy the hypothesis of Lemma 4.1.31 (with our choice of K), and hence we can deduce the existence of two other curves a, b : [−ε, ε] −→ M with endpoints a(−ε) = α(−ε), a(ε) = β(ε) and b(−ε) = β(−ε), b(ε) = α(ε) such that: ALη (a) + ALη (b) < ALη (α) + ALη (β). But then: φη,c(Lη ) (a(−ε), a(ε)) + φη,c(Lη ) (b(−ε), b(ε)) ≤ ALη (a) + ALη (b) < ALη (α) + ALη (β) = −φη,c(Lη ) (α(ε), α(−ε)) − φη,c(Lη ) (β(ε), β(−ε)), where in the last equality we used that α and β are c semi-static. The above inequality and the triangle inequality for φη,c(Lη ) (see Proposition 4.1.9 (1)) lead to a contradiction of Proposition 4.1.9 (5): φη,c(Lη ) (α(−ε), β(ε)) = φη,c(Lη ) (a(−ε), a(ε))
< − φη,c(Lη ) (b(−ε), b(ε)) + φη,c(Lη ) (α(ε), α(−ε)) + φη,c(Lη ) (β(ε), β(−ε))
= − φη,c(Lη ) (β(−ε), α(ε)) + φη,c(Lη ) (α(ε), α(−ε)) + φη,c(Lη ) (β(ε), β(−ε))
≤ − φη,c(Lη ) (β(−ε), α(−ε)) + φη,c(Lη ) (β(ε), β(−ε)) ≤ −φη,c(Lη ) (β(ε), α(−ε)). Therefore, the inverse of the projection is locally Lipschitz, and this concludes the proof.  Remark 4.1.32. (i ) Actually, it follows from the proof (in the choice of K) that the graphs of the Aubry sets (or Mather sets) corresponding to compact sets of cohomology classes are equi-Lipschitz. (ii ) An alternative proof of the graph property will be presented in chapter 5, following Fathi’s weak KAM theory.

66

CHAPTER 4

One can show several other properties of these sets. For instance, we have remarked that the Mather sets, being the support of invariant probability measures, are recurrent under the flow. This is not true anymore for the Aubry and Mañé sets, but something can still be said. Let us first recall the definition of an ε-pseudo orbit. Given a (compact) metric space X and a flow ϕ on it, we say that there exists an ε-pseudo orbit between two points x, y ∈ X if we can find ε ⊂ X and positive times t1 , . . . , tkε > 0 such that x0 = x, xkε = y and {xn }kn=0
dist ϕti+1 (xi ) , xi+1 ≤ ε for all i = 0, . . . , kε . ec is chain transitive, i.e., for each ε > 0 Proposition 4.1.33. (i) ΦL N e and for all (x, v), (y, w) ∈ Nc , there exists an ε-pseudo-orbit for the flow ΦL connecting them. (ii) ΦL Aec is chain recurrent, i.e., for each ε > 0 and for all (x, v) ∈ Aec , there exists an ε-pseudo-orbit for the flow ΦL connecting (x, y) to itself. The proof of this result can be found for instance in [26, Theorem V]. As a consequence of the chain transitivity it follows that the Mañé set must be connected (the Aubry set in general is not). Corollary 4.1.34. The Mañé set is connected. 4.2

SOME TOPOLOGICAL AND SYMPLECTIC PROPERTIES OF THE AUBRY AND MAÑÉ SETS

In this section we want to discuss (without proofs) some topological and symplectic properties of the Aubry and Mañé sets, similarly to what we have already seen and proven for the Mather sets. In Proposition 3.3.6 we have related the intersection of Mather sets corresponding to different cohomology classes to the “flatness” of the α-function. The same result holds for the Aubry set and has been proven by Daniel Massart in [55, Proposition 6]. However, the proof in this case is less straightforward. Proposition 4.2.1 (Massart, [55]). Let c ∈ H1 (M ; R) and denote by Fc the maximal face of the epigraph of α containing c in its interior. (i) If a cohomology class c1 belongs Fc , then Ac ⊆ Ac1 . In particular, if c1 belongs to the interior of Fc , then they coincide, i.e., Ac = Ac1 . (ii) Conversely, if two cohomology classes c and c1 are such that Aec ∩ Aec1 6= ∅, then for each λ ∈ [0, 1] we have α(c) = α(λc + (1 − λ)c1 ), i.e., the epigraph of α has a face containing c and c1 . In particular, Massart proved that it is possible to relate the dimension of a “face” of the epigraph of the α-function to the topological complexity of the Aubry sets corresponding to cohomologies in that face (see [55, Theorem 1]). More precisely, for any sufficiently small ε > 0, let us define Cc (ε) to be the set of integer homology classes which are represented by a piecewise C 1 closed curve made with arcs contained in Ac except for a remainder of total length less

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

67

T than ε. Let Cc := ε>0 Cc (ε). Let Vc be the space spanned in H1 (M ; R) by Cc . Note that Vc is an integer subspace of H1 (M ; R), that is, it has a basis of integer elements (images in H1 (M ; R) of elements in H1 (M ; Z)). We denote by: - Fc the maximal face (flat piece) of the epigraph of α containing c in its interior; - Vect Fc the underlying vector space of the affine subspace generated by Fc in H 1 (M ; R); - Vc⊥ the vector space of cohomology classes of C 1 1- forms that vanish on Vc ; - Gc the vector space of cohomology classes of C 1 1-forms that vanish in Tx M for each x ∈ Ac ; - Ec the space of cohomology classes of 1-forms of class C 1 , the supports of which are disjoint from Ac . Theorem 4.2.2 (Massart, [55]). Ec ⊆ Vect Fc ⊆ Gc ⊆ Vc⊥ . Moreover, as we have already proven for the Mather sets (see section 3.4, Proposition 3.4.4), they are symplectic invariant. The same holds also for these other sets, but the proof in this case is definitely less trivial and requires a more subtle study of the action-minimizing orbits. It can be deduced, for instance, as a special case of [13, Theorem 1.10] (which also applies to the non-autonomous case). Let us denote by A∗c (H) and Nc∗ (H) the Aubry and Mañé sets associated with a Tonelli Hamiltonian H (in the sense of the Legendre transform of the corresponding ones for the associated Lagrangian). Then: Theorem 4.2.3 (Bernard, [13]). Let L : TM −→ R be a Tonelli Lagrangian and H : T∗ M −→ R the associated Hamiltonian. If Φ : T∗ M −→ T∗ M is an exact symplectomorphism, then A∗0 (H ◦ Φ) = Φ−1 (A∗0 (H))

and

N0∗ (H ◦ Φ) = Φ−1 (N0∗ (H)) .

This result can be easily extended to non-exact symplectomorphisms and to other cohomology classes using Lemma 3.4.1, under the assumption that Φ is isotopic to the identity (as we have already done in the case of Proposition 3.4.2; see also [69], where the case of simplectomorphisms non-isotopic to the identity is also discussed). Theorem 4.2.4. Let L : TM −→ R be a Tonelli Lagrangian and H : T∗ M −→ R the associated Hamiltonian. If Ψ : T∗ M −→ T∗ M is a symplectomorphism of class [Ψ] isotopic to the identity, then     ∗ A∗c (H ◦ Ψ) = Ψ−1 A∗c+[Ψ] (H) and Nc∗ (H ◦ Ψ) = Ψ−1 Nc+[Ψ] (H) . Remark 4.2.5. More geometric proofs of this result can be obtained using weak KAM theory, for instance [77, Lemma 1] (which does not apply to all symplectomorphism, but only to those that are Hamiltonian isotopic to the identity) and [18]. See also [72].

68 4.3

CHAPTER 4

AN EXAMPLE: THE SIMPLE PENDULUM (PART II)

In section 3.5 we discussed Mather sets, the α-function, and the β-function for the special case of a simple pendulum, described by the Lagrangian: L : TT −→ (x, v) 7−→

R
1 2 |v| + 1 − cos(2πx) . 2

In particular, we proved that: • for all − π4 ≤ c ≤ • if c >

4 π,

fc = {(0, 0)}; M

4 π,

p f±c = {(x, v) : v = ± 2[(1 + α(c)) − cos(2πx)], ∀ x ∈ T}. M We want to see which ones are the Mañé and Aubry sets in this case. Let us start by recalling the following fact, which we have proven, in a slightly different form, in section 2.1. Proposition 4.3.1. Let Λ be a c-invariant (Lipschitz) Lagrangian graph in T∗ M . Then, the projection on TM of each orbit on Λ is c semi-static. See for instance [40, 75]. The proof is essentially the same as that of Proposition 2.1.8 (see also Remarks 2.1.9 (ii ) and 4.1.14 (iii )). The proof extends to the case of Lipschitz c-Lagrangian graphs (i.e., Lipschitz sections that are locally the graphs of closed 1-forms of cohomology class c). Observe in fact that also in the Lipschitz case, the Hamiltonian stays constant on invariant Lagrangian graphs (see for instance [77]). Hence, it follows from Proposition 4.3.1 that for c > π4 , p e±c ⊇ {(x, v) : v = ± 2[(1 + α(±c)) − cos(2πx)], ∀ x ∈ T} = M f±c . N p In fact, it is easy to check that {(x, v) : v = ± 2[(1 + α(c)) − cos(2πx)], ∀ x ∈ T} is the graph of a closed 1-form of cohomology class ±c (the cohomology is just the signed area enclosed between this graph and the x-axis). See also the discussion in section 3.4. e±c must be equal to M f±c , since it is connected (Corollary 4.1.34) Moreover, N and it must be contained in the energy level corresponding to the value α(±c). Therefore, recalling the inclusions in Theorem 4.1.21, we can conclude that e±c = Ae±c = M f±c N

for all c >

4 . π

Let us see what happens for |c| ≤ π4 . Observe that they all correspond to the same energy level, namely the one of the separatrices (α(c) = 0 in this case). In this energy level there are exactly three orbits:

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

69

- The fixed point (0, 0), i.e., γ0 (t) ≡ 0. - The upper separatrix γ+ ; for instance, let us choose the parametrization given by
γ+ (t) = π ΦL t (1/2, 2) . - The lower separatrix γ − ; for instance, let us choose the parametrization given by
γ− (t) = π ΦL t (1/2, −2) . Observe that because of the symmetry of L and the chosen parametrizations, we have that γ+ (t) = γ− (−t) for all t ∈ R. First of all, let us show that for |c| < π4 , neither γ+ nor γ− can be c semistatic. In fact, let us consider a closed 1-form ηc whose graph is contained in the region between the separatrices. This is possible since c is less than 4/π and the cohomology represents the signed area of the region between the curve and the x-axis. Moreover, since |c| is strictly less that 4/π, there will be a positive measure subset of T on which H(x, ηc (x)) < 0. If γ+ were c semi-static (similarly for γ− ), then using that γ+ is asymptotic in the past and in the future to 0, that α(c) = 0, and that the Mañé potential φηc ,0 is Lipschitz continuous (Proposition 4.1.9 (3)), we obtain: φηc ,0 (0, 0)

= =

lim φηc ,0 (γ+ (−T ), γ+ (T )) Z T lim Lηc (γ+ (t), γ˙ + (t)) dt

T →+∞ T →+∞

−T T

Z ≥


ηc (γ+ (t)) · γ˙ + (t) − H(γ+ (t), ηc (γ+ (t))) dt

lim

T →+∞

−T

Z

T

= − lim

T →+∞

H(γ+ (t), ηc (γ+ (t))) dt > 0, −T

where the third inequality comes from the Legendre-Fenchel inequality and the last one from the fact that there exists a positive measure set of T in which H(γ+ (t), ηc (γ+ (t))) is strictly negative. But this leads to a contradiction, since the action of the constant path γ0 is zero: ALηc (γ0 ) = 0. We have just proven that (use also Theorem 4.1.21): ec = Aec = M fc = {(0, 0)} for all |c| < 4 . N π Finally, let us consider the case c± := ± π4 . As above, it follows from Proposition 4.3.1 (taking the graphs of the separatrices as invariant Lipschitz Lagrangian graphs) that: p ec = {(x, v) : v = ± 2[1 − cos(2πx)], ∀ x ∈ T} ⊃ M fc . N ± ±

70

CHAPTER 4

ec (but observe that this time We want to show that also in this case Aec± = N ± they contain the Mather set properly). Let us denote by η± the closed 1-forms given by the graphs of (respectively) the upper and lower separatrices. We have already pointed out that [η± ] = c± . The key observation is the following lemma. Lemma 4.3.2. For every x, y ∈ T we have φη+ ,0 (x, y) = −φη− ,0 (y, x). Observe that it follows immediately from this lemma that the upper (or lower) separatrix is not only c+ semi-static (or c− semi-static), but is also c+ ec = Aec . static (or c− static). Therefore, N ± ± Proof. The above equality is always true if x = y (see Proposition 4.1.9 (4)). Suppose without any loss of generality that x < y and let S < T such that γ+ (S) = x = γ− (−S) and γ+ (T ) = y = γ− (−T ) (this is true only for the parametrization that we chose above). Now, using the fact that these curves are semi-static (for their respective cohomologies), the symmetry of L, the relation between γ+ and γ− , and the fact that η+ = −η− , we obtain: Z φη+ ,0 (x, y)

T


L(γ+ (t), γ˙ + (t)) − η+ (γ+ (t)) · γ˙ + (t) dt

= S T

Z


L(γ− (−t), −γ˙ − (−t)) + η− (γ− (t)) · (−γ˙ − (−t)) dt

= S

Z = −

−S

−T


L(γ− (s), γ˙ − (s)) − η− (γ− (s)) · γ˙ − (s) dt

= −φη− ,0 (y, x).  Summarizing: • For c >

4 π:

p e±c = Ae±c = M f±c = ± 2[(1 + α(c)) − cos(2πx)], ∀ x ∈ T} . N • For |c|
0 and x, y ∈ M , let us consider : Z t hη,t (x, y) = min Lη (γ(s), γ(s)) ˙ ds , (4.3) 0

where the minimum is taken over all piecewise C 1 paths γ : [0, t] −→ M , such that γ(0) = x and γ(t) = y. This minimum is achieved because of the Tonelli theorem (Theorem 4.1.1). We define Peierls’ barrier as: hη (x, y) = lim inf (hη,t (x, y) + α(c)t) . t→+∞

(4.4)

Remark 4.4.1. (i ) Observe that hη depends not only on the cohomology class c, but also on the choice of the representative η; namely, if η 0 = η + df , then hη0 (x, y) = hη (x, y) + f (y) − f (x). Anyhow, this dependence will not be harmful for what we are going to do in the following (it will not affect the set of action-minimizing curves). (ii ) The main difference between Peierls’ barrier and the Mañé potential is that in this case we consider curves defined over longer and longer time intervals. In particular, it is easy to check that: hη (x, y) ≥ φη,α(c) (x, y) ∀ x, y ∈ M. (iii ) This function hη is a generalization of Peierls’ barrier introduced by Aubry [7] and Mather [58, 60, 63] in their study of twist maps. In some sense we are comparing, in the limit, the action of Tonelli minimizers of time length T with the corresponding average c-minimal action −α(c)T . Remember, in fact, that −α(c) is the “average action” of a c-minimal measure. (iv ) Albert Fathi [37] showed that—in the autonomous case—this lim inf can be replaced by a limit. This is not generally true in the non-autonomous timeperiodic case (see for instance [41] for some counterexamples). Tonelli Lagrangians for which this convergence result holds are called regular ; Patrick Bernard [10] showed that under suitable assumptions on the Mather set, it is possible to prove that the Lagrangian is regular. For instance, if the Mather set fc is the union of 1-periodic orbits, then Lη is regular. This problem turned out M to be strictly related to the convergence of the so-called Lax-Oleinik semigroup (see [37] for its definition). 3 The function that we are defining here is actually a variant of h∞ defined in [64]. Pay c attention that throughout article [64], the sign of the α-function is wrong: wherever there is α(c), it should be substituted by −α(c).

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

73

Analogously to Proposition 4.1.9, one can prove the following. Proposition 4.4.2. The values of the map hη are finite. Moreover, the following properties hold: i) for each x, y, z ∈ M and t > 0 hη (x, y) ≤ hη (x, z) + hη,t (z, y) + α(c)t; ii) for each x, y, z ∈ M , hη (x, y) ≤ hη (x, z) + hη (z, y). iii) for each x, y, z ∈ M , hη (x, y) ≤ hη (x, z) + φη,α(c) (z, y). iv) for each x ∈ M , hη (x, x) ≥ 0; v) for each x, y ∈ M , hη (x, y) + hη (y, x) ≥ 0. It is interesting to consider the following symmetrization: δc : M × M −→ (x, y) 7−→

R hη (x, y) + hη (y, x).

(4.5)

Observe that this function does now depend only on the cohomology class c, and that moreover it is nonnegative, is symmetric, and satisfies the triangle inequality. An interesting property of δc is the following one (see [64, Section 8]). If d denotes the distance induced on M by the Riemannian metric g, then there exists C > 0 such that for each x, y ∈ M we have δc (x, y) ≤ Cd(x, y)2 . Remark 4.4.3. The same estimate continues to be true for the non-autonomous time-periodic case. In this case we have that δc ((x, τ0 ), (y, τ1 )) ≤ C[d(x, y) + kτ1 − τ0 k]2 for each (x, τ0 ), (y, τ1 ) ∈ Ac , where kτ1 − τ0 k = inf {|t1 − t0 | : ti ∈ R, ti ≡ τi (mod. 1), i = 0, 1} . Let us see now some relation between this Peierls’ barrier (or equivalently δc ) and c-action-minimizing curves. Let γ : R −→ M be a c-minimizer and consider xα , x0α in the α-limit set4 of γ, and xω , x0ω in the ω-limit set5 of γ. John Mather in [64, Section 6] proved that δc (xα , x0α ) = δc (xω , x0ω ) = 0. In general, it is not true that δc (xα , xω )=0; what one can prove is that this value does not depend 4 Recall that a point z is in the α-limit set of γ if there exists a sequence t → −∞ such n that γ(tn ) → z. 5 Recall that a point z is in the ω-limit set of γ if there exists a sequence t → +∞ such n that γ(tn ) → z.

74

CHAPTER 4

on the particular xα and xω , i.e., δc (xα , xω ) = δc (x0α , x0ω ): it is a property of the limit sets rather than of their elements. Nevertheless, there will exist particular c-minimizers for which this value is equal to 0 and these will be the c-minimizers that we want to single out. Definition 4.4.4 (c-regular minimizers). A c-minimizer γ : R −→ M is called a c-regular minimizer if δc (xα , xω ) = 0 for each xα in the α-limit set of γ and for each xω in the ω-limit set of γ. Mather defined the Aubry set as the union of the support of all these c-regular minimizers. Definition 4.4.5 (Aubry set). The Aubry set (with cohomology class c) is: [ ˙ : γ is a c-regular minimizer and t ∈ R} . {(γ(t), γ(t)) Aec = It turns out that this set coincides exactly with the one that we have defined in section 4.1. In fact, one can prove that: Proposition 4.4.6. γ is a c-regular minimizer of L if and only if γ is a c static curve of L. A proof of this can be found in [37, Proposition 9.2.5]. The essential ingredient is that φη,α(c) (x, y) = hη (x, y) if x, y ∈ Ac (see also Remark 5.1.28). Moreover, one can also provide another alternative definition of the (projected) Aubry set: Proposition 4.4.7 (See [37, Proposition 5.3.8]). The following properties are equivalent. i) x ∈ Ac ; ii) hη (x, x) = 0; iii) there exists a sequence of absolutely continuous curves γn : [0, tn ] → M such that: - for each n, we have γn (0) = γn (tn ) = x; - the sequence tn → +∞, as n → +∞; Rt - as n → +∞, 0 n Lη (γn (s), γ˙ n (s)) ds + α(c)tn → 0. Remark 4.4.8. (i ) Therefore, the Aubry set consists of points that are contained in loops with periods as long as we wish and with actions as close as we want to the minimal average one. (ii ) Moreover, it follows from ii) in Proposition 4.4.7 that δc is a pseudometric on the projected Aubry set Ac = {x ∈ M : δc (x, x) = 0}.

ACTION-MINIMIZING CURVES FOR TONELLI LAGRANGIANS

75

(iii ) One can easily construct a metric space out of (Ac , δc ). We call quotient Aubry set (or Mather quotient; see [39]) the metric space (A¯c , δ¯c ) obtained by identifying two points in Ac if their δc -pseudodistance is zero. This set plays quite an interesting role in the study of the dynamics; see, for example, [66, 67, 76, 39] for more details.

Chapter Five The Hamilton-Jacobi Equation and Weak KAM Theory Another interesting approach to the study of action-minimizing invariant sets is provided by the so-called weak KAM theory, which represents the functional analytic counterpart to the variational methods discussed in the previous sections. In section 2.1, in fact, we pointed out the relation between KAM tori (or more generally, invariant Lagrangian graphs) and classical solutions and subsolutions of the Hamilton-Jacobi equation (see Remark 2.1.2 (iv )). The approach that we are going to describe will be based on the study of “weak” (non-classical) solutions of the Hamilton-Jacobi equation and some special class of subsolutions (critical subsolutions). From a more geometrical point of view, this can be interpreted as the study of particular Lagrangian graphs (not necessarily invariant) and their non-removable intersection (see also [72]). This point of view makes this approach particularly interesting, since it relates the dynamics of the system to the geometry of the space and might potentially open the way to a “symplectic” definition of the Aubry-Mather theory (see also sections 3.4, 4.2, appendix A, and [13, 77, 18]). In this chapter we want to provide a brief presentation of this theory, omitting most of the proofs, for which we refer the reader to the excellent—and selfcontained—presentation [37]. 5.1

WEAK SOLUTIONS AND SUBSOLUTIONS OF HAMILTON-JACOBI AND FATHI’S WEAK KAM THEORY

The main object of investigation is represented by the Hamilton-Jacobi (H-J) equation: Hη (x, dx u) = H(x, η(x) + dx u) = k , where η is a closed 1-form on M with a certain cohomology class c. Observe that considering H-J equations for different 1-forms corresponding to different cohomology classes is equivalent to Mather’s idea of modifying the Lagrangian (see section 3.1). From now on, we will consider L to be a Tonelli Lagrangian on a compact manifold M , and H its associated Hamiltonian. Let us fix η to be a closed 1-form on M with cohomology class c, and, as before, denote by Lη and Hη the modified Lagrangian and Hamiltonian. In classical mechanics, one is interested in studying solutions of this equation, i.e., C 1 functions u : M → R such that Hη (x, dx u) = k. It is easy to check that for any given cohomology class there

THE HAMILTON-JACOBI EQUATION AND WEAK KAM THEORY

77

exists at most one value of k for which these C 1 solutions may exist. In fact, it is enough to observe that if u and v are two C 1 functions on a compact manifold, there will exist a point x0 at which their differentials coincide (take any critical point of u − v). We will see (Theorem 5.1.16) that this value of k, for which solutions may exist, coincides with α(c) or the Mañé critical value c(Lη ) (defined in chapters 3 and 4). Remark 5.1.1. The existence of such solutions has significant implications for the dynamics of the system and it is, consequently, quite rare. In particular, the solutions correspond to Lagrangian graphs, which are invariant under the Hamiltonian flow ΦH t (Hamilton-Jacobi theorem). For instance, in the case of M = Td and nearly-integrable systems, these solutions correspond to KAM tori (this might give an idea of their rareness in systems that are not close to an integrable one). One of the main results of the weak KAM theory is that, in the case of Tonelli Hamiltonians, solutions of a weaker kind always exist. In the following we are going to define these generalized solutions and their relation with the dynamics of the system. It is important to point out that one of the main ingredient in the proof of all these results is provided by the Fenchel inequality (cf. (1.3) in section 1.2). Let us start by generalizing the concept of subsolution. In the C 1 case it is easy to check—using Fenchel inequality—that the following property holds (the proof is essentially the same as for (2.4) in section 2.1). Proposition 5.1.2. Let u : M → R be C 1 ; u satisfies H(x, η(x) + dx u) ≤ k for all x ∈ M if and only if, for all a < b and γ : [a, b] → M , b

Z u(γ(b)) − u(γ(a)) ≤

Lη (γ(t), γ(t)) ˙ dt + k(b − a). a

This last inequality provides the grounds for a definition of subsolutions in the C 0 case. Definition 5.1.3 (Dominated functions). Let u : M → R be a continuous function; u is dominated by Lη + k, and we will write u ≺ Lη + k if, for all a < b and γ : [a, b] → M , Z u(γ(b)) − u(γ(a)) ≤

b

Lη (γ(t), γ(t)) ˙ dt + k(b − a).

(5.1)

a

One can check that if u ≺ Lη + k, then u is Lipschitz and its Lipschitz constant can be bounded by a constant C(k) independent of u; in fact, it is sufficient to apply the definition of dominated function with the speed-one geodesic connecting any two points x and y and consider the maximum of L over the unit tangent ball (see [37, Proposition 4.2.1 (iii)]). In particular, all dominated functions for values of k in a compact set are equi-Lipschitz. On the other hand,

78

CHAPTER 5

it is easy to check that each Lipschitz function is dominated by Lη + k for a suitable k depending on its Lipschitz constant: this shows that dominated functions exist. Dominated functions generalize subsolutions of H-J to the continuous case. In fact: Proposition 5.1.4 (see [37, Theorem 4.25]). If u ≺ Lη + k and dx u exists, then H(x, η(x)+dx u) ≤ k. Moreover, if u : M → R is Lipschitz and H(x, η(x)+ dx u) ≤ k almost everywhere., then u ≺ Lη + k. Remark 5.1.5. Using the fact that any Lipschitz function is differentiable almost everywhere (Rademacher theorem), one could equivalently define subsolutions in the following way: a locally Lipschitz function u : M −→ R is a subsolution of Hη (x, dx u) = k, with k ∈ R, if Hη (x, dx u) ≤ k for almost every x ∈ M. Remark 5.1.6. One interesting question is: for which values of k do there exist functions dominated by Lη + k (or equivalently subsolutions of H(x, η(x) + dx u) = k)? It is possible to show that there exists a value kc ∈ R such that H(x, η + dx u) = k does not admit any subsolution for k < kc , while it has subsolutions for k ≥ kc (see [50, 37]). In particular, if k > kc , there exist C ∞ subsolutions. It turns out that the constant kc coincides with α(c) and the Mañé’s critical value (where c = [η]). See [28, 37]. Functions corresponding to this “critical domination” play an important role, since they encode significant information about the dynamics of the system. Definition 5.1.7 (Critical subsolutions). A function u ≺ Lη + α(c) is said to be critically dominated. Equivalently, we will also call it an η-critical subsolution, since H(x, η(x) + dx u) ≤ α(c) for almost every x ∈ M . Remark 5.1.8. The above observation provides another definition of α(c): α(c) =

inf

max H(x, η(x) + dx u) .

u∈C ∞ (M ) x∈M

This theorem has been proven in [28, Theorem A] (see also [37]). This infimum is not a minimum, but it becomes a minimum over the set of Lipschitz functions on M (and also over the smaller set of C 1,1 functions; see the section 5.2 and [42, 11]). This characterization has the following geometric interpretation. If we consider the space T∗ M equipped with the canonical symplectic form, the graph of the differential of a C 1 η-critical subsolution (plus the 1-form η) is nothing else than a c-Lagrangian graph (i.e., a Lagrangian graph with cohomology class c). Therefore, the Mañé c-critical energy level Ec∗ = {(x, p) ∈ T∗ M : H(x, p) = α(c)} corresponds to a (2d − 1)-dimensional hypersurface such that the region it bounds is convex in each fiber and does not contain in its interior any cLagrangian graph, while all of its neighborhoods do. Analogously to what we have already seen for subsolutions, it would be interesting to investigate if there exists an equivalent characterization of classical

THE HAMILTON-JACOBI EQUATION AND WEAK KAM THEORY

79

solutions of Hamilton-Jacobi that does not involve the regularity of the solution. This would allow us to define weak solutions, which hopefully are not as rare as the classical ones. Let us now recall some properties of classical solutions, which will allow us to provide a weaker definition of solution (the proof of this proposition is essentially the same as that for (2.4) in section 2.1). Proposition 5.1.9 (see [37, Theorem 4.1.10]). Let u : M → R be a C 1 function and k ∈ R. The following conditions are equivalent: 1. u is solution of H(x, η(x) + dx u) = k; 2. u ≺ Lη + k and for each x ∈ M there exists γx : (−∞, +∞) → M such that γx (0) = x and for any [a, b]: Z b u(γx (b)) − u(γx (a)) = Lη (γx (t), γ˙ x (t)) dt + k(b − a). a

3. u ≺ Lη + k and for each x ∈ M there exists γx : (−∞, 0] → M such that γx (0) = x and for any a < b ≤ 0: Z b Lη (γx (t), γ˙ x (t)) dt + k(b − a). u(γx (b)) − u(γx (a)) = a

4. u ≺ Lη + k and for each x ∈ M there exists γx : [0, +∞) → M such that γx (0) = x and for any 0 ≥ a < b: Z b u(γx (b)) − u(γx (a)) = Lη (γx (t), γ˙ x (t)) dt + k(b − a). a

Inspired by this fact, let us consider the curves for which equality in (5.1) holds. Definition 5.1.10 (Calibrated curves). Let u ≺ Lη + k. A curve γ : I → M is (u, Lη , k)-calibrated if, for any [a, b] ⊆ I, Z b u(γ(b)) − u(γ(a)) = Lη (γ(t), γ(t)) ˙ dt + k(b − a). a

These curves are very special curves, and it turns out that they are orbits of the Euler-Lagrange flow. In fact: Proposition 5.1.11. If u ≺ Lη +k and γ : [a, b] → M is (u, Lη , k)-calibrated, then γ is a c-Tonelli minimizer, i.e., Z b Z b Lη (γ(t), γ(t)) ˙ dt ≤ Lη (σ(t), σ(t)) ˙ dt a

a

for any σ : [a, b] → M such that σ(a) = γ(a) and σ(b) = γ(b). Most of all, this implies that γ is a solution of the Euler-Lagrange flow and therefore it is C r (if L is C r ).

80

CHAPTER 5

The proof of this result is the same as the one of Proposition 2.1.8 (see also [37, Proposition 4.3.2 and Corollary 4.3.3]). Moreover, the following differentiability result holds. Proposition 5.1.12 (see [37, Theorem 4.3.8]). Let u ≺ Lη + k and γ : [a, b] → M be (u, Lη , k)-calibrated. i) If dγ(t) u exists for some t ∈ [a, b], then H(γ(t), η(γ(t)) + dγ(t) u) = k and ∂L (γ(t), γ(t)). ˙ dγ(t) u = ∂v ii) If t ∈ (a, b), then dγ(t) u exists. Remark 5.1.13. Calibrated curves are “Lagrangian gradient lines” of gradL u (where gradL u is a multivalued vector field given by the equation dx u =

∂L (x, gradL u) . ∂v

Therefore, there is only one possibility for calibrated curves at each point of differentiability of u. This suggests the following definitions. Definition 5.1.14 (weak KAM solutions). Let u ≺ Lη + k. • u is a weak KAM solution of negative type (or a backward weak KAM solution) if for each x ∈ M there exists γx : (−∞, 0] → M such that γx (0) = x and γx is (u, Lη , k)-calibrated; • u is a weak KAM solution of positive type (or a forward weak KAM solution) if for each x ∈ M there exists γx : [0, +∞) → M such that γx (0) = x and γx is (u, Lη , k)-calibrated. Remark 5.1.15. Observe that any weak KAM solution of negative type u− (or of positive type u+ ) for a given Lagrangian L can be seen as a weak KAM solution of positive type (or of negative type) for the anti-symmetrical ˜ v) := L(x, −v). Lagrangian L(x, Let us denote with Sη− the set of weak KAM solutions of negative type and with Sη+ the ones of positive type. Albert Fathi [34, 37] proved that these sets are always nonempty. Theorem 5.1.16 (Weak KAM theorem). There is only one value of k for which weak KAM solutions of positive or negative type of H(x, η(x) + dx u) = k exist. This value coincides with α(c), where α : H1 (M ; R) → R is Mather’s αfunction. In particular, for any u ≺ Lη + α(c) there exist a weak KAM solution of negative type u− and a weak KAM solution of positive type u+ , such that u− = u = u+ on the projected Aubry set Ac .

THE HAMILTON-JACOBI EQUATION AND WEAK KAM THEORY

81

Therefore, for any given weak KAM solution of negative type u− (or of positive type u+ ) there exists a weak KAM solution of positive type of positive type u+ (or of negative type u− ) such that u− = u+ on the projected Aubry set Ac . In particular: Proposition 5.1.17 (see [37, Theorem 4.12.6]). The projected Mather set Mc is the uniqueness set for weak KAM solutions of the same type. Namely, if u− , v− are weak KAM solutions of negative type (or u+ , v+ are weak KAM solutions of positive type) and u− = v− on Mc (or u+ = v+ on Mc ), then they coincide everywhere on M . Two solutions u− and u+ that coincide on the (projected) Mather set are said to be conjugate. We will denote by (u− , u+ ) a couple of conjugate subsolutions. Let us try to understand the dynamical meaning of such solutions. Albert Fathi [37] —using these generalized solutions—proved a weak version of the Hamilton-Jacobi theorem, showing the relation between these weak solutions and the dynamics of the associated Hamiltonian system. We will state it for a weak KAM solution of negative type, but—using Remark 5.1.15—one can deduce an analogous statement for weak KAM solutions of positive type. Theorem 5.1.18 (Weak Hamilton-Jacobi theorem [37, Theorem 4.13.2]). Let u− : M → R be a weak KAM solution of negative type and consider Graph(η + du− ) := {(x, η(x) + dx u− ), where dx u− exists}. Then: i) Graph(η + du− ) is compact and is contained in the energy level Ec∗ = {(x, p) ∈ T∗ M : H(x, p) = α(c)};   ii) ΦH −t Graph(η + du− ) ⊆ Graph(η + du− ) for each t > 0;   iii) M = π Graph(η + du− ) , where π : T∗ M → M is the canonical projection. Moreover, let us define: I∗ (u− ) :=

\ t≥0

  Graph(η + du ) . ΦH − −t

I∗ (u− ) is nonempty, compact, and invariant under ΦH t . Furthermore, its “unstable set” contains Graph(η + du− ), i.e., n
t→+∞ o ∗ Graph(η + du− ) ⊆ W u (I∗ (u− )) := (y, p) : dist ΦH −→ 0 . −t (y, p), I (u− ) There is a relation between these invariant sets I∗ (u− ) (or I∗ (u+ )) and the Aubry set Aec (recall that I∗ (u− ), I∗ (u+ ) ⊂ T∗ M , while Aec ⊂ TM ).

82

CHAPTER 5

Theorem 5.1.19.   A∗c := L Aec

=

\ u− ∈Sη−

=

\

I∗ (u− ) =

\

u− ∈Sη−

\

I∗ (u+ ) =

u+ ∈Sη+

=

\

Graph(η + du− ) Graph(η + du+ )

u+ ∈Sη+

{x ∈ M : u− (x) = u+ (x)},

(5.2)

(u− ,u+ )

where L : TM → T∗ M denotes the Legendre transform of L and (u− , u+ ) are conjugate solutions. Since it is easier to work with subsolutions rather than weak solutions, we want to discuss now how η-critical subsolutions, although they contain less dynamical information than weak KAM solutions, can be used to characterize Aubry and Mañé sets in a similar way. Consider u ≺ Lη + α(c). For t ≥ 0 define  e It (u) := (x, v) ∈ TM : γ(x,v) (s) := πΦL s ((x, v)) is (u, Lη , α(c)) − calibr. on (−∞, t]} . T We will call the Aubry set of u: e I(u) := t≥0 e It (u), which can be also defined as  e I(u) := (x, v) ∈ TM : γ(x,v) (s) := πΦL s ((x, v)) is (u, Lη , α(c)) − calibr. on R . These sets e I(u) are nonempty, compact, and invariant. Moreover, here are some properties of these sets (compare with Theorem 5.1.18). Proposition 5.1.20. Let u ≺ Lη + α(c). 1. e It (u) is compact; It (u) ⊆ e I0 (u) for all t0 ≥ t ≥ 0; 2. e It0 (u) ⊆ e   3. L e I0 (u) is contained in the energy level Ec∗ corresponding to α(c);   4. L e It (u) ⊆ Graph(η + du) for all t > 0;   5. L e I0 (u) ⊆ Graph(η + du) (observe that for weak solutions these two sets coincide);   e 6. ΦL I (u) =e It (u) for all t > 0; 0 −t 7.

S

t>0

e It (u) = e I0 (u);

THE HAMILTON-JACOBI EQUATION AND WEAK KAM THEORY

83

  T e 8. e I(u) = t≥0 ΦL −t I0 (u) ;   9. e I0 (u) ⊆ W u e I(u) . g is a bi-Lipschitz homeomorphism [analogous to 10. π : e I(u) −→ π(I(u)) Mather’s graph theorem]. The same is true for e It (u) for each t > 0. Theorem 5.1.21 (Fathi). The Aubry and Mañé sets defined in (4.2) and (4.1) can be equivalently defined in the following ways: \ \ e Aec = L−1 (Graph(η + du)) I(u) = u≺Lη +α(c)

ec N

=

[

u≺Lη +α(c)

e I(u).

u≺Lη +α(c)

Moreover, there exists u∞ ≺ Lη + α(c) such that Aec = e I(u∞ ) ∩ L−1 (Ec∗ ). The proof of this theorem follows from the results in [37, Chapter 9]. For the last statement it is sufficient to observe that the set of critically dominated functions is a separable subset of C(M ). Let {un } be a countable dense family of such functions and define u∞ as a convex combination of their normalization (with P∞ respect to a fixed point x0 ∈ M ), e.g., u∞ (x) = n=0 21n (un (x) − un (x0 )). Remark 5.1.22. Using this characterization, the graph property of the Aubry set (Theorem 4.1.30) follows easily from property (10) in Proposition ec is a result of the nonemptiness of 5.1.20. Moreover, the nonemptiness of N e I(u). As far as the nonemptiness of Aec is concerned, one can deduce it from this characterization and Proposition 5.3.2. From Theorem 5.1.21 one can also deduce another interesting property of critically dominated functions: their differentiability on the projected Aubry set (recall that a priori these functions are only Lipschitz, so they are differentiable almost everywhere). Proposition 5.1.23 (see [37, Theorem 4.3.8 (i)]). Let u ≺ Lη + α(c). For each x ∈ Ac , u is differentiable at x and dx u does not depend on u; namely, ∂L dx u = (x, π|−1 (x)). ec A ∂v In addition to the Aubry set and Mañé set, one can also recover the definition of the Mañé potential φη,α(c) (see section 4.1) and Peierls’ barrier hη (see section 4.4 and (4.4)) in terms of these solutions and subsolutions. Let us start by observing that, from definition 5.1.3, if u ≺ Lη + α(c), then for each x, y ∈ M and t > 0 we have that u(y) − u(x) ≤ hη,t (x, y) + α(c)t, where hη,t is defined as Z t hη,t (x, y) = min Lη (γ(s), γ(s)) ˙ ds , 0

84

CHAPTER 5

where the minimum is taken over all piecewise C 1 paths γ : [0, t] −→ M such that γ(0) = x and γ(t) = y. This minimum is achieved because of the Tonelli theorem (Theorem 4.1.1). This implies that for each u ≺ Lη +α(c) and for each x, y ∈ M , u(y)−u(x) ≤ hη (x, y), i.e., φη,α(c) (x, y) ≥

[u(y) − u(x)]

sup

∀ x, y ∈ M.

u≺Lη +α(c)

One can actually show that they are equal. Proposition 5.1.24 (see [37, Corollary 9.1.3]). For each x, y ∈ M , we have the equality φη,α(c) (x, y) = sup [u(y) − u(x)] . u≺Lη +α(c)

Remark 5.1.25. The quantity on the right-hand side is also called “viscosity semi-distance” (see [37, Section 8.4]). As far as Peierls’ barrier is concerned, let us observe that similarly to what happens for the Mañé potential, in this case we also have that for each u ≺ Lη + α(c) and for each x, y ∈ M , u(y) − u(x) ≤ hη (x, y), i.e., hη (x, y) ≥

sup

[u(y) − u(x)]

∀ x, y ∈ M .

u≺Lη +α(c)

Moreover, if u− ∈ Sη− and u+ ∈ Sη+ are conjugate solutions, the same result holds: u− (y) − u+ (x) ≤ hη (x, y) and consequently hη (x, y) ≥

sup [u− (y) − u+ (x)]

∀ x, y ∈ M ,

(u− ,u+ )

where (u− , u+ ) denotes conjugate weak KAM solutions. In addition to this, it is possible to show that the above inequality is actually an equality. In fact: Proposition 5.1.26 (see [33, Théorème 7]). For x ∈ M let us define the function hxη : M → R (or hη,x : M → R) by hxη (y) = hη (x, y) (or hη,x (y) = hη (y, x)). For each x ∈ M , the function hxη (or −hη,x ) is a weak KAM solution of negative (or positive) type. Moreover, its conjugate function ux+ ∈ Sη+ (or ux− ∈ Sη− ) vanishes at x. Therefore: Corollary 5.1.27. For each x, y ∈ M , we have the equality hη (x, y) =

sup [u− (y) − u+ (x)] ,

(u− ,u+ )

where the supremum is taken over pairs of conjugate solutions. Moreover, for any given x, y ∈ M this supremum is actually attained.

85

THE HAMILTON-JACOBI EQUATION AND WEAK KAM THEORY

Observe that, since for any u ≺ Lη + α(c) there exist a weak KAM solution of negative type u− and a weak KAM solution of positive type u+ , such that u− = u = u+ on the projected Aubry set Ac (see Theorem 5.1.16), one can get the following representations for Peierls’ barrier hη on the projected Aubry set: hη (x, y)

=

sup

[u(y) − u(x)]

(5.3)

u≺Lη +α(c)

for all x, y ∈ Ac . This supremum is actually attained for any fixed x ∈ Ac . Remark 5.1.28. In particular, (5.3) shows that Peierls’ barrier and the Mañé potential coincide on the projected Aubry set. 5.2

REGULARITY OF CRITICAL SUBSOLUTIONS

In this section we say more about the regularity of critically dominated functions or η-critical subsolutions. We have remarked before in this chapter that for k < α(c) there do not exist functions dominated by Lη + k, while for k ≥ α(c) they do exist. Moreover, if k > α(c), these functions can be chosen to be C ∞ (see also the characterization of α(c) in Remark 5.1.8). The critical case has totally different features. As a counterpart of their relation with the dynamics of the system, critical dominated functions have very rigid structural properties, which become an obstacle when someone tries to make them smoother. For instance, as we have recalled in Proposition 5.1.23, if u ≺ Lη + α(c), then its differential dx u exists on Ac and it is prescribed over there. This means that although it is quite easy to make these functions smoother (e.g., C ∞ ) out of the projected Aubry set, it is impossible to modify them on this set. Nevertheless, Albert Fathi and Antonio Siconolfi [42] managed to prove that C 1 η-critical subsolutions do exist and are dense, in the following sense: Theorem 5.2.1 (Fathi, Siconolfi). Let u ≺ Lη + α(c). For each ε > 0, there exists a C 1 function u ˜ : M −→ R such that: i) u ˜ ≺ Lη + α(c); ii) u ˜(x) = u(x) on Ac ; iii) |˜ u(x) − u(x)| < ε on M \ Ac . Moreover, one can choose u ˜ so that it is a strict η-critical subsolution, i.e., we have Hη (x, dx u ˜) < a(c) on M \ Ac . This result has been improved by Patrick Bernard [11], who showed that every η-critical subsolution coincides, on the Aubry set, with a C 1,1 η-critical subsolution.

86

CHAPTER 5

Theorem 5.2.2 (Bernard, [11]). Let H be a Tonelli Hamiltonian. If the Hamilton-Jacobi equation has a subsolution, then it has a C 1,1 subsolution. Moreover, the set of C 1,1 subsolutions is dense for the uniform topology in the set of subsolutions. Remark 5.2.3. In general C 1,1 is the best regularity that one can expect: it is easy in fact to construct examples in which C 2 η-critical subsolutions do not exist. For example, consider the case in which the Aubry set projects over all the manifold M and it is not a C 1 graph (e.g., on M = T take L(x, v) = 2 2 1 2 2 kvk +sin (πx) and η = π dx). In this case there is only one critical subsolution (up to constants) that is an actual solution: its differential is Lipschitz but not C 1 . It is therefore clear that the structure of the Aubry set plays a crucial role. Patrick Bernard [12] proved that if the Aubry set is a union of finitely many hyperbolic periodic orbits or hyperbolic fixed points, then smoother subsolutions can be constructed. In particular, if the Hamiltonian is C k , then these subsolutions will be C k too. The proof of this result is based on the hyperbolic structure of the Aubry set and the result is deduced from the regularity of its local stable and unstable manifolds. See also [38] for a survey on this problem and some results concerning Denjoytype obstructions to the existence of regular critical subsolutions on T2 . Using the density of C 1 critically dominated functions, one can improve some of the results in (5.2), Theorem 5.1.21 and (5.3). Let us denote by Sη1 the set of C 1 η-critical subsolutions and by Sη1,1 the set of C 1,1 η-critical subsolutions. Then: \ \ e e Aec := I(u) = I(u) u∈Sη1

=

\

u∈Sη1,1

\

L−1 (Graph(η + du)) =

u∈Sη1

L−1 (Graph(η + du)) . (5.4)

u∈Sη1,1

In particular, there exists a C 1,1 η-critical subsolution u ˜ such that: Aec

= L−1 (Graph(η + d˜ u) ∩ Ec∗ ) = e I(˜ u) ∩ L−1 (E ∗ ) . c

(5.5)

Moreover, for any x, y ∈ Ac : hη (x, y)

=

sup {u(y) − u(x)} = sup {u(y) − u(x)} , u∈Sη1

u∈Sη1,1

where all the above suprema are maxima for any fixed x, y ∈ Ac .

(5.6)

THE HAMILTON-JACOBI EQUATION AND WEAK KAM THEORY

5.3

87

NON-WANDERING POINTS OF THE MAÑÉ SET

In this section we want now to use this approach and the above-mentioned results to show that the non-wandering set of the Euler-Lagrange flow restricted to the Mañé set is contained in the Aubry set (we mentioned this in Proposition 4.1.28). Let us first recall the definition of non-wandering point for a flow Φt : X −→ X. Definition 5.3.1. A point x ∈ X is called non-wandering if, for each neighborhood U and each positive integer n, there exists t > n such that f t (U)∩U = 6 ∅. We will denote the set of non-wandering points for Φt by Ω(Φt ). Proposition If M is a compact manifold and L a Tonelli Lagrangian  5.3.2.  L e on TM , then Ω Φt Nc ⊆ Aec for each c ∈ H1 (M ; R). Remark 5.3.3. 1) Proposition 5.3.2 also shows that the Aubry set is nonempty. In fact, any continuous flow on a compact space possesses nonwandering points. 2) Since every point in the support of an invariant measure µ is nonwandering, fc ⊆ Aec . M   e Proof. Let (x, v) ∈ Ω ΦL t N0 . By the definition of nonwandering point, fc and tk → +∞ such that (xk , vk ) → (x, v) there exist a sequence (xk , vk ) ∈ N (x , v ) → (x, v) as k → +∞. From (5.4), for each (xk , vk ) there and ΦL k k tk
exists an η-critical subsolution uk , such that the curve γk (t) = π ΦL t (xk , vk ) is (uk , Lη , α(c))-calibrated. Moreover, after possibly extracting a subsequence, we 1 can assume that, on
any compact interval, γk converges in the C -topology to γ(t) = π ΦL (x, v) . t Pick now any critical subsolution u. If we show that γ is (u, Lη , α(c))calibrated, using (5.4) we can conclude that (x, v) ∈ Aec . First of all, observe that, by the continuity of u, k→∞

u(γk (tk )) − u(xk ) −→ 0. Using the fact that η-critical subsolutions are equi-Lipschitz (as remarked after definition 5.1.3), we can also conclude that k→∞

uk (γk (tk )) − uk (xk ) −→ 0 and, therefore, Z tk k→∞ Lη (γk (s), γ˙ k (s)) + α(c) ds = uk (γk (tk )) − uk (xk ) −→ 0 . 0

(5.7)

88

CHAPTER 5

Let 0 ≤ a ≤ b and choose tk ≥ b. Observe now that u(γk (b)) − u(γk (a))

= u(γk (tk )) − u(xk ) − [u(γk (tk )) − u(γk (b))] − [u(γk (a)) − u(xk )] Z tk ≥ u(γk (tk )) − u(xk ) − Lη (γk (s), γ˙ k (s)) + α(c) ds b Z a Lη (γk (s), γ˙ k (s)) + α(c) ds − 0

Z b = u(γk (tk )) − u(xk ) + Lη (γk (s), γ˙ k (s)) + α(c) ds a Z tk − Lη (γk (s), γ˙ k (s)) + α(c) ds. 0

Taking the limit as k → ∞ on both sides, one can conclude: Z u(γ(b)) − u(γ(a)) ≥

b

Lη (γ(s), γ(s)) ˙ + α(c) ds a

and therefore, from the fact that u ≺ Lη +α(c), the equality follows. This shows that γ is (u, Lη , α(c))-calibrated on [0, ∞). To show that it is indeed calibrated on all R, one can make a symmetric argument, letting (yk , wk ) = ΦL tk (xk , vk ) play the role of (xk , vk ) in the previous argument. In fact, one has (yk , wk ) → (x, v) and ΦL −tk (yk , wk ) → (x, v) as k → +∞, and the very same argument works. 

Appendix A On the Existence of Invariant Lagrangian Graphs In this appendix we would like to describe some properties of Hamiltonian and Lagrangian systems, with particular attention to the relation between their action-minimizing properties, their symplectic nature, and their dynamics. More specifically, we will illustrate what kind of information Aubry-Mather theory conveys to the study of the integrability of these systems, and, more generally, how this information relates to the existence or nonexistence of invariant Lagrangian graphs. A.1

SYMPLECTIC GEOMETRY OF THE PHASE SPACE

Let us start by recalling some concepts of symplectic geometry (for a more comprehensive introduction we refer the reader to [22]). The cotangent bundle T∗ M may be equipped with a canonical symplectic structure, i.e., a closed nondegenerate 1 2-form ω. Let us see how to define this canonical form. If (U, x1 , . . . , xd ) is a local coordinate chart for M and (T∗ U, x1 , . . . , xd , p1 , . . . , pd ) the associated cotangent coordinates, one can define the 2-form as ω=

d X

dxi ∧ dpi .

i=1

In particular, one can check that ω is intrinsically defined (i.e., independent of the choice of the coordinate charts) by considering the 1-form on T∗ U, λ=

d X

pi dxi ,

i=1

which satisfies ω = −dλ and is coordinate-independent; in fact, in terms of the natural projection π : T∗ M −→ (x, p) 7−→

M x,

1 By nondegeneracy we mean that if for some V ∈ T ∗ (x,p) T M the restriction ω(V, ·) ≡ 0 on T(x,p) T∗ M , then V = 0.

90

APPENDIX A

the form λ may be equivalently defined pointwise by λ(x,p) = (dπ(x,p) )∗ p ∈ T∗(x,p) T∗ M .

(A.1)

The 1-form λ is called the Liouville form (or the tautological form). We also remarked that, since ω is nondegenerate and closed, the Hamiltonian vector field XH is uniquely determined by: ω (XH (x, p), ·) = dx H(·) . One can easily check that, in local coordinates, the above equation is equivalent to Hamilton’s equation (1.2). A distinguished role in the study of the geometry of a symplectic space is played by the so-called Lagrangian submanifolds. Definition A.1.1 (Lagrangian submanifolds). Let Λ be a d-dimensional C 1 submanifold of (T∗ M, ω). We say that Λ is Lagrangian if, for any (x, p) ∈ Λ, T(x,p) Λ is a Lagrangian subspace, i.e., ω T = 0. Λ (x,p)

We will mainly be concerned with Lagrangian graphs, that is, Lagrangian manifolds Λ ⊂ T ∗ M such that Λ = {(x, η(x)) , x ∈ M }. In (T∗ M, ω) there exists an interesting well-known relation between Lagrangian graphs and closed 1-forms (recall that a 1-form can be interpreted as a section of T∗ M ). Proposition A.1.2. Let Λ = {(x, η(x)), x ∈ M } be a smooth section of T∗ M . Λ is Lagrangian if and only if η is a closed 1-form. Proof. Let us consider: sη : M −→ x 7−→

T∗ M (x, η(x)) .

We want to prove first that s∗η λ = η, where λ is the tautological form introduced above and s∗η λ denotes its pullback. Recalling that λ(x, p) = (dπ(x, p))∗ p, we get: (s∗η λ)(x)

= (dsη (x))∗ λ(x, p) = (dsη (x))∗ (dπ(x, p))∗ η(x) = d(π ◦ sη (x, p))∗ η(x) = η(x) ,

where in the last equality we used that π ◦ sη is the identity map. Using this property, the claim follows immediately. In fact: Λ is Lagrangian ⇐⇒ ω TΛ = 0 ⇐⇒ s∗η ω = 0 ⇐⇒ s∗η dλ = 0 ⇐⇒ ds∗η λ = 0 ⇐⇒ dη = 0 ⇐⇒ η is closed.  In the light of this relation, one can define the cohomology class (or Liouville class) of Λ to be the cohomology class of the closed 1-form representing it.

ON THE EXISTENCE OF INVARIANT LAGRANGIAN GRAPHS

91

Remark A.1.3. Observe that this cohomology class can be defined in a more intrinsic way; in fact, the projection π|ΛL : ΛL ⊂ T∗ M −→ M induces an isomorphism between the cohomology groups H1 (M ; R) and H1 (ΛL ; R). The preimage of [λ|ΛL ] under this isomorphism is called the Liouville class of ΛL , and one can easily check that these two definitions coincide. This characterization motivates the following extension of the notion of Lagrangian graph to the Lipschitz case. Definition A.1.4 (Lipschitz Lagrangian graphs). A Lipschitz section Λ of T∗ M is a C 0 -Lagrangian graph if it locally coincides with the graph of an exact differential. As above, one can define its cohomology class. Let us now consider a Hamiltonian H : T∗ M −→ R and the associated Hamiltonian flow ΦH t . We want to analyze the relation between the property of being Lagrangian and the dynamics on it. Proposition A.1.5. Let Λ be a Lagrangian submanifold. Then Λ is invari ant if and only if H Λ ≡ const. Proof. (=⇒) The Hamiltonian vector field XH is defined by ω(XH , ·) = dH. Since Λ is invariant, XH Λ is tangent to Λ. But Λ is Lagrangian; therefore 0 = ω(XH , V ) = dH · V for any V ∈ T Λ, and this implies that H is constant on Λ. (⇐=) Since H is constant on Λ, we have that 0 = dH · V = ω(XH , V ) for every V ∈ TΛ. Since Λ is Lagrangian, XH belongs to TΛ itself, and therefore Λ is invariant.  Remark A.1.6. It follows immediately from the above property that invariant Lagrangian graphs correspond to η-critical subsolutions of the HamiltonJacobi equation for some 1-form η whose cohomology class coincides with that of Λ. Therefore, if Λ is an invariant Lagrangian graph, then the value of the constant in Proposition A.1.5 is given by α(cΛ ), where α is the α-function associated with H and cΛ is the cohomology class of Λ. See also Remark 5.1.8. A.2

EXISTENCE AND NONEXISTENCE OF INVARIANT LAGRANGIAN GRAPHS

In this section we would to address the core question of this appendix, namely: what kind of information does the principle of least Lagrangian action convey to the study of the integrability of these systems, and, more generally, how does this information relate to the existence or nonexistence of invariant Lagrangian graphs? Hereafter we will address this very interesting (and difficult) question from two different perspectives. The main results are all contained in [21], [56], and

92

APPENDIX A

[77] (see also [78]), to which we will refer the reader for a more comprehensive presentation and for the proofs. A.2.1

Weak Liouville-Arnol0d Theorem

(See [77, 21].) It is natural to expect that “sufficiently” symmetric systems ought to possess an abundance of invariant Lagrangian graphs. This is indeed the content of a very classical result in the study of Hamiltonian systems: what is generally called Liouville-Arnol 0d theorem (see, for example, [5]). This theorem is concerned with the integrability of a Hamiltonian system, i.e., with the existence of a regular foliation of the phase space by invariant Lagrangian submanifolds. This theorem provides sufficient conditions for the existence of such a foliation in terms of the existence of independent “symmetries” that are in involution. In order to make the latter condition clearer, let us recall some terminology. Consider a Hamiltonian system with d degrees of freedom, given by a Hamiltonian H : V → R defined on a 2d-dimensional symplectic manifold (V, ω), and denote by {·, ·} the associated Poisson bracket, defined as follows: if f, g ∈ C 1 (M ), then {f, g} = ω(Xf , Xg ) = df · Xg , where Xf and Xg denote the Hamiltonian vector fields associated with f and g (see for instance [5]). An important role in the study of the dynamics is played by the functions F : V → R that are in involution with the Hamiltonian, i.e., whose Poisson bracket {H, F } ≡ 0 on V (equivalently we can say that H and F Poisson-commute). Such functions, whenever they exist, are called integrals of motion (or first integrals) of H. It is quite easy to check that the condition of being in involution is equivalent to F being constant along the orbits of the Hamiltonian flow of H and vice versa; moreover, this implies that the associated Hamiltonian vector fields XH and XF commute. The Liouville-Arnol0d theorem relates the integrability of a given Hamiltonian system to the existence of “sufficiently many” integrals of motion in involution. Theorem A.2.1 (Liouville-Arnol0d). Let (V, ω) be a symplectic manifold of dimension 2d and let H : V −→ R be a proper Hamiltonian. Suppose that there exist d integrals of motion F1 , . . . , Fd : V −→ R such that: i) F1 , . . . , Fd are C 2 and functionally independent almost everywhere on V ; ii) F1 , . . . , Fd are pairwise in involution, i.e., {Fi , Fj } = 0 for all i, j = 1, . . . d. Suppose the nonempty regular level set Λa := {F1 = a1 , . . . , Fd = ad } is connected. Then Λa is an d-torus, Td , and there is a neighbourhood O of 0 ∈ H 1 (Λa ; R) such that for each c0 ∈ O there is a unique smooth Lagrangian Λc0 that is a graph over Λa with cohomology class c0 . Moreover, the flow of XH |Λc0 is conjugated to a rigid rotation. See for instance [5, Section 49] for a proof of this theorem.

ON THE EXISTENCE OF INVARIANT LAGRANGIAN GRAPHS

93

Remark A.2.2. The map F := (F1 , . . . , Fd ) is referred to as an integral map, a first-integral map, or a momentum map. The invariance of the level set Λa simply follows from the Fi ’s being integrals of motion; the fact that it is a Lagrangian torus and that the Hamiltonian flow is conjugate to a rigid rotation strongly relies on these integrals being pairwise in involution and independent. A natural question is then the following: Is it possible to weaken the assumptions in the Liouville-Arnol 0d theorem? In particular: What happens when the involution hypothesis on the integrals of motion is dropped? It is clear from Remark A.2.2 that the involution hypothesis is essential for deducing the nontrivial fact that these invariant sets are Lagrangian and that the motion on them is conjugate to a rigid rotation. Roughly speaking this is not just a sufficient condition, it is also somehow necessary: these submanifolds’ property of being Lagrangian is in fact essentially equivalent to the involution hypothesis. Hence, it seems almost impossible to deduce interesting results without assuming it. However, in the case of Tonelli Hamiltonians, it turns out to be possible. In [77] the following definition was introduced. Definition A.2.3 (Weak integrability [77]). Let H ∈ C 2 (T ∗ M ) and let d = dim M . If there is a C 2 map F : T ∗ M −→ Rd whose singular set is nowhere dense, and F Poisson-commutes with H, then we say that H is weakly integrable. Remark A.2.4. There exist examples of Hamiltonian systems that are weakly integrable but not Liouville integrable. See, for example, [77, Appendix A]. Moreover, Butler and Paternain [20] proved the following. Let G be a compact semi-simple Lie group of rank at least 2. In any neighbourhood of the bi-invariant metrics, there are left-invariant metrics with positive topological entropy that are not completely integrable. Nevertheless, these metrics are weakly integrable. In [77] and in a subsequent joint work with Leo Butler [21], we proved a version of the Liouville-Arnol0d theorem for weakly integrable Tonelli Hamiltonians. We named this theorem the weak Liouville-Arnol 0d theorem because it drops the involutivity hypothesis of the classical theorem and still obtains results that are quite analogous. Theorem A.2.5 (Weak Liouville-Arnol0d). Let M be a closed manifold of dimension d, and H : T ∗ M −→ R a weakly integrable Tonelli Hamiltonian with integral map F : T ∗ M −→ Rd . If for some cohomology class c ∈ H 1 (M ; R) the Mather set M∗c ⊂ regF (where regF denotes the set of non-singular points of F ), then there exists an open neighbourhood O of c in H 1 (M ; R) such that the following holds. i) For each c0 ∈ O there exists a smooth invariant Lagrangian graph Λc0 of cohomology class c0 , which admits the structure of a smooth Tk -bundle over a base B d−k that is parallelizable, for some k > 0.

94

APPENDIX A

ii) The motion on each Λc0 is Schwartzman strictly ergodic (see [40, 75] and appendix B), i.e., all invariant probability measures have the same rotation vector and the union of their supports equals Λc0 . In particular, all orbits are conjugate by a smooth diffeomorphism isotopic to the identity. iii) Mather’s α-function (or minimal average action) α : H 1 (M ; R) −→ R is differentiable at all c0 ∈ O and its convex conjugate β : H1 (M ; R) −→ R is differentiable at all rotation vectors h ∈ ∂α(O), where ∂α(O) denotes the set of subderivatives of α at some element of O. In particular, for c ∈ O, orbits on Λc0 have rotation vector ∂α(c0 ). Remark A.2.6. Observe that the flow of XH |Λc0 is a rotation on the Tk – fibers of Λc0 with rotation vector hc0 = ∂α(c0 ), where ∂α(c0 ) is the derivative of α at c0 and k = b1 (M ), the first Betti number of M . This is analogous to what happens in the classical Liouville-Arnol0d theorem, where the rotation vector is the derivative of H (in action-angle coordinates) at c0 . This theorem shows the existence of a family of smooth invariant Lagrangian graphs {Λc0 }c0 ∈O , which form a lamination of the space. However, in general, these graphs cannot be expected to foliate any open set of the phase space. In fact, there are obvious dimensional obstructions: {Λc0 }c0 ∈O is a (dim H 1 (M ; R))dimensional family of graphs of dimension dim M . Hence, a necessary condition for this to happen is that dim H 1 (M ; R) ≥ dim M . Is this condition the unique obstruction? What can be said if these graphs foliate an open set? The following result answers these questions (see [77, 21]). Theorem A.2.7. Assume the hypotheses of Theorem A.2.5. If dim H 1 (M ; R) ≥ dim M , then dim H 1 (M ; R) = dim M and M is diffeomorphic to Td = Rd /Zd . In particular, it follows that H is integrable in the sense of Liouville and therefore the integrals of motion are in involution. Remark A.2.8. (i ) In other words, if dim H 1 (M ; R) ≥ dim M then the notion of weak integrability is equivalent to the classical notion of Liouville integrability. This hypothesis is satisfied, for example, if M = Td . (ii ) Actually, in [21] we proved something more. We showed that under the hypotheses of Theorem A.2.5: a) if dim M ≤ 3, then M is diffeomorphic to a torus; b) if dim M = 4, then—assuming the virtual Haken conjecture2 —M is diffeomorphic to either T4 or T1 × E, where E is an orientable 3-manifold finitely covered by S 3 . 2 In 3-manifold topology, a central role is played by those closed 3-manifolds which contain a nonseparating incompressible surface, or dually, which have a nonvanishing first Betti number. Such manifolds are called Haken; it is an unproved conjecture that every irreducible 3-manifold with an infinite fundamental group has a finite covering that is Haken. This conjecture is implied by the virtually fibered conjecture [1]. Given the proof of the geometrization conjecture, the virtual Haken conjecture is proven for all cases but hyperbolic 3-manifolds. Thurston and Dunfield have shown there is good reason to believe the conjecture is true in this case [31].

ON THE EXISTENCE OF INVARIANT LAGRANGIAN GRAPHS

95

See [21, Theorem 1.2] for a detailed proof. (iii ) Moreover, we investigated the case in which the system’s symmetries are not classical and do not come from conserved quantities, but are induced by invariance under the action of an amenable Lie group on the universal cover of the manifold. This action need not descend to the quotient and is generally only evident in statistical properties of orbits. In particular, these symmetries may only manifest themselves in the structure of the action-minimizing sets. Analogous results to the ones discussed above can be also proven in this case; see [21, Theorem 1.3] for a precise statement. A.2.2

Differentiability of the Minimal Average Action and Integrability

In this section we discuss another possible approach to the study of the existence of invariant Lagrangian graphs and the integrability of the system (see [56]). As we have recalled in sections 3.2 and 3.3, in the study of Tonelli Lagrangian and Hamiltonian systems, a central role in understanding the dynamical and topological properties of the action-minimizing sets is played by the so-called Mather’s β-function, with particular attention on its differentiability and nondifferentiability properties. Understanding whether or not this function is differentiable, or even smoother, and what the implications of its regularity are on the dynamics of the system is a formidable problem, still far from being understood. Examples of Lagrangians admitting a smooth β-function are easy to construct; trivially, if the base manifold M is such that dim H1 (M ; R) = 0 then β is a function defined on a single-point set and it is therefore smooth. Furthermore, if dim H1 (M ; R) = 1 then a result by Carneiro [23] allows one to conclude that β is differentiable everywhere, except possibly at the origin. As soon as dim H1 (M ; R) ≥ 2 the situation becomes definitely less clear and the smoothness of β becomes a more “untypical” phenomenon. Nevertheless, it is still possible to find some interesting examples in which it is smooth. For instance, let H : T∗ Td −→ R be a completely integrable Tonelli Hamiltonian system, given by H(x, p) = h(p), and consider the associated Lagrangian L(x, v) = `(v) on TTd . It is easy to check (see section 2.1) that in this case, after identifying H1 (Td ; R) with Rd , one has β(h) = `(h) and therefore β is as smooth as the Lagrangian. One can weaken the assumption on the complete integrability of the system and consider C 0 -integrable systems, i.e., Hamiltonian systems that admit a foliation of the phase space by disjoint invariant continuous Lagrangian graphs, one for each possible cohomology class (see [2]). It is then possible to prove that also in this case the associated β-function is C 1 . These observations raise the following question: If dim H1 (M ; R) ≥ 2, does the regularity of β imply the integrability of the system? Or more generally, is the existence of an invariant Lagrangian graph detected by some regularity property of this function? In a joint work with Daniel Massart [56], we addressed the above problem in the case of Tonelli Lagrangians on closed surfaces, not necessarily orientable (in

96

APPENDIX A

which case one considers the Lagrangian lifted to the orientable double cover). The main results can be summarized as follows. Theorem A.2.9. Let M be a closed surface and L : TM −→ R a Tonelli Lagrangian. (i) If M is not the sphere, the projective plane, the Klein bottle, or the torus, then β cannot be C 1 on the whole of H1 (M ; R). (ii) If M is not the torus then the Lagrangian cannot be C 0 -integrable. (iii) If M is the torus, then β is C 1 if and only if the system is C 0 -integrable. In particular, in the orientable case it is possible to relate the differentiability of β at (non-singular) 1-irrational homology classes to the existence of invariant Lagrangian graphs foliated by periodic orbits; this can be thought of as a generalization of a result by Mather for twist maps (see [61]). Theorem A.2.10. Let M be a closed, oriented surface and L : T M −→ R an autonomous Tonelli Lagrangian. Let h0 be a 1-irrational, non-singular homology class, and let h∂β(h0 )i denote the underlying vector space of the affine subspace generated by ∂β(h0 ) in H 1 (M, R). Then, the dimension of h∂β(h0 )i is at least the genus of M . Moreover, if M = T2 and β is differentiable at h0 , then there exists an invariant Lagrangian graph foliated by periodic orbits, whose homology is a multiple of h0 . Remark A.2.11. Observe that the above result is not true if h is singular (see example in [56, Remark 2]). Moreover, when M is not orientable, the situation is different because β may have flats of maximal dimension (that is, of dimension equal to the first Betti number of M ). So β may well be differentiable at some 1-irrational, non-singular homology class without having any invariant Lagrangian graph foliated by periodic orbits (see [56, pages 11-12] for a more precise discussion of this issue). The above results consider the C 0 -integrable case. An open question is whether similar results can be obtained with C 0 –integrability replaced by integrability in the sense of Liouville. In the case of mechanical systems we can bridge this gap. A mechanical Lagrangian is a Lagrangian of the form L(x, v) = 1/2 gx (v, v) + f (x), where g is a Riemannian metric and f is a C 2 function on T2 . Proposition A.2.12. Let L be a mechanical Lagrangian on a two-dimensional torus, whose β-function is C 1 . Then, the potential f is identically constant and the metric g is flat. In particular, L is integrable in the sense of Arnol 0d-Liouville. See [56, Proposition 6] for a complete proof.

Appendix B Schwartzman Asymptotic Cycle and Dynamics In section 3.2 we introduced the concept of rotation vector of a measure. This is closely related to the notion of the Schwartzman asymptotic cycle of a flow, introduced by Sol Schwartzman in [74] as a first attempt to develop an algebraic topological approach to the study of dynamics. In this chapter, we would like to provide some examples and investigate some properties of what we have called Schwartzman uniquely ergodic flows (see also [40, 75]). B.1

SCHWARTZMAN ASYMPTOTIC CYCLE

First, we would like to discuss more in depth the concepts of rotation vector and Schwartzman asymptotic cycle. One can provide a different description of the Schwartzman asymptotic cycle of a flow. This is also known as the flux homomorphism in volume preserving and symplectic geometry; see [9, Chapter 3]. We will use the description given in [32, pages 67–70]. This definition has the technical advantadge of not relying on the Krylov-Bogolioubov theory of generic orbits in a dynamical system, although a more geometrical definition showing that “averaged” pieces of long orbits converge almost everywhere in the first homology group for any invariant measure is certainly more heuristic and intuitive. Let us start with some standard facts. As usual, we set T = R/Z. The space T is a topological group for addition. An important feature of T is that the canonical projection π : R → T is a covering map. Therefore, given any continuous path γ : [a, b] → T, with a ≤ b, we can find a continuous lift γ¯ : [a, b] → R such that γ = π¯ γ . Any two such lifts differ by an integer. It follows that the quantity γ¯ (b) − γ¯ (a) does not depend on the lift. We will set V(γ) = γ¯ (b) − γ¯ (a) ∈ R. This quantity remains constant on the homotopy class, with fixed endpoints, of the path γ. Moreover, if γ is a closed path, i.e., we have γ(a) = γ(b), then V(γ) ∈ Z, and, if such a closed path is homotopic to 0 (with fixed endpoints) then V(γ) = 0. It is also clear that for c ∈ [a, b], we have V(γ|[a, b]) = V(γ|[a, c]) + V(γ|[c, b]). Note also that two continuous paths γ1 , γ2 : [a, b] → T can be added by the formula (γ1 + γ2 )(t) = γ1 (t) + γ2 (t).

98

APPENDIX B

For this addition we have V(γ1 + γ2 ) = V(γ1 ) + V(γ2 ). Another important property of the map V is its continuity on the functional space C 0 ([a, b], T), endowed with the topology of uniform convergence. Let θ be the closed 1-form on T whose lift to R is the usual differential form dt on R, where dt is the differential of the identity map R → R, t 7→ t. It is well-known that when γ : [a, b] → T is C 1 , we have Z V(γ) =

Z θ=

γ

b

θγ(t) (γ(t)) ˙ dt. a

If X is a topological space and F : X × [a, b] → T is a given map, we will define V(F ) : X → R by ∀x ∈ X, V(F )(x) = V(Fx ), where Fx : [a, b] → T is defined by Fx (t) = F (x, t). The continuity of V on C 0 ([a, b], T) implies that V(F ) is continuous. Furthermore, the continuity of V on C 0 ([a, b], T) also implies that the map C 0 (X × [a, b], T) → C 0 (X, T), F 7→ V(F ) is continuous when we provide the spaces of continuous maps with the compact open topology. If F can be lifted to a continuous map F¯ : X × [a, b] → R with F = π F¯ , then V(F )(x) = F¯ (x, b) − F¯ (x, a). Suppose now that X is a topological space and that (φt )t∈R is a continuous flow on X. We will define Φ : X × [0, 1] → X by Φ(x, t) = φt (x). If f : X → T is continuous, we set V(f, φt ) = V(f ◦ Φ) : X → R. There is another way to define V(f, φt ) that is used in [32]. The function F (f, Φ) : X × [0, 1] → T, defined by F (f, Φ)(x, t) = f (φt (x)) − f (x), is continuous and identically 0 on X ×{0}; it is therefore homotopic to a constant and can be lifted to a continuous map F (f, Φ) : X ×[0, 1] → R, with F (f, Φ)|X × {0} identically 0. We have V(f, φt )(x) = F (f, Φ)(x, 1). Note that if f is homotopic to 0 then it can be lifted continuously to f¯ : X → R. In that case F¯ (f, Φ) = f¯Φ − f¯ and V(f, φt )(x) = f¯(φ1 (x)) − f¯.

99

SCHWARTZMAN ASYMPTOTIC CYCLE AND DYNAMICS

If µ is a measure with compact support and invariant under the flow φt , for a continuous f : X → T we can define S(µ, φt )(f ), or simply S(µ)(f ) when φt is fixed, by Z S(µ)(f ) =

V(f, φt )(x) dµ(x). X

It is not difficult to verify that for f1 , f2 : X → T, we have S(µ)(f1 + f2 ) = S(µ)(f1 ) + S(µ)(f2 ). Moreover, if f : X → T is homotopic to 0, it can be lifted to f¯ : X → R and Z Z Z S(µ)(f ) = [f¯(φ1 (x)) − f¯(x)] dµ(x) = f¯(φ1 (x)) dµ(x) − f¯(x) dµ(x) = 0, X

X

X

since µ is invariant by φ1 . Therefore, if we denote by [X, T] the set of homotopy classes of continuous maps from X to T, which is an additive group, the map S(µ) is a well-defined additive homomorphism from the additive group [X, T] to R. When X is a good space (like a manifold or a locally finite polyhedron), it is well known that [X, T] is canonically identified with the first cohomology group H1 (X; Z). In that case S(µ) is in Hom(H1 (X; Z), R). Since the first cohomology group with real coefficients H1 (X; R) is H1 (X; Z) ⊗ R, we can view S(µ) as an element of the dual H1 (X; R)∗ of the R-vector space H1 (X; R). When H1 (X; R) is finite-dimensional then H1 (X; R)∗ is in fact equal to the first homology group H1 (X; R), and therefore S(µ) defines an element of H1 (X; R), i.e., a 1-cycle. This 1-cycle S(µ) is called the Schwartzman asymptotic cycle of µ. Note that H1 (X; R) is finite-dimensional when X is a finite polyhedron or a compact manifold. It should be also noted that for a manifold M the projection T M → M is a homotopy equivalence. Therefore H1 (T M ; R) = H1 (M ; R) is finite-dimensional when M is a compact manifold. B.2

DYNAMICAL PROPERTIES

We want now to study the behavior of Schwartzman asymptotic cycles under semi-conjugacy. Proposition B.2.1. Suppose that φit : Xi → Xi , i = 1, 2 are two continuous flows. Suppose also that ψ : X1 → X2 is a continuous semi-conjugation between the flows, i.e., ψ ◦ φ1t = φ2t ◦ ψ for every t ∈ R. Given a probability measure µ with compact support on X1 invariant under φ1t , for every continuous map f : X2 → T, we have S(ψ∗ µ, φ2t )([f ]) = S(µ, φ1t )([f ◦ ψ]), where ψ∗ µ is the image of µ under ψ. In particular, if we are in the situation where Hom([Xi , T]) ≡ H1 (Xi ; R), i = 1, 2, we obtain S(ψ∗ µ, φ2t ) = H1 (ψ)(S(µ, φ1t )).

100

APPENDIX B

Proof. Notice that f ψφ1t (x)−f ψ(x) = f φ2t (ψ(x))−f (ψ(x)). Therefore, by uniqueness of the liftings, V(f ψ, φ1t )(x) = V(f, φ2t )(ψ(x)). An integration with respect to µ finishes the proof.  We would like now to relate the Schwartzman asymptotic cycles to the rotation vectors ρ(µ) defined for Lagrangian flows. We first consider the case of a C 1 flow φt on the manifold N . We call X the continuous vector field on N generating φt , i.e., dφt (x) ∀x ∈ N, X(x) = . dt t=0 By the flow property φt+t0 = φt ◦ φt0 , this implies dφt (x) = X(φt (x)). dt

∀x ∈ N, ∀t ∈ R,

In the case of a manifold N , the identification of [N, T] with H1 (N ; Z) is best described with the de Rham cohomology. We consider the natural map IN : [X, T] → H1 (X; R) defined by IN ([f ]) = [f ∗ θ], where [f ] on the left-hand side denotes the homotopy class of the C ∞ map f : N → T, and [f ∗ θ] on the right-hand side is the cohomology class of the pullback by f of the closed 1-form on T whose lift to R is dt. Note that any homotopy class in [N, T] contains smooth maps because C ∞ maps are dense in C 0 maps (for the Whitney topology). Therefore the map IN is indeed defined on the whole of [N, T]. As is well known, this map IN induces an isomorphism of [N, T] on H1 (N ; Z) ⊂ H1 (N ; R) = H1 (N ; Z) ⊗ R. Given a C ∞ map f : N → T, the C 1 flow φt on N , and x ∈ N , we compute V(f, φt )(x). If γx : [0, 1] → N is the path t 7→ φt (x), by definition we have V(f, φt )(x) = V(f ◦ γx ). Since γx is C 1 , we get Z Z V(f, φt )(x) = θ= f ∗ θ. f ◦γx

γx

Since γx (t) = φt (x), we have γ˙ x (t) = X(φt (x)). It follows that Z 1 V(f, φt )(x) = (f ∗ θ)φt (x) (X[φt (x)]) dt. 0

Recall that the interior product iX ω of a differential form ω with X is given by (iX ω)x (· · · ) = ωx (X(x), · · · ). When ω is a differential 1-form then iX ω is a function. With this notation, we get Z 1 V(f, φt )(x) = (iX f ∗ θ)(φt (x)) dt. 0

101

SCHWARTZMAN ASYMPTOTIC CYCLE AND DYNAMICS

Therefore if µ is an invariant measure for φt , which we will assume to have a compact support, we obtain Z Z 1 (iX f ∗ θ)(φt (x)) dtdµ(x). S(µ) = N

0

∗

Since iX f θ is continuous, and since we are assuming that µ has a compact support, we have Z 1Z S(µ) = (iX f ∗ θ)(φt (x)) dµ(x)dt. 0

N

R R By invariance of µ under φt , we get N (iX f ∗ θ)(φt (x)) dµ(x) = N (iX f ∗ θ)(x) dµ(x), and therefore Z 1Z Z ∗ S(µ) = (iX f θ)(x) dµ(x)dt = (iX f ∗ θ) dµ. 0

N

N

This shows that as an element of H1 (M ; R)∗ , the Schwarztman asymptotic cycle S(µ) is given by Z S(µ)([ω]) =

iX ω dµ. N

Examples I. We can now easily compute Schwartzman asymptotic cycles for linear flows on Td . Such a flow is determined by a constant vector field α ∈ Rd on Td (here we use the canonical trivialization of the tangent bundle of Td ); the associated flow Rtα : Td → Td is defined by Rtα (x) = x + [tα], where [tα] is the class in Td = Rd /Zd of the vector tα ∈ Rd . If ω is a 1-form with constant coefficients, Pd i.e., ω = i=1 ai dxi , with ai ∈ R, the interior product iα ω is the constant funcPd tion i=1 αi ai . Therefore, it follows that S(µ) = α ∈ Rd ≡ H1 (Td ; R). II. We now compute Schwartzman asymptotic cycles for Euler-Lagrange flows. In this case N = T M and φt is an Euler-Lagrange flow φL t of some Lagrangian L. If we call XL the vector field generating φL t , since this flow is obtained from a second order ODE on M , we get ∀x ∈ M, ∀v ∈ Tx M,

T π(XL (x, v)) = v,

where T π : T (T M ) → T M denotes the canonical projection. Since this projection π is a homotopy equivalence, to compute S(µ) we only need to consider forms of the type π ∗ ω, where ω is a closed 1-form on the base M . In this case, (iXL π ∗ ω)(x, v) = ωx (T π(XL (x, v)) = ωx (v). Therefore, for any probability measure µ on T M with compact support and invariant under φL t , we obtain Z Z S(µ)[π ∗ ω] = ωx (v) dµ(x, v) = ω ˆ dµ. TM

TM

102

APPENDIX B

This is precisely ρ(µ) as it was defined in section 3.2. Note that the only property we have used is the fact that φt is the flow of a second order ODE on the base M. To simplify things, in the remainder of this section, we will assume that X is a compact space, for which we have [X, T] = H1 (X; Z), and H1 (X; Z) is finitely generated. In that case, the dual space H1 (X; R)∗ is H1 (X; R), and for every flow φt on X and every probability measure µ on X invariant under φt , the Schwartzman asymptotic cycle is an element of the finite-dimensional vector space H1 (X; R). Suppose that x is a periodic point of φt or period T > 0. One can define an invariant probability measure µx,t0 for φt by Z Z 1 t0 g(x) dµx,t0 = g((φt (x)) dt, t0 0 X where g : X → R is a measurable function. It is easy to verify that S(µx,t0 ) is equal in H1 (X; R) to the homology class [γx,t0 ]/t0 , where γx,t0 is the loop t 7→ φt (x), t ∈ [0, t0 ]. When x is a fixed point of φt , then the Dirac mass δx at x is invariant under φt , and in that case S(δx ) = 0. Definition B.2.2. For a flow φt on X, we denote by S(φt ) the set of all Schwartzman asymptotic cycles S(µ), where µ is an arbitrary probability measure on X invariant under φt . Since X is compact, note that for the weak topology the set M(X) of probability Borel measures on X is compact and convex. It is even metrizable, since we are assuming X is metrizable. Furthermore, the subset M(X, φt ) ⊆ M(X) of probability measures invariant under φt is, as is well known, compact convex and nonempty. Therefore S(φt ) is a compact convex nonempty subset of H1 (X; R). For the case of a linear flow Rα on Td , we have shown above that S(Rtα ) = {α} ⊂ Rd ≡ H1 (Td ; R). The following corollary is an easy consequence of Proposition B.2.1. Corollary B.2.3. For i = 1, 2, suppose that φit is a continuous flow on the compact space Xi , which satisfies Hom([Xi , T], R) ≡ H1 (Xi ; R). If ψ : X1 → X2 is a topological conjugacy between φ1t and φ2t (i.e., the map ψ is a homeomorphism that satisfies ψφ1t = φ2t ψ, for all t ∈ R), then we have S(φ2t ) = H1 (ψ)[S(φ1t )]. We denote by F(X) the set of continuous flows on X. We can embed F(X) in C 0 (X × [0, 1], X) by the map φt 7→ F φt ∈ C 0 (X × [0, 1], X), where F φt (x, t) = φt (x). The topology on C 0 (X × [0, 1], X) is the compact open (or uniform) topology, and we endow F(X) with the topology inherited from the embedding given above.

103

SCHWARTZMAN ASYMPTOTIC CYCLE AND DYNAMICS

Lemma B.2.4. The map φt 7→ S(φt ) is upper semicontinuous on F(X). This means that for each open subset U ⊆ H1 (X; R), the set {φt ∈ F(X) | S(φt ) ⊂ U } is open in F(X). Proof. Since the topology on C 0 (X ×[0, 1], X) is metrizable, if this were not true we could find an open set U ⊂ H1 (X; R) and a sequence φnt of continuous flows on X converging uniformly to a flow φt , with S(φt ) ⊂ U , and S(φnt ) not contained in U . This means that for each n we can find a probability measure µn on X invariant under φnt and such that its Schwartzman asymptotic cycle S(µn , φnt ) for φtn is not in the open set U . Since M(X) is compact for the weak topology, extracting a subsequence if necessary, we can assume that µn → µ. It is not difficult to show that µ is invariant under the flow φt . We now show that S(µn , φnt ) → S(µ, φt ). This will yield a contradiction and finish the proof because S(µn , φnt ) is in the closed set H1 (X; R)\U for every n, and S(µ, φt ) ∈ U . To show that the linear maps S(µn , φnt ) ∈ H1 (X; R) = H1 (X; R)∗ converge to the linear map S(µ, φt ), it suffices to show that S(µn , φnt )([f ]) → S(µ, φt )([f ]), for every [f ] ∈ [X, T] = H1 (X; Z) ⊂ H1 (X; R) = H1 (X; Z) ⊗ R. Fix now a continuous map f : X → T. Denote by Fn , F : X × [0, 1] → T the maps defined by Fn (x, t) = f (φnt (x)) − f (x)and F (x, t) = f (φt (x)) − f (x). By the uniform continuity of f on the compact metric space X, the sequence Fn converges uniformly to F . Since Fn |X × {0} ≡ 0, if we call F˜n : X × [0, 1] → R the lift of Fn such that F˜n |X × {0} ≡ 0, then the sequence F˜n also converges uniformly to F˜ , which is the lift of F such that F˜ |X × {0} ≡ 0. Since the µn are probability measures, we have Z Z ˜ (x, 1)µn (x) ≤ kF˜n − F˜ k∞ −→ 0. F˜n (x, 1)µn (x) − F X

X

Since µn → µ weakly, we also have Z Z ˜ F˜ (x, 1)µn (x) − F (x, 1)µ(x) −→ 0. X

Therefore S(µn , φnt )([f ]) =

X

R X

F˜n (x, 1)µn (x) → S(µ, φt )([f ]) =

R X

F˜ (x, 1)µ0 (x). 

Definition B.2.5 (Schwartzman unique ergodicity). We say that a flow φt is Schwartzman uniquely ergodic if S(φt ) is reduced to one point. By the computation done above, linear flows on the torus Td are Schwartzman uniquely ergodic. Of course, all uniquely ergodic flows (i.e., flows having exactly one invariant probability measure) are also Schwartzman uniquely ergodic. Moreover, by Corollary B.2.3, any flow topologically conjugate to a Schwartzman uniquely ergodic flow is itself Schwartzman uniquely ergodic.

104

APPENDIX B

Theorem B.2.6. The set S(N ) of Schwartzman uniquely ergodic flows is a Gδ in F(X). Proof. Fix some norm on H1 (X; R). We will measure diameters of subsets of H1 (X; R) with respect to that norm. Fix  > 0. Call U the set of flows φt such that the diameter of S(φt ) ⊂ H1 (X; R) is < . If φ0t ∈ U , we can find U , an open subset of H1 (X; R) of diameter <  and containing S(φ0t ). By the lemma above, the set {φt ∈ F(X) | S(φt ) ⊂ U } is open in F(X), contains φ0t , and T is contained in U . The set of Schwartzman uniquely ergodic flows is then  n≥1 U1/n . Proposition B.2.7. Let φt : X −→ X be a continuous flow on the compact path-connected space X. Suppose that there exist ti ↑ +∞ such that φti −→φ in C(X, X) (with the C 0 -topology). Then, φt is Schwartzman uniquely ergodic. In particular, periodic flows and (uniformly) recurrent flows are Schwartzman uniquely ergodic (in both cases φ = Id). Proof. Fix a continuous map f : X → T. Consider the function F : X × [0, +∞) → T, (x, t) 7→ f (φt (x)) − f (x). We have F (x, 0) = 0 for every x ∈ X. Call F¯ : X × [0, +∞) → R the (unique) continuous lift of F such that F¯ (x, 0) = 0 for every x ∈ X. The definition of the Schwartzman asymptotic cycle gives Z S(µ)([f ]) = F¯ (x, 1) dµ(x), X

for every probability measure invariant under φt . We claim that we have ∀t, t0 ≥ 0, ∀x ∈ X,

F¯ (x, t + t0 ) = F¯ (φt (x), t0 ) + F (x, t)

In fact, if we fix t and we consider each side of the equality above as a (continuous) function of (x, t0 ) with values in R, we see that the two sides are equal for t0 = 0, and that they both lift the function (x, t0 ) 7→ f (φt+t0 (x)) − f (x) = f (φ0t (φt (x)) − f (φt (x)) + f (φt (x)) − f (x) with values in T. By induction, it follows easily that ∀k ∈ N,

F¯ (x, k) =

k−1 X

F¯ (φj (x), 1).

j=0

Therefore, if t ≥ 0 and [t] is its integer part, we also obtain [t]−1

F¯ (x, t) = F¯ (φ[t] (x), t − [t]) +

X

F¯ (φj (x), 1).

(B.1)

j=0

It follows that ∀t ≥ 0, ∀x ∈ X,

|F¯ (x, t)| ≤ ([t] + 1)kF¯ |X × [0, 1]k∞ .

(B.2)

105

SCHWARTZMAN ASYMPTOTIC CYCLE AND DYNAMICS

By compactness, kF¯ |X × [0, 1]k∞ is finite. If we integrate equality (B.1) with respect to a probability measure µ on X invariant under the flow φt , we obtain Z Z Z F¯ (x, t) dµ(x) = F¯ (x, t − [t]) dµ(x) + [t] F¯ (x, 1) dµ(x). X

X

X

Therefore we have Z S(µ)([f ]) = lim

t→+∞

X

F¯ (x, t) dµ(x). t

(B.3)

Suppose now that we set γx (s) = φs (x); we have F¯ (x, t) = V(f γx |[0, t]). Fix now some point x0 ∈ X, and consider ti → +∞ such that φti → φ in the C 0 topology. Since F¯ (x0 , t)/t is bounded in absolute value by 2kF¯ |X × [0, 1]k∞ for t ≥ 1, extracting a subsequence if necessary, we can assume that F¯ (x0 , ti )/ti → c ∈ R. If x ∈ X, we can find a continuous path γ : [0, 1] → M with γ(0) = x0 and γ(1) = x. The map Γ : [0, 1] × [0, t] → T, (s, s0 ) → φs0 (γ(s)) is continuous; therefore we can lift it to a continuous function with values in R, and this implies the equality V(Γ|[0, 1] × {0}) + V(Γ|{1} × [0, t]) − V(Γ|[0, 1] × {1}) − V(Γ|{0} × [0, t]) = 0. This can be rewritten as V(f γx |[0, t]) − V(f γx0 |[0, t]) = V(f φt γ) − V(f γ), which translates to F¯ (x, t) − F¯ (x0 , t) = V(f φt γ) − V(f γ). Since φti → φ uniformly, by continuity of V, the left-hand side remains bounded as t = ti → +∞. It follows that (F¯ (x, ti ) − F¯ (x0 , ti ))/ti → 0. Hence, for every x ∈ X, we also have that F¯ (x, ti )/ti tends to the same limit c as F¯ (x0 , ti )/ti . Since, by (B.2), F¯ (x, t)/t is uniformly bounded for t ≥ 1, by Lebesgue’s dominated convergence we obtain from (B.3) that S(µ)([f ]) = c, where c is independent of the invariant measure µ. This is of course true for any f : X → T. Therefore S(µ) does not depend on the invariant measure µ.  An interesting property of Schwartzman uniquely ergodic flows (which also shows that they have some kind of rigidity) is the following proposition, which follows easily from the definition of Schwartzman unique ergodicity and what we remarked above about the asymptotic cycles of fixed and periodic points (see also [74]). Proposition B.2.8. Suppose that φt is a Schwartzman uniquely ergodic flow on X. If there exists either a fixed point or a closed orbit homologous to zero, then all closed orbits are homologous to zero. In the remaining case, if C1 and C2 are closed orbits with periods τ1 and τ2 , then Cτ11 and Cτ22 are homologous. Since [C1 ] and [C2 ] are in H1 (X; Z), it follows in this case that the ratio of the periods of any two closed orbits must be rational. Consequently, for any continuous family of periodic orbits of φt , all orbits have the same period.

106

APPENDIX B

Definition B.2.9 (Schwartzman strict ergodicity). We say that a flow φt is Schwartzman strictly ergodic if it is Schwartzman uniquely ergodic and it has an invariant measure µ of full support (i.e., µ(U ) > 0 for every nonempty open subset U of X). We call the Schwartzman class of φt the unique element in S(φt ). Remark B.2.10. Linear flows on the torus Td are Schwartzman strictly ergodic (they preserve the Lebesgue measure). Of course, all strictly ergodic flows (i.e., flows having exactly one invariant probability measure, for which the support is full) are also Schwartzman strictly ergodic. A minimal flow which is Schwartzman uniquely ergodic is in fact Schwartzman strictly ergodic (because all invariant measures have full support). Moreover, any flow topologically conjugate to a Schwartzman strictly ergodic flow is also Schwartzman strictly ergodic. In particular, it is possible to prove the following action-minimizing properties of Schwartzman strictly ergodic Lagrangian graphs of a Tonelli Hamiltonian H (see [40] for more details). More specifically, we say that an invariant Lagrangian graph Λ is Schwartzman strictly ergodic for H if it is invariant for the H Hamiltonian flow ΦH t , and the restriction of Φt on Λ is Schwartzman strictly ergodic. Theorem B.2.11. Let us consider H : T ∗ M −→ R a Tonelli Hamiltonian and let L : T M −→ R be its associated Lagrangian. Let Λ be a Schwartzman strictly ergodic Lagrangian graph with Schwartzman class (or homology class) ρ. The following properties are satisfied:

(i) if Λ ∩ L Aec 6= ∅, then Λ = L Aec and c = cΛ , where cΛ is the cohomology class of Λ. (ii) the Mather function α is differentiable at cΛ and ∂α(cΛ ) = ρ. Therefore, (iii) any invariant Lagrangian graph of H that carries a measure with rotation vector ρ is equal to the graph Λ; (iv) any invariant Lagrangian graph of H is either disjoint from Λ or equal to Λ.

Bibliography [1]

Ian Agol. Criteria for virtual fibering. J. Topol., 1(2):269–284, 2008.

[2]

Marie-Claude Arnaud. Fibrés de Green et régularité des graphes C 0 lagrangiens invariants par un flot de Tonelli. Ann. Henri Poincaré, 9(5):881–926, 2008.

[3]

Vladimir I. Arnol0d. Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Uspehi Mat. Nauk, 18(5 (113)):13–40, 1963.

[4]

Vladimir I. Arnol0d. Instability of dynamical systems with many degrees of freedom. Dokl. Akad. Nauk SSSR, 156:9–12, 1964.

[5]

Vladimir I. Arnol0d. Mathematical methods of classical mechanics, volume 60 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition.

[6]

Serge Aubry. The twist map, the extended Frenkel-Kontorova model and the devil’s staircase. Phys. D, 7(1-3):240–258, 1983. Order in chaos (Los Alamos, N.M., 1982).

[7]

Serge Aubry and P. Y. Le Daeron. The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states. Phys. D, 8(3):381– 422, 1983.

[8]

J. M. Ball and V. J. Mizel. One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. Arch. Rational Mech. Anal., 90(4):325–388, 1985.

[9]

Augustin Banyaga. The structure of classical diffeomorphism groups, volume 400 of Mathematics and Its Applications. Kluwer Academic Publishers Group, Dordrecht, 1997.

[10] Patrick Bernard. Connecting orbits of time dependent Lagrangian systems. Ann. Inst. Fourier (Grenoble), 52(5):1533–1568, 2002. [11] Patrick Bernard. Existence of C 1,1 critical sub-solutions of the HamiltonJacobi equation on compact manifolds. Ann. Sci. École Norm. Sup. (4), 40(3):445–452, 2007.

108

BIBLIOGRAPHY

[12] Patrick Bernard. Smooth critical sub-solutons of the Hamilton-Jacobi equation. Math. Res. Lett., 14(3):503–511, 2007. [13] Patrick Bernard. Symplectic aspects of Mather theory. Duke Math. J., 136(3):401–420, 2007. [14] Patrick Bernard. The dynamics of pseudographs in convex Hamiltonian systems. J. Amer. Math. Soc., 21(3):615–669, 2008. [15] Patrick Bernard. Young measures, superposition and transport. Indiana Univ. Math. J., 57(1):247–275, 2008. [16] Patrick Bernard. Introduction to weak KAM theory. Course at Ecole d’Automne sur la Theorie KAM faible, 19-23/10/2009, Centre International de Rencontres Mathématiques, Luminy (France), 2009. [17] Patrick Bernard and Gonzalo Contreras. A generic property of families of Lagrangian systems. Ann. of Math. (2), 167(3):1099–1108, 2008. [18] Patrick Bernard and Joana Oliveira dos Santos. A geometric definition of the Mañé-Mather set and a theorem of Marie-Claude Arnaud. Math. Proc. Cambridge Philos. Soc., 152(1):167–178, 2012. [19] Patrick Bernard, Vadim Kaloshin, and Ke Zhang. Arnold diffusion in arbitrary degrees of freedom and 3-dimensional normally hyperbolic invariant cylinders. Preprint, 2011. [20] Leo T. Butler and Gabriel P. Paternain. Collective geodesic flows. Ann. Inst. Fourier (Grenoble), 53(1):265–308, 2003. [21] Leo T. Butler and Alfonso Sorrentino. Weak Liouville-Arnol0 d theorems and their implications. Comm. Math. Phys., 315(1):109–133, 2012. [22] Ana Cannas da Silva. Lectures on symplectic geometry, volume 1764 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2001. [23] Mario J. Dias Carneiro. On minimizing measures of the action of autonomous Lagrangians. Nonlinearity, 8(6):1077–1085, 1995. [24] Chong-Qing Cheng. Arnold diffusion in nearly integrable Hamiltonian systems. Preprint, 2012. [25] Chong-Qing Cheng and Jun Yan. Existence of diffusion orbits in a priori unstable Hamiltonian systems. J. Differential Geom., 67(3):457–517, 2004. [26] Gonzalo Contreras, Jorge Delgado, and Renato Iturriaga. Lagrangian flows: The dynamics of globally minimizing orbits. II. Bol. Soc. Brasil. Mat. (N.S.), 28(2):155–196, 1997.

BIBLIOGRAPHY

109

[27] Gonzalo Contreras and Renato Iturriaga. Global minimizers of autonomous Lagrangians. 22o Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium]. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999. [28] Gonzalo Contreras, Renato Iturriaga, Gabriel P. Paternain, and Miguel Paternain. Lagrangian graphs, minimizing measures and Mañé’s critical values. Geom. Funct. Anal., 8(5):788–809, 1998. [29] Gonzalo Contreras, Leonardo Macarini, and Gabriel P. Paternain. Periodic orbits for exact magnetic flows on surfaces. Int. Math. Res. Not., (8):361– 387, 2004. [30] Gonzalo Contreras and Gabriel P. Paternain. Connecting orbits between static classes for generic Lagrangian systems. Topology, 41(4):645–666, 2002. [31] Nathan M. Dunfield and William P. Thurston. The virtual Haken conjecture: Experiments and examples. Geom. Topol., 7:399–441, 2003. [32] Albert Fathi. Structure of the group of homeomorphisms preserving a good measure on a compact manifold. Ann. Sci. École Norm. Sup. (4), 13(1):45– 93, 1980. [33] Albert Fathi. Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math., 325(6):649–652, 1997. [34] Albert Fathi. Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sér. I Math., 324(9):1043–1046, 1997. [35] Albert Fathi. Orbites hétéroclines et ensemble de Peierls. C. R. Acad. Sci. Paris Sér. I Math., 326(10):1213–1216, 1998. [36] Albert Fathi. Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math., 327(3):267–270, 1998. [37] Albert Fathi. Weak KAM theorem and Lagrangian dynamics. 10th Preliminary version, 2009. [38] Albert Fathi. On existence of smooth critical subsolutions of the HamiltonJacobi equation. Publ. Mat. Urug., 12:87–98, 2011. [39] Albert Fathi, Alessio Figalli, and Ludovic Rifford. On the Hausdorff dimension of the Mather quotient. Comm. Pure Appl. Math., 62(4):445–500, 2009. [40] Albert Fathi, Alessandro Giuliani, and Alfonso Sorrentino. Uniqueness of invariant Lagrangian graphs in a homology class or a cohomology class. Ann. Sc. Norm. Super. Pisa Cl. Sci., VIII(4):659–680, 2009.

110

BIBLIOGRAPHY

[41] Albert Fathi and John N. Mather. Failure of convergence of the Lax-Oleinik semi-group in the time-periodic case. Bull. Soc. Math. France, 128(3):473– 483, 2000. [42] Albert Fathi and Antonio Siconolfi. Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation. Invent. Math., 155(2):363–388, 2004. [43] Giovanni Forni and John N. Mather. Action minimizing orbits in Hamiltonian systems. In Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991), volume 1589 of Lecture Notes in Math., pages 92–186. Springer, Berlin, 1994. [44] Mark I. Freidlin and Alexander D. Wentzell. Random perturbations of dynamical systems, volume 260 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. SpringerVerlag, New York, 1984. Translated from the Russian by Joseph Szücs. [45] Gustav A. Hedlund. Geodesics on a two-dimensional Riemannian manifold with periodic coefficients. Ann. of Math. (2), 33(4):719–739, 1932. [46] Michael R. Herman. Inégalités “a priori” pour des tores lagrangiens invariants par des difféomorphismes symplectiques. Inst. Hautes Études Sci. Publ. Math., (70):47–101 (1990), 1989. [47] Vadim Kaloshin and Ke Zhang. A strong form of Arnold diffusion for two and a half degrees of freedom. Preprint, 2012. [48] Andrey N. Kolmogorov. On conservation of conditionally periodic motions for a small change in Hamilton’s function. Dokl. Akad. Nauk SSSR (N.S.), 98:527–530, 1954. [49] Nicolas Kryloff and Nicolas Bogoliouboff. La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire. Ann. of Math. (2), 38(1):65–113, 1937. [50] Pierre-Louis Lions, George C. Papanicolaou, and Srinivasa R. S. Varadhan. Homogenization of Hamilton-Jacobi equation. Unpublished preprint, 1987. [51] Ricardo Mañé. On the minimizing measures of Lagrangian dynamical systems. Nonlinearity, 5(3):623–638, 1992. [52] Ricardo Mañé. Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity, 9(2):273–310, 1996. [53] Ricardo Mañé. Lagrangian flows: The dynamics of globally minimizing orbits. Bol. Soc. Brasil. Mat. (N.S.), 28(2):141–153, 1997. [54] Daniel Massart. Normes stables des surfaces. C. R. Acad. Sci. Paris Sér. I Math., 324(2):221–224, 1997.

BIBLIOGRAPHY

111

[55] Daniel Massart. On Aubry sets and Mather’s action functional. Israel J. Math., 134:157–171, 2003. [56] Daniel Massart and Alfonso Sorrentino. Differentiability of Mather’s average action and integrability on closed surfaces. Nonlinearity, 24(6):1777– 1793, 2011. [57] John N. Mather. Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology, 21(4):457–467, 1982. [58] John N. Mather. More Denjoy minimal sets for area preserving diffeomorphisms. Comment. Math. Helv., 60(4):508–557, 1985. [59] John N. Mather. A criterion for the nonexistence of invariant circles. Inst. Hautes Études Sci. Publ. Math., (63):153–204, 1986. [60] John N. Mather. Modulus of continuity for Peierls’ barrier. In Periodic solutions of Hamiltonian systems and related topics (Il Ciocco, 1986), volume 209 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 177–202. Reidel, Dordrecht, 1987. [61] John N. Mather. Differentiability of the minimal average action as a function of the rotation number. Bol. Soc. Brasil. Mat. (N.S.), 21(1):59–70, 1990. [62] John N. Mather. Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z., 207(2):169–207, 1991. [63] John N. Mather. Variational construction of orbits of twist diffeomorphisms. J. Amer. Math. Soc., 4(2):207–263, 1991. [64] John N. Mather. Variational construction of connecting orbits. Ann. Inst. Fourier (Grenoble), 43(5):1349–1386, 1993. [65] John N. Mather. Arnol0 d diffusion. I. Announcement of results. Sovrem. Mat. Fundam. Napravl., 2:116–130 (electronic), 2003. [66] John N. Mather. Total disconnectedness of the quotient Aubry set in low dimensions. Comm. Pure Appl. Math., 56(8):1178–1183, 2003. Dedicated to the memory of Jürgen K. Moser. [67] John N. Mather. Examples of Aubry sets. Ergodic Theory Dynam. Systems, 24(5):1667–1723, 2004. [68] John N. Mather. Arnol0 d diffusion. II. Unpublished manuscript, 2007. [69] Marco Mazzucchelli and Alfonso Sorrentino. Remarks on the symplectic invariance of Aubry-Mather sets. Preprint, 2014.

112

BIBLIOGRAPHY

[70] Jürgen Moser. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962:1–20, 1962. [71] Jürgen Moser. Monotone twist mappings and the calculus of variations. Ergodic Theory Dynam. Systems, 6(3):401–413, 1986. [72] Gabriel P. Paternain, Leonid Polterovich, and Karl Friedrich Siburg. Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory. Mosc. Math. J., 3(2):593–619, 745, 2003. [73] R. Tyrrell Rockafellar. Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J., 1970. [74] Sol Schwartzman. Asymptotic cycles. Ann. of Math. (2), 66:270–284, 1957. [75] Alfonso Sorrentino. On the structure of action-minimizing sets for Lagrangian systems. ProQuest LLC, Ann Arbor, MI, 2008. Thesis (PhD)– Princeton University. [76] Alfonso Sorrentino. On the total disconnectedness of the quotient Aubry set. Ergodic Theory Dynam. Systems, 28(1):267–290, 2008. [77] Alfonso Sorrentino. On the integrability of Tonelli Hamiltonians. Trans. Amer. Math. Soc., 363(10):5071–5089, 2011. [78] Alfonso Sorrentino. A variational approach to the study of the existence of invariant Lagrangian graphs. Boll. Unione Mat. Ital. (9), 6(2):405–440, 2013. [79] Alfonso Sorrentino and Claude Viterbo. Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms. Geom. Topol., 14(4):2383–2403, 2010.

Index α-function, 22, 28, 59, 77, 78 flats (or facets), 32, 66 subdifferential, 30 symplectic invariance, 38 action-minimizing action-minimizing curve, 48 action-minimizing measure, 20 see also invariant probability measure c-action-minimizing measure, 22 c-minimizer, 50 c-regular minimizer, 74 c-semi-static curve, 55 c-static curve, 56 c-time-free minimizer, 50 cohomology class of an action-minimizing measure, 22 homology class of an action-minimizing measure, 26 Tonelli theorem, 48 Arnol0d, Vladimir I., vii Arnol0d diffusion, vii KAM theorem, vii see also KAM theory Liouville-Arnol0d theorem, 92 weak Liouville-Arnol0d theorem, 93 Aubry, Serge, viii, 72 Aubry set, 57, 83 c-static curve, 56 Carneiro’s theorem, 58 graph theorem, 64 relation with flats of the α-function, 66 symplectic invariance, 67 projected Aubry set, 57 quotient Aubry set, 74 β-function, 26, 28 (non)differentiability on surfaces, 96 flats (or facets), 32, 96 integrability and β-function, 96 subdifferential, 30 symplectic invariance, 38

c-minimizer, 50 c-regular minimizer, 74 c-semi-static curve, 55 c-static curve, 56 Carneiro’s theorem, 24, 58 closed measure, 22, 46 see also measure effective Hamiltonian, 23 see also α-function see also Mañé critical value Euler-Lagrange see also Lagrangian Euler-Lagrange equation, 1 Euler-Lagrange flow, 1 Fathi, Albert, viii, x, 51, 57, 66, 72, 76, 80, 85 see also weak KAM theory Hamilton-Jacobi equation, 76 classical solution, 79 classical subsolution, 77 critical subsolution, 78 regularity of critical subsolutions, 85 weak Hamilton-Jacobi theorem, 81 weak KAM solution, 80 weak KAM theorem, 80 weak KAM theory, 76 Hamiltonian, 4 Hamilton equation, 5 Hamilton-Jacobi equation, 76 see also Hamilton-Jacobi equation see also weak KAM theory Hamiltonian flow, 5 Hamiltonian vector field, 5, 90 integrability, 92 see also integrability Mañé Hamiltonian, 5 mechanical Hamiltonian, 5 optical Hamiltonian, 4 see also Tonelli Hamiltonian Riemannian Hamiltonian, 5

114 Hamiltonian (continued) Tonelli Hamiltonian, 4 holonomic measure, 45 see also measure integrability, 92 first integral, 92 integrability and β-function, 96 integral of motion, 92 involution (or commutativity), 92 Liouville-Arnol0d theorem, 92 weak integrability, 93 weak Liouville-Arnol0d theorem, 93 invariant probability measure, 18 see also action-minimizing measure action-minimizing measure w.r.t. rotation vector/homology class, 26 c-action-minimizing measure, 22 closed measure, 22, 46 ergodic action-minimizing measures, 34 holonomic measure, 45 homology class of a measure, 24 see also Schwartzman asymptotic cycle Kryloff-Bogoliouboff theorem, 18 Lagrangian action of a measure, 19 rotation vector, 24 Schwartzman asymptotic cycle, 24, 97 see also Schwartzman asymptotic cycle KAM theory, vii, 8 KAM torus, 8 Kolmogorov-Arnol0d-Moser theorem, vii weak KAM theory, 76 see also weak KAM theory Kolmogorov, Andrey N., vii KAM theorem, vii see also KAM theory Lagrangian, 1 Euler-Lagrange flow, 1 Euler-Lagrange equation, 1 Lagrangian action, 1, 19 Lagrangian graph, 90 see also Lagrangian submanifold Lagrangian submanifold, 90 see also Lagrangian submanifold Mañé Lagrangian, 3 mechanical Lagrangian, 3

INDEX

modified Lagrangian (by closed oneforms), 21, 31 Riemannian Lagrangian, 3 Tonelli Lagrangian, 2 Lagrangian action action of a curve, 1 action of a measure, 19 action-minimizing measure, 20 action-minimizing measure w.r.t. rotation vector/homology class, 26 c-action-minimizing measure, 22 principle of least action, viii semicontinuity of the action functional, 20 Lagrangian submanifold, 90 Lagrangian graph, 90 cohomology class of a Lagrangian graph, 90 Lipschitz Lagrangian graph, 91 Schwartzman class of a Lagrangian graph, 106 Schwartzman strictly ergodic Lagrangian graph, 106 uniqueness results, 106 Liouville class, 91 Legendre transform, 4 see also Legendre-Fenchel transform Legendre-Fenchel Legendre-Fenchel inequality, 6 Legendre-Fenchel transform, 4 Liouville form, 35, 90 see also symplectic form Liouville-Arnol0d theorem, 92 see also integrability Mañé, Ricardo, viii, 3, 47, 51, 57, 58 genericity in the sense of Mañé, 47 Mañé critical value, 23, 51, 59, 77, 78 see also α-function Mañé potential, 51, 84 Mañé set, 55, 57, 83 c-semi-static curve, 55 Carneiro’s theorem, 58 connectedness, 66 non-wandering points, 87 symplectic invariance, 67 Mañé strict critical value, 31 Mather, John N., viii, 2, 17, 46, 51, 56, 72, 73 Mather’s crossing lemma, 64 generic properties of Mather measures, 47 Mather measure, 20, 22, 26

115

INDEX

Mather quotient, 74 Mather set, 23, 57 Carneiro’s theorem, 24, 58 cohomology class, 23 graph theorem, 23, 27, 64 projected Mather set, 23, 27 rotation vector or homology class, 27 symplectic invariance, 36 Mather’s α-function, 22, 59, 77, 78 see also α-function Mather’s β-function, 26 see also β-function Mather’s pseudometric, 73 Maupertuis’ principle, viii see also principle of least action measure action-minimizing measure, 20 c-action-minimizing measure, 22 closed measure, 22, 46 holonomic measure, 45 homology class, 24 invariant probability measure, 18 Kryloff-Bogoliouboff theorem, 18 Lagrangian action of a measure, 19 rotation vector, 24 Schwartzman asymptotic cycle, 24, 97 see also Schwartzman asymptotic cycle Moser, Jürgen, vii KAM theorem, vii see also KAM theory Peierls’ barrier, 72, 84 principle of least action, viii see also Maupertuis’ principle Schwartzman, Sol, 97 Schwartzman asymptotic cycle, 97 asymptotic cycle of a measure, 102 dynamical properties, 99 examples, 101 semicontinuity, 103 Schwartzman strict ergodicity, 94, 106 Schwartzman strictly ergodic Lagrangian graph, 106 Schwartzman unique ergodicity, 103 simple pendulum, 39 α-function, 44 Action-minimizing measures, 40

Aubry sets, 70 β-function, 44 Mañé sets, 70 Mather sets, 44 subdifferential, 30 see also α-function see also β-function symplectic Liouville form, 35, 90 Poisson bracket, 92 symplectic form, 35, 89 symplectic gradient, 6, 90 see also Hamiltonian vector field symplectic invariance α- and β-functions, 38 Aubry set, 67 Mañé set, 67 Mather set, 36 symplectomorphism, 35 tautological form, 35, 90 tautological form, 35, 90 see also Liouville form Tonelli, Leonida Tonelli Hamiltonian, 4 see also Hamiltonian Tonelli Lagrangian, 2 see also Lagrangian Tonelli theorem, 48 virtual Haken conjecture, 94 weak integrability, 93 see also integrability weak Liouville-Arnol0d theorem, 93 weak KAM theory, 76 see also Hamilton-Jacobi equation 81 Aubry set, 83 calibrated curve, 79 conjugate solutions, 81 critical subsolution, 78 dominated function, 77 Mañé potential, 84 Mañé set, 83 Peierls’ barrier, 84 regularity of critical subsolutions, 85 uniqueness set of solutions, 81 viscosity semi-distance, 84 weak Hamilton-Jacobi theorem, 81 weak KAM solution, 80 weak KAM theorem, 80

Mathematical Notes 34. Lie Groups, Lie Algebras, and Cohomology, by Anthony W. Knapp 35. Plateau’s Problem and the Calculus of Variations, by Michael Struwe 36. Casson’s Invariant for Oriented Homology 3-Spheres: An Exposition, by Selman Akbulut and John D. McCarthy 37. Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability, by Daryl Geller 38. Several Complex Variables: Proceedings of the Mittag-Leffler Institute, 1987–1988, by John E. Fornaess 39. D-modules and Spherical Representations, by Frédéric V. Bien 40. Elliptic Curves, by Anthony W. Knapp 41. Geometry on Poincaré Spaces, by Jean-Claude Hausmann and Pierre Vogel 42. Lectures on Hermite and Laguerre Expansions, by Sundaram Thangavelu 43. The Classical and Quantum 6j-symbols, by J. Scott Carter, Daniel E. Flath, and Masahico Saito 44. The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds, by John W. Morgan 45. Blow-up Theory for Elliptic PDEs in Riemannian Geometry, by Olivier Druet, Emmanuel Hebey, and Frédéric Robert 46. Quadrangular Algebras, by Richard M. Weiss 47. Diffusion, Quantum Theory, and Radically Elementary Mathematics, edited by William G. Faris 48. Thurston’s Work on Surfaces, by Albert Fathi, François Laudenbach, and Valentin Poénaru 49. Hodge Theory, edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê D˜ ung Tráng. 50. Action-minimizing Methods in Hamiltonian Dynamics, by Alfonso Sorrentino. 51. Complex Ball Quotients and Line Arrangements in the Projective Plane, by Paula Tretkoff.