Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors in Dynamics 0521475724, 9780521475723

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CHAPTER

1

EXAMPLES OF HOMOCLINIC ORBITS IN DYNAMICAL SYSTEMS

We discuss a number of dynamical systems with (transverse) homoclinic orbits, just to have some examples in order to motivate the following chapters. First we need some definitions. We deal with diffeomorphisms cp:M - M of a compact manifold to itself. In this chapter it is enough to assume that cpis of class C1, but for some of the later results we need cpto be C 2 or C 3 . Also the compactness of M is not always needed-some of the examples in this chapter will be on R.2 • We say that p E M is a hyperbolic fixed point of cp if cp(p) = p and if (dcp)phas no eigenvalue of norm 1. For such a hyperbolic fixed point, one defines the stable and the unstable manifold as W 8 (p) = {x E M)cp\x) - p for i - +oo}

and W"(p)

= {x E M)cpi(x) -

p for i -

-oo}.

According to the invariant manifold theorem (see [HPS,1977] and also Appendix 1) both W 8 (p) and W"(p) are injectively immersed submanifolds of M, are as differentiable as cp, and have dimensions equal to the number of eigenvalues of (dcp)pwith norm smaller than 1, respectively bigger than l. One can give the corresponding definitions for periodic points, i.e. fixed points of some power of cp. If cp:R_n - R_n is a linear map with no eigenvalues of norm 1, then the origin O is a hyperbolic fixed point and W 8 (0), W"(O) are complementary linear subspaces: R.n = W 8 (0) EBW"(O). We say that if p is a hyperbolic fixed point of cp, q is homoclinic to p if p -1-q E W 8 (p) n W"(p), i.e. if q -1-p and if Hm t.pi(q) = p (this last z--,±oo

form of the definition makes clear why Poincare called such points "biasymptotique"). We say that q is a transverse homoclinic point if W 8 (p) and W"(p) intersect transversally at q, i.e. if

It is clear that linear diff eomorphisms have no homoclinic points .

12

§1

1 Examples of Homoclinic Orbits in Dynamical Systems

Homoclinic

orbits in a deformed

linear map

2 2 We start with the linear map cp:JR. --> JR., cp(x,y) = (2x,½y). The stable manifold is the y-axis, the unstable manifold is the x-axis. Next consider 2 2 --> JR. is a diffeomorphism of the form the composition 1llo . and a with 0 < I.XI < 1 < lal. For simplicity we assume that O < >.< 1 < a. From the theory of hyperbolicity (see Appendix 1) we know that: - the stable and unstable separatrices of p, w•(p) and wu(p), are 02, - there are 0 1 linearizing coordinates in a neighbourhood U of p, i.e . 0 1 coordinates x 1 , x2 such that p = (0, 0) and such that cp(x 1 , x 2 ) = (>.· X1,a· x2). See Figure 2.1. This linearization follows at once from the existence of cp-invariant stable and unstable foliations near p which are of class 01; see also [H,1964]. We assume that W 8 (p) and wu(p) have points of transverse intersection different from p---such points, or their orbits, are called homoclinic or bi-asymptotic top. In the two-dimensional situation we consider mainly primary homoclinic points in order to simplify the figures and the geometric

2.1 Description of the situation

19

Figure 2.1 arguments. A homoclinic point q is primary if the arcs l", joining p and q in wu, and f_S,joining p and q in w•, form a double point free closed curve. Note that whenever p has homoclinic points, it has primary homoclinic points-if all intersections of wu(p) and W"(p) are transverse, then the number of primary homoclinic orbits is finite. See Figure 2.2. We also note that the notion of primary homoclinic orbit does not extend to dimensions greater than 2. Still the results which we discuss below extend without much difficulty to the nonprimary or n-dimensional case .

primary homoclinic points

nonprimary homoclinic points (encircled)

Figure 2.2 Let the linearizing coordinates be defined on U and let their image be the square ( -1, +1) x ( -1, + 1) c IR2 • We consider extensions of the domain of definition of these linearizing coordinates. Identifying points in U with the corresponding points in IR.2 , we have: if cp-1 ([>.,1) x ( -1, + 1)) n U = ¢>,we

20 2 Dynamical Consequences of a Transverse Homoclinic Intersection

can extend the domain of the linearizing coordinates x 1 , x 2 to r.p-1 ([>.,1) x (-1, +1)) using the formulas X1 = >.-1 . (x1 or.p), X2 = a- 1 · (x2 or.p). Repeating this construction one can extend the linearizing coordinates along any segment in W 8 (p) starting in p: one only has to take the original domain U sufficiently small. This follows from the fact that ws(p) has no self intersections. In the same way one can extend the domain of these linearizing coordinates along the unstable separatrix wu(p). Homoclinic intersections, however, form an obstruction to a simultaneous extension of such coordinates along both the stable and the unstable separatrix. In our situation where q is a primary homoclinic point, we extend the linearizing coordinates along both f" and £8, the arcs in wu(p) and W 8 (p) joining p and q; however, these coordinates will be bi-valued near q as indicated in Figure 2.3.

shaded area : bi-valued linearizing coordinates Figure 2.3 Figure 2.4 shows the situation in IR.2 : the shaded area denotes the two sets corresponding to the above neighbourhood of q. Now we consider in the domain of the extended coordinates a rectangle R = {-a ::S:x1 ::S:b, -a ::S:X2 ::S:,6}, a, b, a, ,6 > 0, containing gs, the arc in W 8 (p) joining p and q, and such that for some N - Rn r.pn(R)consists of one rectangle containing p for 0 :=:;:n < N, - Rn r.pN(R) consists of two connected components, one containing q, as indicated in Figure 2.5, i.e.

= b, -a

::;: x2 ::;:,6} n r.pN (R)

= , r.pN( { -a ::;:x1 ::;:b, x2 = ,6}) n R = ..

{ x1

2.1 Description of the situation

21

p

Figure 2.4 For what follows it is important that one can choose R so that N is arbitrarily big: just take ,6 small. By taking ,6 small and hence N big, r.pN(R) will become a very narrow strip around f". So, transversality of wu(p) and W 8 (p) at q implies transverse intersection of the sides of Rand S1 =>S2 =>· · ·: So = Qo; S1 is the vertical strip W(Q_1) n So; S 2 2 is the vertical strip '11 (Q_2 ) n S1; ... ; Boo= S;.

PROOF:

n

For each point r E S00 , w- i(r) E Q_i, i ~ 0. Horizontal strips are similarly defined and we have horizontal strips T1 ::> T 2 =>T3 =>· · · such that for r E T 00 = wi(r) E Q; for all i ~ l. Now

nT;,

i>l

S00 n T00 =f:.¢. Otherwise, for some io, S~ n T;0 = ¢, but Sio contains a lin e from side (1,2) to the side (3,4) and T;0 contains a line from (1,4) to (2,3). These lines have to intersect.

2.3 The maximal invariant subset of R

For any point r E S00 follows that r E A.

nT

00 ,

23

wi(r) E Qi for all i E Z. From this, it also D

shaded area : vertical strip Figure 2.7

§3 The maximal invariant subset of and invariant foliations

R -

hyperbolicity

In this section we impose more conditions on W = cpN restricted to R. In the linearizing coordinates on a neighbourhood of es,the arc in W 8 (p) joining p and q, we have

see Section 1. We only have to describe 'IJ!in those points of R which are mapped back into R, i.e. in w-1 (R) n R. In the component of 111-1 (R) n R containing p and q, W is linear and in fact w(x1, x2) = (.XNX1, aNx2) with 0 < .X< 1 < a.

w• (p)

Figure 2.8

24 2 Dynamical Consequences of a Transverse Homoclinic Intersection The other component of w-1(R) n R is mapped to the component of RnW(R) containing q. This component of RniI!(R) is the region where the linearizing coordinates, constructed in Section 1, were bi-valued, or rather where we have apart from the linearizing coordinates following W 5 (p), and which are in Figure 2.8 the Cartesian coordinates of the plane, also the linearizing coordinates following wu(p). We denote by e 1,e 2 the coordinate vector fields of the linearizing coordinates following ws(p) and by e;, e2 the coordinate vector fields of the linearizing coordinates following wu(p); see Figure 2.9.

ws(p) Figure 2.9 For r in the component of Rnw- 1(R) of q, we have (d1l!)e1(r) = >..N • e~(w(r)) Due to the transversality of wu(p) w(R), for N big, e1 and e2 are linearly and W(R) thin, we may assume that the (or its inverse) is almost constant.

which is mapped on a neigbourhood and (dw)e 2(r) = aN. e2(w(r)). and W 5 (p) and to the thinness of independent. Also, by choosing R matrix transforming e 1, e2 into e;, e 2

THEOREM 1. For R sufficiently thin, and hence N big, the maximal invariant subset A = wn(R) in R is hyperbolic. (The technical definition of

n

nEZ

hyperbolicity is given in Appendix I). PROOF: A continuous cone field Con Rn w(R) is a map which assigns to each r E Rn w(R) a two-sided cone C(r) in Tr(M), given by two linearl y independent vectors W1(r), Wz(r):

Continuity of C means that w1 and w2 depend continuously on r. Let 0 ,. denote the zero vector in Tr(M). An unstable cone field is a continuous con e field on Rn W(R) such that - for each r ER n W(R) n ir,- 1(R), (dw)(C(r))

c Int (C(w(r))) U {Or},

2.3 The maximal invariant subset of R

25

- for each r ER n w(R) n w-1(R) and VE C(r), jjdlll(v)II ~ v llvll, for some v > 1, where both norms are taken with respect to the basis e1, ez. Below we construct such an unstable cone field. From the existence of such a cone field it follows that there is a continuous direction field Eu(r) , wi(R), such that defined for r E

n

- Eu(r) C C(r), - dw maps Eu(r) to Eu(w(r)), - for some v > 1, and all v E Eu(r) with r E

n

wi(R), lldw(v)II ~ v•llull.

Eu, restricted to A, is obtained by taking the intersections of the forward images of the cone field C under dw. Replacing 1lJby ir,- 1, one constructs in the same way a stable cone field and the direction field E 8 , which is invariant under and contracted by dw. Then TA(M) = E'f..EBEA is the required splitting for hyperbolicity-see Appendix 1 and Moser [M,1973a]. Now we come to the construction of the unstable cone field on Rn w(R). In the component of Rn w(R) containing p we simply take cones around e2 extending 45° to both sides. In the other component of Rn w(R) there is (assuming R and w(R) sufficiently thin) an angle a, smaller than 90°, so that for each point r in that component of Rn W(R), the cone around e2(r), extending over an angle a to both sides, contains e2(r) in its interior. This is due to the fact that e 1(r) and e2(r) are linearly independent. The unstable cone field C is just defined as the field of cones, centred on e2 and extending 45°, respectively a, to both sides of e2 depending on the component of Rn w(R). In order to show that this cone field has the required properties, we introduce constants A, B and B' so that: whenever r E Rn w(R) and v =vi· e1(r) + vz · e2(r) E Tr(M) then for v E C(r) we have lv1I :SA· lv2I, on the other hand whenever lv1I :S B · lv2I, then v E C(r); whenever v = v~ • e;(r) + v~ • e2(r) and jv~I :S B' · jv~I, then v E C(r). If N is so -A < min(B, B'), then dw maps cones to the interior of cones. big that Also for N sufficiently big, the lengths of vectors in our cones are strictly increased by diI!. So for N big enough our cone field has the required properties. But the cone field was constructed after choosing N, Wis defined in terms of N(w = cpN), and the domain of the cone field is defined in terms of W: domain = Rnw(R). However, the way to raise N is to make R thinner. This decreases the domain where the cone field has to be defined. The fact that 1J, changes from cpN to cpN', N' > N, has no influence on the arguments: the vector

I~(

26 2 Dynamical Consequences of a Transverse Homoclinic Intersection fields e1, e2, e~ and e2do not change. So Rand N may be ajusted afterwards. This completes the proof of the hyperbolicity of A. □ Observe that we proved slightly more: there are vector fields eu E eX and es E e11and a constant v > 1, such that for all r E A, l!dlJ!(eu(r))II :::O: v · lleu(r)II and lldlJ!(es(r))II ::; v- 1 · l!es(r)IINow we come to the second subject of this section. The cone fields just constructed will now be used to construct the stable and the unstable foliation. We only describe the construction of the unstable foliation. First we present the definition. An unstable foliation for A

=

nw\R)

is a foliation P ofa neighbourhood

iEZ

of A (here we take Rn IJ!(R)) such that (a) for each r EA, P(r), the leaf of P containing r, is tangent to Eu(r), (b) for each r, sufficiently near A, IJ!(P(r)) :) P(IJ!(r)). We require the tangent directions of leaves of P to vary continuously. CONSTRUCTION

OF THE UNSTABLE FOLIATION:

sitions of R, w(R), and

~

w-1(R)

Rr,

We recall the relative po-

in Figure 2.10.

~

'l'(R)

(R r, q,- 1 (R))I 'I' (R)

Figure 2.10 2

We take a C foliation w- 1(R)) n R so that:

J-u(not

yet the unstable foliation) on (IJ!(R) U

(a) in Rn W(R) the tangent directions of leave are contained in the unstable cones, (b) the images under W of leaves in (Rn w- 1(R)) \ w(R) have tangent directions contained in the unstable cones, (c) the four arcs of aR n w- 1(R) are leaves of arcs denoted by eo, (d) the four arcs of a(w(R)) n Rare arcs is denoted by e1,

leaves of

J-u,the

union of these four

J-u,the

union of these four

2.4 The maximal invariant subset of R - structure

27

(e) w maps leaves of J-unear eo to leaves of J-unear e1. Since all the cones of the unstable cone field are centred around the vertical vector field e2 and contain (where defined) e2, it is clear that such a foliation J::uexists. For foliations as described above we define an operator IJ!* as follows: in (Rn w- 1 (R)) \ w(R), the leaves of J-uand w*(P) are the same; in (Rnw(R)), the leaves of w*(P) are connected components of '11-images of leaves of J-uintersected with (Rn IJ!(R)). Due to the above conditions (c), (d) and (e), w*(P) is also C 2. From the invariant manifold theory it follows that the limit lim w:(J-u) i-+oo

=r

exists. This limit depends on the choice of the "initial foliation" J-u.The limit is C1; if however cp is C 3 then this limit is C 1H"; see Appendix 1. Observe that we can extend our vector fields eu and es in Eu, respectively 5 E , to tangent vector fields of F' and ;:s (F 8 is just an unstable foliation for w- 1) so that for some constant v > 1, and all r ER n w(R) n w- 1(R), lldlJ!(eu(r))JI 2::v · lleu(r)II and

l!dlJ!(es(r))II ::; v- 1 · lles(r)II -

Stable and unstable foliations can be constructed for any basic set of a C 1 diffeomorphism in dimension 2 [M,1973b]. In higher dimensions the existence of such foliations constitutes an interesting open problem except when the basic set is zero-dimensional [P,1983]; for futher details see Appendix 1. We stress that in this comment we are insisting on foliations defined in a neighbourhood of the basic set! Sometimes in the literature one also refers to stable (or unstable) foliation as the set of stable manifolds ("leaves") through the points of the basic set, and in this sense the foliation always exist for any hyperbolic set; see Appendix 1.

§4

The maximal invariant subset of R - structure

We divide A, the maximal invariant subset of R, into blocks. For each sequence A = (a-k, a-k+l, · · · , ak-i, ak) with a; = 0 or 1, we define the Ablock as AA = {r E AIIJ!i(r) E Q; for i = -k, · · · , k }; we call k the radius of A. The diameter of AA is just the supremum of the distances between points in AA, measured in R 2 . As we saw in the last section, expansions and contractions of vectors along unstable, respectively stable, foliations are at least by a factor v > 1,

28 2 Dynamical Consequences of a Transverse Homoclinic Intersection

respectively i;- 1 . Let c be the maximal length of a component of a stable or unstable leaf in Rn IJ!(R). PROPOSITION l. Let AA be an A-block and let A have radius k. Then the diameter of AA is at most 2 · c · i;-k (for the definition of c, ii see above). PROOF: For any two points p,p' in the same component of Rn IJ!(R) there are unique arcs f"'(p, p") and f,8 (p', p") in leaves of P, respectively F 5 , joining p, respectively p', and the intersection p" of the unstable leaf through p and the stable leaf through p'. (Figure 2.11). When p,p' are both in AA, then this whole configuration will remain in the same component of Rn IJ!(R) when we apply IJ!i,i = - k, · · · , +k. This implies that the lengths of eu(p, p") and f 5 (p',p") are both at most c · i;-k_ This implies the proposition. □

p

Figure 2.11 From the above proposition and the result of Section 2, we obtain the following THEOREM l. The size of an A-block AA goes to zero as the radius of A goes to infinity. For each infinite sequence··· ,a-2,a- 1,ao,a1,a2,· ·· ,ai = 0 or 1, there is exactly one point r E A such that IJ!i(r) E fh for all i. There is a homeomorphism h: A - (Z2)Z (product topology on (Z2 )Z) such that forr EA, h(r) = ··· ,a-2,a-1,ao,a1,a2,··· with IJ!i(r) E Q;- Ifa:(Z 2 )Z--+ (Z2)z is the shift operator, i.e. a( { ai};EZ = {a:hEz with a: = ai+l, then WIA

---> a --->

commutes.

REMARK 1: It follows from the above theorem and its proof that if ii, is C 1 -close to IJ!, then the same conclusions hold for the maximal invariant

2.4 The maximal invariant subset of R - structure

29

set A of ii, in R. Namely, if C is an unstable cone field for IJ!whose domain is slightly extended beyond Rn IJ!(R), we conclude that C is also an unstable cone field for 111if 111is sufficiently C 1 close to IJ!. Then all the above arguments apply, with the obvious modifications, to ll!. This implies that for such ii, and A,there is a conjugacy H: A - A, i.e. a homeomorphism such that the diagram below commutes: WIA

--->

REMARK 2: The periodic points are dense in A. This follows from the corresponding statement for (Z2 )Z and a: any finite sequence can be completed to an infinite periodic sequence. It is clear that all these periodic points are of saddle type (one expanding and one contracting direction). We add one more observation about these periodic orbits to be used in the next chapters. In general we say that a fixed point p of a diffeomorphism cp is dissipative if ldet(dcp)(p)I < 1. The same applies to periodic points, say of period k: just replace

.n . a2n _ an, x =an· (x - 1), 'fl = a2n . y _ an.

Note that these are not yet the final reparametrizations and coordinate transformations but the final ones differ from these just by constant factors,

3 Homoclinic Tangencies

50

as made explicit at the end of the proof. Also, a and .X depend on µ and hence on µ although this is not expressed in the above formulas. Finall y, note that a fixed box in the (x, y)-coordinates, gives for n --+ oo boxes converging to q in the (x, y)-coordinates. {.x = O}

a

p

\~

'

n+N( ({)ii= 0

{x = O})

Figure 3.18

cp:+N

We now start with our main calculation: expressing in terms ofµ, x and y. See Figure 3.18. Let (µ,x, y) denote a point. The (µ, x, y) variables of this point are (see above)

µ=a -2n -µAfter applying change):

1

\n+ a -n ,

cp:to X

Next we apply

·,I\

this point we get as (x, y)-coordinates

(µ does not

= _xn· (1 + a-nx),

cp:and

find

x = 1 +a· a-n · y + Hi(µ, _xn· (1 + a-nx), a-n · y), y = /3. a-2n. 11 2 + (a-2nµ _ 1 . _xn+ a-n)

+ 1. _xn.(1 + a-nx) + iI2(µ, _xn.(1 + a-n), a-n. y).

Transforming this back to the (x, y)-coordinates, and denoting the values of these coordinates of the new point by x,y we have

= x=ay+a = r.r.;2 Y = /JY

+a-

2n

n · H-1 (a -2n -µ-"(·/\ -

+-µ + 'Y · \n ,I\

· H-2 (a -2n

•

n

a ·x

·µ-"(·/\

\ n + a -n \ n (1 + a -n-) x ,a -n ·y-) ,,I\·

-

\n

\ (1 + 0- -n-)X ,a--n +a--n ,/\·

-) ·y.

(2)

3.4 Homoclinic tangencies, scaling and quadratic maps

51

Next, we need to show that in the above expression certain parts converge to zero for n --+ oo in the C 2 topology (uniformly on compacta in the (µ, x, y)-coordinates). In the expression for y, the term 1 · _xn· o-n· x goes clearly to zero because .Xa < l. The terms involving H; are more complicated. We first observe that when

(µ,x,y) remains bounded, the corresponding values of

(µ, x, y which are substituted in

-

1)

H;satisfy µ X

= O(a-n) = O(.Xn)

y - 1 = O(o--n)

(3)

as n goes to infinity. Next we define

Then

which converges to zero for n --+ oo. Next, the first and second order derivatives of H 1 (µ,x,y) converge to zero (uniformly on compacta); this follows from (1), (3) and O < >..a< l. In fact, the derivatives of H 1 are easier to estimate than H 1 itself. This is the way in which one proves that H 1 goes to zero as announced. The same procedure works for the corresponding expression in the formula for y. So, for n --+ oo, the transformation formulas (2) converge to

By the substitution

µ = 13-1µ,

x = af3-1x, y = 13-1f;,

52

3 Homoclinic Tangencies

this limiting transformation

becomes

Now the theorem is proved: we have the announced transformation as limit of ,p~+N, composed with suitable coordinate transformations and □ reparametrizations of µ. We observe that the calculation above, under stronger hypotheses , was carried out independently in [TY, 1986]. 1: One can also consider the effect of our (rescaling) coordinat e transformations on stable and unstable foliations. Let F! and I'µ be stable 5 and unstable foliations defined in a neighbourhood of Pµ,: i.e. W (pµ,) is a leaf of F!, 'Pµ,maps leaves of F! into such leaves _or ~aps the~ outside the domain of definition of F!, and the tangent d1rect10ns of Fµ, depend continuously on x, y, and µ; similarly for Pµ. Then we can extend F! by o)f negative iterates of 'Pµ,and I'µ by positi~e iterate~ of 'Pµ,till the dofmWai~(s definition of both these foliations contam the pomt of tangency o Po and W 5 (po) in their interior. Next we consider these ~oliations with respect to the (x, y, µ)-or the (x,y, µ,)-coordinates (dep~ndmg on n). _Itturns 0 t by estimates which are much like the above estimates, that m these u , 2 I coordinates F 5 and F'!: converge in the C topology for n --> oo. n the - y- µ-)-coordinates the limiting leaves of :F'!:consist of horizontal curves. (x, ' ' µ, - -2 . the limiting leaves of :F'!:consist of the parabolas {y = x + a\a E JR}. This topic is discussed in Ch;pter 6, at the end of Appendix 1 and in Appendix 4. REMARK

REMARK

2: If the maps ,pr;,t:{µ.) are not orientation preserving, then we may

,p~:1:i)

which is orientable: after reparame trizreplace them by the square ing according to Wn,µ, these maps converge to ((p,,)2,and, due to renorm alization, we know that ((p,,)2 has, after rescalingµ, and x, the same proper ties as (p,,.

CHAPTER

4

CANTOR SETS IN DYNAMICS FRACTAL DIMENSIONS

AND

As already indicated in earlier chapters, the closure of a set of homoclinic intersections is often a Cantor set. In the following chapters, concerning the study of homoclinic bifurcations, we shall have to impose, in the formulations of several results, conditions on such Cantor sets. These conditions will involve numerical invariants like Hausdorff dimension, limit capacity, (local) thickness and denseness, which we discuss in this chapter. When these invariants are nonintegers, one speaks of sets of fractal dimensions; often nowadays the name fractal is associated to sets whose topological dimension is smaller than their Hausdorff dimension, like the Cantor sets we deal with here. Since these Cantor sets are not of the most general type, we begin our discussion in the present chapter with the description of "dynamically defined (or regular) Cantor sets"; they form the class of Cantor sets in which we are mainly interested in Dynamical Systems. We prove several results concerning the relations between Hausdorff dimension, limit capacity and (local) thickness and denseness and show that they vary continuously with the maps defining the Cantor sets. These Cantor sets have Hausdorff dimension smaller than 1, and thus their Lebesgue measure is zero. This is not in general the case when the surface diffeomorphism giving rise to the Cantor set is of class C 1 , as shown by a counterexample due to Bowen and presented here. Apart from these results that are needed in the next chapters, we derive some additional information concerning the regularity of these Cantor sets, which in our view is of independent interest and bound to play a role in dynamics. We observe that many of the results in this chapter concern Cantor sets that arise from hyperbolic invariant sets of two-dimensional diffeomorphisms. For hyperbolic sets in higher dimensions, our present knowledge is rather more limited: it is not even known if the definition of local Hausdorff dimension is independent of the initial point.

§1

Dynamically

,p:

defined Cantor sets

Let M--> M be a diffeomorphism of class C 3 of a 2-manifold M with a hyperbolic fixed point p of saddle type which is part of a (nontrivial) basic set A. (Actually, we will see that it is enough to take ,p of class C2; see

54

4 Cantor Sets in Dynamics and Fractal Dimensions

the remark at the end of this section). By a basic set we mean a comp act hyperbolic invariant set with a dense orbit, whose periodic orbits are dense and which is the maximal invariant set in a neighbourhood of it. Nontri vial here means that it does not consist of (finitely many) periodic orbits. As an example, one may think of the "maximal invariant subset of R" rela t ed to a transversal homoclinic orbit, as analysed in Chapter 2. For p E A, the sub set An W 8 (p) of W 8 (p) is what we want to call a dynamically defined Cantor W 8 (p) be a smooth identification , su ch set. To be more precise, let et : 1R----> 1 8 that n- o ( 1, then one of the following three alternatives occurs: Ki is conta ined in a gap of K2; K2 is contained in a gap of Ki; Kin K2 =/=¢. PROOF: We assume that neither of the two Cantor sets is contained in a gap of the other and we assume that Ki nK2 = ¢,and derive a contradiction from this. If Ui, U2 are bounded gaps of Ki, K2, we call (Ui, U2) a gap pair if U2 contains exactly one boundary point of Ui (and vice versa); Ui and U2 are said to be linked in this case. Since neither of the Cantor sets is contained in a gap of the other and since they are disjoint, there is a gap pair. Given such a gap pair (Ui, U2) we construct: a point in Ki n K2; or a different gap pair (U{, U2) with f(U{) < f(Ui); or a different gap pair (Ui, U2) with f(U 2) < f(U2). This leads to a contradiction: even if we don't find a point in Kin K2 after apply ing this construction a finite number of times, we get a sequence of gap pairs (U?l, uJi)) such that £(U?l) or f(UJi)) decreases and hence, since the sum of all the lengths of bounded gaps is finite, it goes to zero. Assuming f( ufil) goes to zero, take Qi E ufil: any accumulation point of {qi} belongs to Kin K2. Now we come to the announced construction. Let the relative position of U1 and U2 be as indicated in Figure 4.6.

Figure 4.6 Let Cf and CJ be the bridges of Kj at the boundary points of Uj, j = 1, 2. Since T1 · r2 > 1, ~~~~? . ~i~~\ > 1. So £(Cr) > £(U2) or i(C~) > £(U1), or both. (See Figure 4.7). Therefore the right endpoint of U2 is in Cf or the left endpoint of Ui is in or both. Suppose the first. Let u be the right

ct

64

4 Cantor Sets in Dynamics and Fractal Dimensions

endpoint of U2. If u E K1 then we are done, since u E K2 anyway. If u €/.K 1. then u is contained in a gap U{ of K 1 with e(U{) < e(U1) and (U{, U2) is the c required gap pair. This completes the proof.

Figure 4.7 REMARK 1: Let now 11 and 12 be minimal closed intervals such that K 1 C 11 and K2 C h We say that K1 and K2 are linked if Ii and h are linked. If T(K 1 ) • T(K 2 ) > 1 and if K 1 and K2 are linked, then K 1 n K2 -/- ¢ (since neither can K 1 be contained in a gap of K2 nor K2 in a gap of K 1). Since being linked is an open condition, it follows that whenever T(K 1 ) ·T(K2) > 1. then K 1 - K 2 has interior points. THEOREM l. Let K 1 , K 2 be Cantor sets in JR with Hausdorff dimension h 1, h2. If h 1 + h2 > 1 then (K1 - >.K2) has positive Lebesgue measure for almost every>. E JR (in the Lebesgue measure sense). Before going into the proof of the theorem, we first observe that from the assumption on h 1, h2 it follows that HD(K 1 xK2) 2::HD(K1)+HD(K2) > 1 (see Falconer [F,1985]). Also, let us see how we can state this result in a similar but slightly different way. For >. E JR take 0 E ( -1r /2, +1r/2) such that >.= -tan 0. Let 1redenote the orthogonal projection of JR2 onto the straight line Le which contains ve = (cos 0, sin 0). If we identify JR with Le through JR 3 x 1--+ x · ve then 1re(k) = k · ve = cos 0 · k1 + sin 0 · k2, for k = (k 1, k2) E lR.2 . By our choice of 0 we get 1re(K1 x K2) = cos0(K 1 ->.K2)Since cos 0 -/- 0 this shows that the theorem above can be rephrased in the following (slightly stronger) form. THEOREM 2. Let K C lR.2 be such that H D(K) > 1 and 1re : JR2 -> JR be as above. Then 1re(K) has positive Lebesgue measure for almost every 0 E (-1r /2, +1r/2) (in the Lebesgue measure sense). This result was first proved by Marstrand [M,1954]. The argument that we present here, which uses ideas from potential theory, is due to Kaufman and can be found in Falconer [F,1985].

65

4-2 Numerical invariants of Cantor sets PROOF: Let d

= H D(K)

> l. We first assume that O < md(K) < oo and

that for some C > 0

(1)

2 and Q < r ~ l. Letµ be the finite measure on JR defined by for all x E JR. 2 µ(A) = md(A n K), for A a measurable subset of JR . For -1r /2 < 0 < 1r/2, let us denote by µe the (unique) measure on JR such ~hat J fd~e = JUO 1re)dµfor every continuous function f. The theorem will follow, 1f we show that the support of µ 0 has positive Lebesgue measure for almost all 0 E (-1r/2, 1r/2), since this support is clearly contained in 1re(K). To do this 2

we use the following fact. LEMMA1. Let 17be a finite measure with compact support on JRand fJ(p) = _1_ J+ooe~ixpd1J(x),for p E JR (fJ is the Fourier transform of 17). If O < -./2i -oo 2 dp < \f/(p)\ 00 then the support of 17has positive Lebesgue measure.

f~:

PROOF OF THE LEMMA: The assumption that fJ is square-integrable implies (Pla ncherel's theorem) that d(K). Take co > 0 so that for O < c :s;co N 0 (K) :s;c- /3, i.e. there is a covering of K by not more than c- /3 intervals of length c. For every R E nn, the inverse images by (\J!nlR) of these intervals form a covering of R by intervals of length at most c>.;¾. This means that N 0 »-1 (R) :s;c /3for O < c :s;co, or, in other words , n,R

1

N0 (R) :s;>.~ ~ · c /3for O < c :s;>-;:;,¾·co- Then N 0 (K) :s;E-/3(

L

>.~~) for

RERn 1 all O < E :s;>.;;co, where An = sup An,R· Repeating the argument we get

RERn

for all k ?: 1

N 0 (K)

:s;c- /3(

L

>.~~t

if

O< c

:s;>.;:;-kc 0.

RERn

This implies

and so, making

/3-+ d(K),

Since An > 1 this proves that

L

>.~.~K) ?: 1, that is, d(K) :s;f3n-

RERn

Now we derive a contradiction from the assumption that H D(K) < 0 11 • Take H D(K) < o < o,,. Then there are finite coverings U of K with arb itra ril y small diameter for which Ha(U) is also arbitrarily small . We assum e that every element of U intersects at most one R E R,11 • This will be the case if we require that Ha(U) :s;co for some co = co(n , o) > 0. We denote UR = {U E U I Un R i=-qi}. Let , as above, k ?: 0 be such that

70

4 Cantor Sets in Dynamics and Fractal Dimensions

wk+l(Ki n K) = K for all Ki E R 1 . Then, if H 0 (U) and hence diam(U) is sufficiently small, (wn+k I R)(UR) is a well defined covering of K for all RE Rn. Note that

(since a< O'.n< f3n < ~)- We claim that

for some

.RoE nn. Otherwise

we would have

~ (c- 1 which is a contradiction,

I:

A~,R) · Ho(U)

RE'R,n

since, by assumption, a < an and so

In this way we construct, from the initial finite covering U, a new covering U' = (wn+klRo)(UR0 ), with fewer elements than U and such that H 0 (U1) :S c::0 . Repeating this argument we eventually obtain a covering of K with no elements at all. This is the required contradiction. Finally, to prove that (!3n- an)n -, 0 we first note that, by the bound ed distortion property there is a > 0, such that An,R :S a · An,R, for all n ?: 1 O'.nlog a + log C . and RE Rn. Take 8n =------,where)..= mf 1'11'1 > l. Then - log a + n log )..

'"'

L,_, RERn

)..-(on+On)< a(on+on)'"' A-On . A-On n,R L,_, n,R n,R RE'R,n :S a(on+on).)..-n·On. A~.~f RE'R,n = a(on+6nl. >-.-non. = l,

I:

c

by definition of Dn- It follows that f3n :S O'.n+ Dn, i.e.

an log a+ log C H D(K) · log a+ log C < -- -------. {3n - O'.n< ------ n log )..- log a n log )..- log a

4.2 Numerical invariants of Cantor sets

71

This implies the convergence we have claimed and completes the proof of □ the theorem. The above theorem is a consequence of the regularity of dynamically defined Cantor sets. It makes the propositions on the measure of the difference of two Cantor sets, in terms of limit capacity and Hausdorff dimension, cover, for dynamically defined Cantor sets, almost all cases the exceptions being d(K 1 ) + d(K2) = 1 and K 1 - >-.K2for exceptional values of >-..Before proceeding with our discussion on the relations between the invariants (dimensions) of a Cantor set, let us explore some consequences of the ideas involved in the proof of this theorem. First we recall that in the heuristic proof we have the following formula for the Hausdorff dimension and the limit capacity. If K is an affine Cantor set (see the examples in the previous section) with Markov parthen tition {K 1 , ... , Kk} and )..i denotes the (constant) value of l'11'1K.I, H D(K) = d(K) = d, d being the unique number such that I: >-.;d= l. We use this formula to compute the precise value of H D(K) = d(K) in a particular case. Take K to be an affine Cantor set with Markov partition {K 1 , ... , Kk} such that all the Ki have equal length, say {3 • diam K for O < /3 < l/k. Since we are assuming that K is affine (and not just generalized affine), \JImaps each Kin K onto K, so we must have )..i = {3-1 for all i. Therefore H D(K) = d(K) = log k/ log(/3- 1). For k = 2, since {3 = (l - a)/2, we get the formula stated at the beginning of this section. Incidentally, this shows that the dimension of a dynamically defined Cantor set can take any value between O and l. Also, for any p E (0, 1), there are diffeomorphisms exhibiting a saddle point p and a basic set A with p E A such that HD(A n W 5 (p)) = p. Our second remark concerns the role played by the bounded distortion property. Although we made use of it in the last part of the proof this is not strictly necessary for the theorem above. In fact this result is still true for Cantor sets defined by expanding maps which are only C 1 (and so may not have this property); see [T,1988]. Even more so, if O,

b b f3n- - :S an :S d(K) = H D(K) :S f3n :S O'.n+ -, for all n > l. n n We want to explore some important consequences of this estimate.

(2) First

72

4

Cantor Sets in Dynamics and Fractal Dimensions

observe that, denoting A

= sup IIJ!'Iand d = H D(K) = d(K),

and so ~ L., REnn

>.n,R -d < - Ab < oo , for all n >_l.

(3)

In a similar way,

L A;;,t 2: cA-b

> 0, for all n 2: 1.

(4)

REnn

Using these facts we prove the following proposition. PROPOSITION 3. Let K C IR be a dynamically defined Cantor set and let d = HD(K). Then, 0 < md(K) < oo . Moreover, there is c > 0 such that, for all x E K and O < r '.S1, -1

C

'.S

md(Br(x) d r

n K)

'.SC.

(5)

We point out that the bounded distortion property is fundamental her e: contrary to the theorem above, this last proposition wouldn't hold in general if \JI were only C 1. PROOF: We keep the notations from the proof of the above theorem. Observe that by the mean value theorem and condition (3),

Hd(Rn)

=

L

[f(R)]d '.S

REnn

L (>.;;-}·f(K))

d '.SAb(f(K))d,

RERn

where f(K) denotes the diameter of K. Since diam(Rn) this proves that md(K) '.SAb(f(K)l < oo.

----> 0

as n

----> oc.

Proving that md(K) is positive requires a little more effort. First , we claim that for some a 1 > 0 we have

(f(U)l

2: a1 · ~ L., A-d n,R

(6)

for every interval U intersecting K and n 2: 1 sufficiently large dependin g on U. To show this we fix o > 0 such that the a-neighbourhood of K i::, contained in the domain of \JI. Take k = k(U) 2: 0 minimal such that

4.2 Numerical invariants of Cantor sets

73

Let n > k. Then S E nn-k intersects wk(U) if and only if S = wk(R) for some R. E nn intersecting U. Moreover, in such case we have

An,R= sup l(wn)'IRI 2: inf l(wk)'IRI. sup l(wn-k)'lsl

2: inf l(wk)'luuRI. An-k,S· On the other hand, by the mean value theorem we have

Observe that, by construction, IJ!J(U UR) is contained in the domain of \JI for all O '.Sj '.Sk - l. The bounded distortion property implies that

where a is some positive number independent of U, R and k. From all this and the fact that d 2: On-k, we obtain

I:

(e(U))d 2: c- 1

A;;-~k .s · e(u)d

SEnn-k Snwk(U)#ef>

L

2: c- 1

A~~k,\Jik(R) · (f(wk(U))/ sup l(wk)'luuRl)d

HER" RnU#

L

2: c- 1od

A~~k,\Jik(R).a-d. (inf l(wk)'luuRl)-d

RERn RnU-f-¢,

> _ C

-1

O

~

d -d

a

L., REnn

A-d n.R·

RnU-f-¢,

· w1. ·th a1 = c- 1o da -d . This· proves t he c1aim Let now U be any finite covering of K. Take n 2: 1 such that (6) holds for all U EU. Then, by (4),

Hd(U) = L(f(U))d lfEU

2:

L a1 ( L A;;-,t) UEU

RERn RnUN

2: a 1

L A~.t2: a1CA-b. 1/ERn

Since U is arbitrary, this proves

4

74

Cantor Sets in Dynamics and Fractal Dimensions

Now we deal with the second part of the proposition. To make the argument more transparent we first derive an estimate for the d-measure of the intervals RE Rn. For some a 2 > 1, depending only on Kand Ill, we have (7)

for all R E Rn and n 2 1. To show this we observe that wn-imaps R diffeomorphically onto some K; E R 1. From the definition of Hausdorff measure, we have

On the other hand, by the mean value theorem, we have An-1,R

·

£(R) ::; £(K;) ::; An-1,R

·

£(R).

Finally, by the bounded distortion property, it follows that

with a > 0 as above depending only on K and Ill. From all this we get

Clearly, £(K;) can be uniformly bounded from zero and infinity, so to prove (7) we only need to show that the same holds for md(K; n K). The upper bound is trivial since md(K; n K) ::; md(K) < oo. The lower bound follows easily from the fact that, for some k 2 0, wk+ 1(K; n K) =Kand so, again by the definition of Hausdorff measure,

Now we prove (5). For x E K and O < r ::; 1, we let q = q(x, r) 2 0 be minimal such that where, as ·before, a > 0 is such that the domain of Ill contains the oncighbourhood of K. Then, arguing as above with Br(x) and '11qin the place of Rand wn-1 , respectively, we obtain

a

-d m O for all RE RP. It follows that

□

and this completes the proof of the proposition.

Finally, we prove a two-dimensional version of this proposition, which had been remarked following the proof of Marstrand's theorem relating Hausdorff dimension and measure of the difference set. It applies to hyperbolic basic sets of diffeomorphisms on surfaces. 4. Let K 1, K2 be dynamically defined Cantor sets and Jet di= HD(K1), d2 = HD(K2), d = d1 + d2 and K = K 1 x K 2 in JR.2. Then, for some c > 0, (a) 0 < md(K) < oo, PROPOSITION

(b) c- 1 :=;rnd

(K~Br(x))

:=;c for all x

E K and O < r

:=;1.

Takeµ to be the product measureµ= md, x md2 on K. Clearly, (a) and (b) hold ifwe replace there md byµ. Therefore it is now sufficient to show that µ is equivalent to md in the sense that for all Borel subsets A c K µ(A)/md(A) is bounded away from zero and infinity. We consider Markov' partitions R1,R2 for K1,K2 respectively and denote by Rf, i = 1,2, the PROOF:

76

4

Cantor Sets in Dynamics and Fractal Dimensions

family of connected components of ourselves to Borel sets of the form

w;(n-1\Lj),

LJ E R;. We may restric t

since these sets generate the Borel a-algebra of K. Let U = {Ui,j x U2.j l ::; j :S m} be any finite covering of A = R 1 x R 2 by cubes. Fix x;, 1 E U;.J n R ; . i = 1, 2, 1 :S j :Sm (obviously, we may assume U;,J n R; =I=0). Then J

µ(U1,j x U2,j) = md, (U1,j) x md 2 (U2,j) :S md, (B1,J) x md2 (B2,j) 1

:,S:C1C2· (f(U1,j)/

where B;,j denotes the ball in K; centred in Therefore m

·

(f(U2,j))d2

x;,j

and with radius f(U i,J)-

m

I)diam(U1

,j x U2,J))d;:::L(f(Ui

j=l

,J))d' x (f(U 2,j))d2

j=l

;:::(c1c2)- 1

L µ(U1,j x U2,j) j

;:::(c1c2)- 1µ(A). Since U is arbitrary

this proves

To obtain an inequality in the opposite direction we construct coverings Um, of A = R1 x R2, m >> n, as follows. Fix U1 E R 1 , U1 contained in R 1. For each x2 E R2..:_take m(U1, x2) maximal such that if U2(U1, x 2) denot es 1 2 (U,,x ) containing x 2, then f(U 2(U 1, x 2)) ;:=:f(Ui). Clearl y, the element of {U2(U1,x2): x2 E R2} contains a finite covering of R 2 by disjoint int erval s. Since these U2( U1, X2) are elements of Markov partitions R~, j ;:=:1, two o f them either are disjoint or have one contained in the other. Thus, we can extract a finite subcovering by disjoint elements. We now define Urnto be th e family of sets U1 x U2(U1, x2) obtained in this way for all U1 E R 1 contain ed in R1. This is a covering of R1 x R2 by disjoint cubes. Moreover, it is not difficult to deduce from the bounded distortion property that there is O < b < 1 (dependin~ only on K2 and llf2) such that, denoting by Vi(U 1, x 2) the ele2 . x2, we h ave €(U'( U , x ) ) ;:=:b€(U (U , x )) . men t o fR 2rn(U,,x )+l th a t contams 2 1 2 2 1 2 By definition of UHU1,x2), we also have f(UHU 1 , x2)) :S €(U 1). Therefor e,

R;

(8)

4-2

77

Numerical invariants of Cantor sets

Then , by (7) and (8) we get ~)diam(U1

x U2(U1,x2))d

Um

= Lf(U2(U1,x2)/ Um

Um Um

Since the diameter of Um may be taken arbitrarily small (by taking m large), we have md(A) :S b-d,a~a~µ(A) and so our argument is complete. □

Note that O < md(K) < oo ((a) in the proposition) is related to the fact that since K 1 and K2 are dynamically defined, H D(K 1 x K2) = H D(K 1) + H D(K 2). This also follows from the previous theorem stating that d(K;) = H D(K;), i = 1, 2, and the general product formulas H D(K 1 x K2) ;::: H D(K1) + H D(K2) and d(K1 x K2) :S d(K1) + d(K 2) together with the inequality d(K) ;:::H D(K). We now establish an interesting relation between Hausdorff dimension and thickness for Cantor sets in the line. In particular, if the thickness is large then the Hausdorff dimension is close to 1. PROPOSITION 5. If K C 1Ris a Cantor with thickness (log 2/ log(2 + 1/T)).

T

>

then H D(K)

PROOF: Let /3 = (log2/log(2 + 1/T)). We show that H13(U);:=:(diamK)t3 for every finite open covering U of K, which clearly implies the proposition. The key ingredient in this proof is the following elementary fact:

min{x 13+z 13Jx;:::0,z;::: O,x+z :S l,x;::: T(l-x-z),z;:::

T(l-x-z)}

=

1.

(9) We assume from now on that U is a covering with disjoint intervals. This is no restriction because whenever two elements of U have nonempty intersection we can replace them by their union, getting in this way a new covering V such that H/J(V) :S H{J(U). Note that, since U is an open covering of K, it covers all but a finite number of gaps of K. Let U, a gap of K, have minimal length among the gaps of K which are not covered by U. Let and be the bridges of K at the boundary points of U. (See Figure 4.8).

er

ce

78

4 Cantor Sets in Dynamics and Fractal Dimensions

~ ct~ ◄•--([

► +--

u

]()

(

er~

)[

)

Ar---•

Figure 4.8 By construction there are Ae, Ar E U such that ce Take the convex hull A of Ae U Ar. Then

c Ae and er c Ar.

and

f(Ar) 2 f(Cr) 2 T · f(U) 2 T(f(A) - f(At) - f(Ar)) and so, by (9), (f(Ae)).B+ (f(Ar)).6 2 (f(A)).6. This means that the covering U1 of K obtained by replacing Ae and N by A in U is such that H.a(U1 ) ::; Hp(U). Repeating the argument we eventually construct Uk, a covering of th~ convex hull of K with H.a(Uk) :S Hf3(U). Since we must have H (Uk) ;::: 13 (dim K).6, this ends the proof. r,

'-

Note that in general there can be no nontrivial upper estimates for the Hausdorff dimension in terms of the thickness, even in the dynamicalh· defined case. To see this, recall the earlier example of an affine Canto·r set K with Markov partition {K 1 , ... , Kk} with components all of length /3 · diamK, 0 < /3 < ¾, and gaps between K; and K;+i all of length (1-/3- k)-diam(K)/(k-1). As we saw, we have HD(K) = logk/log /3- 1. The thickness can easily be shown to be T(K) = f3(k - 1)/(1 - f3. k). Now consider a sequence of such Cantor sets characterized by k and f3k such that }~~ k · f3k = a E (0, 1). Then, ask-> oa, the Hausdorff dimension tends to 1 while the thickness converges to a/(1 - a). This fact is not really surprising since the thickness was defined as an infimum and so having T(K) small gives very little information concernin g the Cantor set. As mentioned before, this was our main motivation for introducing a variation of the thickness which we called denseness. We shall prove that Cantor sets with small denseness have small Hausdorff dimension . Let us first observe that if K is an affine Cantor set as above and 2f - 1 < k :S 2f, then 0(K) = (e - 1) + f/3(k -1)/(1 - /3k). This follows from th 0 and m(W 5 (p) n A) > 0. In this particular example, W 8 (p) n A is invariant under an expanding C 1 map but the bounded distortion property no longer holds. Given a sequence {f3n} of positive real numbers satisfying

L f3n< 2

and

n~O

we construct in J = [-1, 1], by the standard procedure, a Cantor set K 1 in such way that at the n th step we remove 2n intervals, Jn,k, k E {1, ... , 2n}, f3nis positive. For each of length ~;:. It is then clear that m(K1) = 2 -

L

n 2::1 and k E {2n-l

+ 1, ...

n~O

, 2n}, define

as follows: (i) gn,k is a C 1 orientation preserving homeomorphism; (ii) 9~,k(an,k) = 9~,k(bn.k) = 2, where Jn,k = [an,k, bn,k];

4

82

(iii)

sup xEln,k

Cantor Sets in Dynamics and Fractal Dimensions

12- 9~,k(x)j n-+oo -->

0.

The choice of the f3n's guarantees that (i)-(iii) coexist and makes possibl e this construction. Now, from the above conditions, we can continuously extend all th e 9n,k's to a homeomorphism g of class ci on [/30 /2, l], so that g' IKJ= 2. Finally, let Q be J x J and cp:Q --+ Q be a diffeomorphism given by

cp(x, y) = (g(x) , 9-i(y)), cp(x,y)

i

cp(x,y)

= (g(-x),

if

Q,

if

-g-i(y)),

if

(See Figure 4.10) +I

-------•

), I

-_-_-__---l

/11

],'

-_-__

,, ' /o I

/

/

o

I

I

I

I

I

-7: _

/11

1

/

IJ

I I

It

P0 l2

graph of g, partially defined

Figure 4.10 The reader may easily verify that cpis of class C1, p = (1, -1) is a hypercpn(Q) = K 1 x K 1 is a hyperbolic hors eshoe bolic fixed point of cp, A=

n

nEZ

and that Wi~c(P)nA greater than zero.

§3

= K1.

Local invariants

Besides, both A and K 1 have Lebesgue measure

and continuity

We conclude this chapter with some relevant facts on localized ver sions of the numerical invariants for Cantor sets introduced so far, and on the (continuous) dependence of these invariants on the Cantor set, at leas t for dynamically defined Cantor sets.

4-3 Local invariants and continuity

83

We give the definition of local thickness; local denseness, local Hausdorff dimension and local limit capacity are similarly defined. Let K C 1R be a Cantor set and k EK . The local thickness 7ioc(K, k) of Kat k is defined as

1ioc(K , k) = ,:limsup{ r(K) -,Q

IK

is the intersection of K with an interval

contained in an c:-neighbourhood of k }. For dynamically defined Cantor sets these notions have some additional properties. Let K be a dynamically defined Cantor set with expanding map llt. Th en for every U c K, U open, there is some n so that wn(U) = K. From this and the bounded distortion property it follows that the local invariants 1ioc(K,k), 010 c(K,k), HD1oc(K,k) , and dioc(K,k) are, in the dynamically defined case , all independent of k. Also , since the limit capacity and the Hausdorff dimension are invariant under diffeomorphisms , one has in t his case H D10 c(K, k) = H D(K) = d(K) = d10 c(K , k). The thickness and the denseness are not invariant under diffeomorphisms, and we may have r( K ) < 1ioc(K, k) or 0(K) < 010 c(K, k). For a discussion of the continuous dependence of the invariants on the Cant or set, we restrict ourselves to the dynamically defined case. Bearing in mind the dynamics of basic sets of surface diffeomorphisms , we define when two Cantor sets are near each other as follows. Let K be a Cantor set with expanding map W and Markov partition {Ki, ... , Kt}. Suppos e that 0 llt is c 1+ with Holder constant C, i.e. with l'11'(p)- w'(q)I ::; C IP- qj" for all p , q in a neighbourhood of K. We say that the Cantor set K is near K if K has expanding map 1l1and Markov partition Ki, ... , Kesuch that

- 1l1is

ci+ t and is ci near w, its derivative 'l!' has Holder constant such that (i , C) is near (c:,C),

C

- (Ki , ... , Ke)is near (Ki , .. . , Ke) in the sense that corresponding endpoints are near. An important consequence of this definition is the existence, for nearby Cantor sets K and K as above, of a homeomorphism h : K --+ K, 0close to the identity, such that 1l1 o h = h o W. We construct h as follows. Notice first that, because of the proximity assumptions in the definition, llt(K i) intersects (and then contains) Kj if and only if the same happens with 'l!(K i) and Kj. It follows that, given x E K, there is i; E K such that ii,n(x) E Ki {::}wn(x) E Ki, for all n 2: 0. Since 1l1is expanding, i; must be unique; we define h(x) = x. Clearly 'l!(h(x)) = h(w(x)). On the oth er hand we can obtain h-i by a symmetrical construction, so h is really a bijection. Checking that h is close to the identity presents no particular and Rn for K and K as in difficulty. Just construct Markov partitions the previous section taking connected components of the inverse images of

c

nn

84

4 Cantor Sets in Dynamics and Fractal Dimensions

the Kj, respectively Kj, by wn-i, respectively ii,n-l_ Then, one observes that x and h(x) belong to corresponding intervals of nn and i?,n for all n and that corresponding intervals are uniformly (meaning independentlv of n) close, due to the closeness of k to K, iJ! to \JIand to the bounded distortion property. We are left to show that h is continuous. We do more than that: we prove that it is Holder continuous. Take 8 > O such that d(K;,Kj) > 38 and d(K;,K 1) > 38 for all i =/-j. Now, for x,y EK with Ix - YI :::;8 we let n = n(x, y) 2: 0 be such that

and

1wn(x) - wn(y)I ::::28. By the definition of 8, the interval [wi(x), iJ!i(y)] is contained in some element of the Markov partition, for every O :::;i :::;n - l. On the other hand we may assume that ii,i(x) - W\y) 38 for O :::;i :::;n - l.

I

I :::;

To have this we just take _k close enough to K, in order to have lh(x) - xi :::; 8/2 for all x (note tha~ wi(x) = h(wi(x))). Then again [1J!i(x),ii,i(y)] must be contained in some K 1 , for all O :::;i :::;n - l. By the mean value theorem there are ~i E [wi(x), wi(y)], ~; E [1J!i(x),ii,i(y)] such that n-1

1wn(x) - wn(y)I = Ix - YI

II llll'(~;)I, 0

n-1

lwn(x) - wn(y)I = Ix -yl

II lw'(~;)I. 0

Clearly, we can take O k such that h and h- 1 both are C 7 implies that , · H D(K) :::;H D(K) :::;,- 1 · H D(K) (and analogously for limit capacity). This, in its turn, is a direct consequence of the definitions. This proof is similar to that in [PV,1988]. Now we state and prove the corresponding result for thickness and denseness. 2. The thickness and the denseness of a dynamically defined Cantor set K depend continuously on K. The same holds for local thickness and local denseness. THEOREM

Heuristically, the theorem is proved as follows. The global strategy is to show that the values T(K, U, u), with U = {Un} a presentation of K and u in the boundary of some bounded gap U = Un, depend equicontinuously on K in the sense that if k is close to K then T(K, h(U), h(u)) is close to r(K,U, u) for all U and u. Here h: K --> k is the conjugacy from W to Ill described above (assume k close enough to K to ensure that h exists) and h(U) is the presentation of k given by h(U) = {h(Un)}, where h(Un) is defined by 8h(Un) = h(8Un)- Observe that h, as we constructed it, is monotonic. For any given u and U we can, just by forcing h to be close enough to the identity, make T(K, h(U), h(u)) arbitrarily close to T(K,U, u). We can even make this happen simultaneously for all u (and U) for which the corresponding gap U is big, say with length bigger than some fixed a > 0. However, such a simple argument is insufficient to obtain the uniform closeness that we need. To deal with the small gaps we must use the bounded distortion property. The idea is to iterate the gap U and the U-component C of u until they become big. To be precise we fix /3 > 0 and take k = k(U, C) 2: 0 minimal such that f(iJ!k(U UC)) 2: /3. From the bounded

.

.

d1stort10n property we conclude that T(K,U,u)

:~::~i~~:

f(C) .

= f(U) 1s almost equal to

/3 > 0 and K taking /3 small enough. for k, \JIwe obtain that

their ratio admits a bound depending only on

and which can be made arbitrarily close to 1 by Analogously, from the bounded distortion property

f(h(C)) r(K, h(U), h(u)) = f(h(U))

.

1s almost equal to

f(\J!n(h(C))) _ . Moreover, f(wn(h(U)))

and this is a key point, the bound for the ratio of these last two values may be taken to be independent of k in a neighbourhood of K. This is a consequence of the fact that bounds for the distortion may be taken

86

4 Cantor Sets in Dynamics and Fractal Dimensions

to be uniform in a neighbourhood of any Cantor set. To explain this, let us first observe that the positive numbers c(8) constructed in the proof of the bounded distortion property vary continuously with the dynamically defined Cantor set. In fact, these c( 8) depend only on the Holder constants of the derivative of the expanding map and, by definition, nearby Cantor sets have expanding maps whose derivatives have nearby Holder constants. In particular, it follows that we can take (new) upper bounds c(8) as in the statement of the bounded distortion property which are uniform, i.e. independent of the Cantor set in a neighbourhood of K. We assume in what follows that k belongs to this neighbourhood. Now, if .e(IJ.!k(U))is big, that is larger than a, we can argue as before , -k

i.e. use the proximity of h to the identity to conclude that .e(~ (h(C)))

=

.e(iJ!k(h(C)))

.e(h(IJ.!k(C))). .e(IJ.!k(C)) . This, together with the estimates obf(h(IJ.!k(U))) is close to .e(IJ.!k(U)) tained above with the aid of the bounded distortion property, proves that T(K, h(U), h(u)) is close to T(K,U, u), as we wanted to show. Of course, we still have the problem that IJ.!k(U)may be small. Iterating further is no solution: it may not be possible to do it, if iJ,i ( C) gets out of the domain of IJ! before IJ.!i(U) gets large. Even if this does not happen , as we iterate the length of IJ.!i( U U C) gets bigger and so the bounds given by the bounded distortion property get rougher. Clearly, for the preceding argument we needed these bounds to be close to 1. Instead, what we do is to show that for our purposes this situation doesn't need to be taken into consideration. First, we observe that since .e(wk(U)) ::; a and .e(IJ.!k(U u C)) 2:

/3, if

we have chosen from the beginning

/3 >> a,

then

:~::~~~~

must be very big. The conjugacy h being close to the identity, the same

f(IJ.!k(h(C))) . . Usmg the bounded distortion property as above we .e(IJ.!k(h(U))) conclude that T(K,U, u) and T(K, h(U), h(u)) are very big. Since in the calculation of both the thickness and the denseness one must at some point take an infimum, these values are irrelevant for this calculation and so may be disregarded when proving the continuity of 0(K) and T(K). We now come to a formal proof. holds for

PROOF: Let A= sup JIJl'J and B = 20(K) + 8. Let c > 0, 8 > 0 and a> 0. Suppose that k is close enough to K so that Jh(x) - xi ::; a8 for all x EK. We prove that if a> 0 and 8 > 0 are chosen appropriately small (the precis e

4.3 Local invariants and continuity

87

condit ions are given below) then this implies 0(K)::; (1 + c) 2 0(K)

(a) (b) (c) (d)

+ c(l + c),

+ c)- 1, T(K)::; (1 + c) 2 T(K) + c:(1+ c), T(K) 2: (1 + c)- 2T(K) - c:(1+ c)- 1 . 0(K) 2: (1 + c)- 0(K) - c(l 2

This proves the first part of the theorem. Then we show that the second part. is an easy consequence of the first one. First we take a > O small enough so that the 2ABa-neighbourhood of K is contained in the domain of IJ!. Clearly, we may assume that the same holds fork and IJ.!.For U = {Un} a presentation of K, u a boundary point of a bounded gap U = Un and C the U component of K at u, take k 2: 0 minimal such that .e(IJ.!k(UUC)) 2: Ba. Then .e(wk(UUC)) ::; ABa (because f(wk- 1 (U u C)) ::; Ba) and so .e(IJ.!k(h(U) U h(C))) ::; ABa + 2a8 ::; 2ABa (as long as 8 ::; '\8 ). By the bounded distortion property we have

(1) and -c(2ABa)

e

< [(.e(IJ.!k(h(C)))) ;.e(h(C))l -

.e(IJ.!k(h(U))) ,


lik(C))·

I

t(h(>lik(U)))-t(>lik(U)) t(>li

< ABa-2a8+ABa-2a8 a • (a - 208)

+t(>llk (U))•t(h(>li

(V)>• lt(h(>lik

(C)))-t(>llk

(C))

(U)))

= 8 4AB

1 - 28

(3)

88

4 Cantor Sets in Dynamics and Fractal Dimensions

If 8 > 0 is sufficiently small this implies

I

e(wk(C)) £(\Jlk(h(C))) l'(IJ!k(U)) - £(\Jlk(h(U))) $

l

(3a )

€.

From (la), (2a) and (3a) it immediately follows that

T(K, h(U), h(u)) $ (1 + e) • ((1 + e) • T(K ,U,u) and

+ e)

T(K, h(U), h(u)) 2: (1 + e)- 1 ((1 + e)- 1T(K,U , u) - e).

(4a )

(4b )

!,et now l'(wk(U)) $ a . Then we must have l'(wk(C)) 2: Ba-a. Moreover l'(wk(h(U))) $ a+ 2a8 and £(\Jlk(h(C))) 2: Ba - a - 2a8 . This together with (1) and (2) implies l'(C) > l'(U) -

e- c(ABo).

Ba - a _ (B _ l) -c(ABo a e

)

(5)

and

l'(h(C)) > e- c(2AB o) . Ba - 2a8 - a l'(h(U)) a+2a8

=B-

28 - 1 . -c( 2AB o) 1+28 e

(6 )

Since we have chosen B = 20(K) + 8, we can suppose a and 8 small enough so that these relations imply

T(K,U, u) 2: (B(K)

+ 3)

(5a )

and

T(K, h(U), h(u)) 2: (B(K)

+ 3).

(6a )

Now we proceed to prove the affirmatives (a) through (d) stated near th e beginning of the proof. Recall that by definition

T(K)

= sup inf T(K,U, u),

B(K)

= inf

u

u

u

sup T(K,U, u). u

To prove (a) we must find for any given U a presentation U of k such that s\lp T(K,U, u) $ (1 + e}2sup T(K,U, u) + e(l + e) (al. ) u

u

There is no loss of generality if we assume that

supT(K,U, u) $ B(K) u

+ l.

(a2 )

4-3 Local invariants and continuity

89

Take U = h(U). From (a2) it follows that T(K,U, u) $ B(K) + 1 for all u and so (5a) never holds . Then , by the previous discussion we must have

(as well as (4b)) for all u. This immediately implies (al) and so (a) is proved . The proof of (b) is almost dual to the preceding one so we don 't writ e it down in detail. The only asymmetry comes from the fact that (6a) involves 0(K ) and not B(K). This is bypassed as follows. First , we may as above suppose that (b2) sup T(K,U, u) $ 0(K) + l. u

Now, from (a) (which we have already proved) we get that if k is sufficiently near K then B(K) $ B(K) + l. Then (b2) implies sup T(K ,U, u) $ B(K) + 2

(b3)

u

and now th e argument proceeds as before. To prove (r) we take , for each U, U = h- 1 (U) and show that (cl)

To do this we must to each u associate

u such

that (c2)

Again, it is sufficient to consider the points u for which

T(f(,U,u) :S inf T(K,U,u) + l. u

(c3)

Take u = h(u) and observe that if (c3) holds then T(K ,U,u) $ T(K) + 1 :S 0(K) + 1 and so (5a) doesn't hold. Therefore (4a) is true , and this is just (c2). T he proof of (c) is complete. The proof of (d) is dual to the one of (c) (recall also the remark in the proof of (b)) so we are done with proving the continuity of (global) thickness and denseness . Finally, recall that the local thickness of a Cantor set K at a point k E K is defined by Ti oc

(K, k)

= lim(sup{ T(K1)\K1 CK n B 0 (k) a Cantor set }). o-o

~et e > O be small. Given 8 > 0, take 8 > 0 such that h(K n B 0 (k)) c Kn B1;(h(k)). Let K 1 be a Cantor set in Kn B 0 (k) and let k 1 = h(K 1). If

90

4

Cantor Sets in Dynamics and Fractal Dimensions

h is close enough to the identity (i.e. if k is close enough to K ) th en the arguments above imply r(K 1) ;:::r(Ki) - c:. Since K1 is arbitrary it follows that

sup{r(K1)IK1 CK n B6(k) a Cantor set} ::; sup{r(K1)!K1 Ck n B 6(h(k)) a Cantor set}+ By making

8-+ 0

(and sob-+ TJ oc

c:.

0) we get

(K , k) ::; TJoc (K, h(k))

+€.

In the same way one shows TJ oc

(K, k);:::TJoc (K, h(k)) -

€.

This shows the continuity of local thickness. For local denseness the argument is the same. The proof of the theorem is now complete. □ REMARK 1: Consider a C 3 diffeomorphism r.pof a surface, with a basic set A and a saddle point p E A. For (p a C 3 nearby diffeomorphism th ere are A, a basic set, and p E A, a saddle point (near A and p, respectivel y), and the dynamically defined Cantor sets W"(p) n A and W"(p) n A are near in the above sense (if we take nearby parametrizations for W "(p) and wu(p) as in Section 1 of this present chapter). This follows from the cont inuous dependence on the diffeomorphism of basic sets and .their ci+ e:-stabl e and unstable foliations; see Appendix 1 and Remark 2 in Appendix 2, concern ing continuous dependence of Markov partitions. From this and the propo sitions that we just proved, we deduce the continuous dependence , with respect to the diffeomorphism in the C 3 -topology , of all the invariants of w u(p) n A that we have discussed , namely Hausdorff dimension, limit capacit y, thickness and denseness. To show this , one uses the arguments above together with the observation that C 3 diffeomorphisms and C 3 closeness are used only to obtain c 1+< expanding maps with nearby Holder constant s for the derivatives. This in turn provides bounds for the distortion of distances which are uniform in neighbourhoods of the diffeomorphism and the Cantor set. But, as we remarked before, at the end of Section 1, C 2 diffeomorphi sms induce Cantor sets satisfying the bounded distortion property (and the resulting expanding maps are indeed ci+e: for some c: > 0). The ar gument that we used there also yields uniform estimates for the distortion in a C 2 neigbourhood of the original diffeomorphism. Thus , all the above in variants of W"(p) n A depend continuously on r.p in the C 2 topology.

4-3

Local invariants and continuity

91

For Hausdorff dimension and limit capacity, one can go even further: in [PV ,1988] it is proved that the Hausdorff dimension and limit capacity of W"(p) n A depend continuously on the diffeomorphism in th e C 1 topology. This is done by using, as above, conjugacies with Holder constants near 1. We observe that this result had been obtained in [MM ,1983] as a consequence of a variational principle of the thermodynamical formalism.

CHAPTER

5

HOMOCLINIC BIFURCATIONS: FRACTAL DIMENSIONS AND MEASURE OF BIFURCATION SETS

In this chapter we bring together the Theory of Fractal Dimension s and Bifurcation Theory in Dynamical Systems. A number of (mostly recent) results, leading to further questions and conjectures as laid out in Chapter 7, shows that the first theory is of fundamental importance to the second, at least in the context of homoclini c bifurcations of nonconservative (say dissipative) systems. The result s ca n be stated in great generality if we focus our attention on the maxi mal invariant set of the restriction of the dynamics to a neighbourhood of the orbit of homoclinic tangency and an associated basic set. These results become of a global nature when global assumptions are made concerning filtrat ions and hyperbolicity of the positive or negative limit set. We now explain further the results, but leave the formal statements for t he sections following this introduction. So, in this chapter, we will consid er oneparameter families 'Pµ of surface diffeomorphisms which, as the param ete r varies, go through a homoclinic tangency say at µ = 0 which we assum e to be parabolic and to unfold generically; see Chapter 3. We want to know how big in the parameter space, near this bifurcating point, is the set of values that correspond to diff eomorphisms with a hyperbolic limit set. A main resul t here states that this set has a relatively large Lebesgue measure if we assu me small limit capacities (or Hausdorff dimensions) of the stable and un stab le sets of A, where A is the basic set associated with the homoclinic tan ge ncy. In this chapter we discuss the main ideas of the proof, leaving a compl ete presentation of it to Appendix V. We recall that the stable set W 8 (A) of a basic set A is the uni on of the stable leaves through points of A. In the present context of surfa ce diffcomorphisms, which we now assume to be at least C 2 , this stable foliati on is C 1. So the essential structure of W 8 (A) appears in its intersection with a curve e transverse to the stable leaves- since the foliation is C 1, Hau sd or ff dimension and limit capacity of W"'(A) n fl, are independent of fl, (and t hey are equal to each other as we saw in the previous chapter). In parti cul ar we may take, for a saddle point p E A, e = wu(p). We then define the stable limit capacity or Hausdorff dimension d5 (A) as d(W 5 (A) n w u(p)): the unstable limit capacity is similarly defined. Similar results for heteroclinic cycles as well as corresponding resul ts in higher dimensions are Htated in the last section. We also state a par tial

5.1 Construction of bifurcating families of diffeomorphisms

93

converse concerning nonhyperbolicity of maximal invariant sets when the ab ove Hausdorff dimensions are large (sum bigger than 1). As mentioned before, the result when the Hausdorff dimension is small (sum less than 1) becomes more of a global na~u:e when th~ ho1:1o~linic t angency occurs as first bifurcation and the pos1t1ve or negativ~ limit set of t he bifurcating diffeomorphism is hyperbolic: for a set of relatively large me asure in the parameter space the corresponding diffeomorphisms have thei r global limit set hyperbolic. The first result concerning relative measure of bifurcation sets (but not dea ling with fractal dimensions) was obtained in [NP ,1976], where homoclinic bifurcations from Morse-Smale diffeomorphisms was treated. The res ult we present here when applied to that case is somewhat stronger.

§1 Construction phisms

of bifurcating

families

of diffeomor-

We begin this section by indicating an interesting example of a persistently hy perbolic diffeomorphism on the 2-sphere 8 2 with infinitely many periodic or bits: the diffeomorphism itself as well as every small perturbation of it has a hyperbolic limit set. It is in fact similar to the horseshoe example from 2 Ch apter 1 but it is constructed in 8 2 ~ JR2 U oo instead of in JR . Many ot her examples, and in fact a full discussion about constructing a homoclinic tan gency as a "first" dynamic bifurcation, can be found in Appendix V. We take in 8 2 the diffeomorphic image of a square Q with two semicir cular discs D 1 and D 2 attached as indicated in Figure 5.1.

Q

Figure 5.1

94

5 Homoclinic Bifurcations

We let cp map Q U D1 U D2 inside itself as indicated in Figure 5.2 , i.e. so that in Q we have the above horseshoe example and in D 1 we have one hyperbolic sink S1 , attracting all points in D 1 .

(/J (Q)

Q

Figure 5.2 We extend cp to the complement of Q U D 1 U D 2 in S 2 in such a way that there is only one hyperbolic source S0 and such that for each x E 2 S - (Q U D1 U D2), lim cp-n(x) = So. It is easy to verify that in this case n ...... oo the positive limit set of cpconsists of S1,So and the maximal invariant subset in Q. This last set can be analysed as in Chapter 2 and it is hyperbolic. We let p denote the fixed saddle point in Q as indicated in Figure 5.2 and let A denote the maximal invariant subset of Q. The Cantor sets W 8 (p) n A and wu(p) n A are clearly dynamically defined (if cp is C 3 and if we take a correct identification of W 8 (p) and wu(p) with IR); these are the Cantor sets to which the results of Chapter 4 will be applied. The first bifurcating family of diffeomorphisms that we construct in this section is based on the above example of a diffeomorphism cp of S2 with a horseshoe, a source, and a sink. For simplicity we assume cp to be of class C 00 , although most results in this chapter are true for C 2 diffeomorphism s. Let C be a curve from rs E W 8 (p) to r,, E wu(p) as indicated in Figur e 5.3. U denotes a small neighbourhood of c which is divided by the loc al components of W 5 (p) nu and wu(p) nu containing rs and r,, in the region s U1, Uu, and Uu1- We shall obtain our one-parameter family by modif ying the map cp in U, i.e. by composing cp with Wµ, Wµ a one-parameter familv · of diffeomorphisms which are, outside U, equal to the identity. Before we describe Wµ, we analyse the dynamic properties of orbits pas sing through U; we assume that this neighbourhood U of c is sufficiently small

5.1 Construction of bifurcating families of diffeomorphisms

95

p

Figure 5.3 so that the following considerations are valid; see Figure 5.4. If x E U1UU11 then cpn(x) tends, for n --+ +oo, to the sink S1 and if x E Uu U Uni then .pn(x) tends, for n --+ -oo, to the source So. For x E Uu1, the positive iterates cpn(x), n --+ +oo, will stay near wu(p), but apart from that they may go to the sink or may stay near A (notice that A= wu(p) n W 8 (p); see the construction of cp above); in any case there are points x E U111such that ;pn(x) E U1 for some positive n. Similarly for x E U1, the negative iterates .p-n(x), n --+ +oo, will stay near w•(p), but apart from that they may go to the source or may stay near A; in any case there are points x E U1 such that 0.

- - ....

.... /

/

Ui

I

I

''

\

~

I I

I ,c

Uu I I

~W'(p)

'

Um

- - - .... Figure 5.4

/

wu(p)

96

5 Homoclinic Bifurcations

Now we come to the description of the one-parameter the points in U. We take wµ so that - for µ :S -1,

wµ

family

wµ

moving

is the identity,

-1, Wµ pushes points down in U (in the direction of Uur) so that, for µ < 0, Ur is still mapped inside Ur U Uu,

- forµ>

8 - for µ = 0 there is a tangency of '1>" 0 (Wu(p)) and W (p), or more precisely of Wo(Ur n Un) and Un n Uru; this tangency has quadratic order of contact and unfolds generically for µ > O into two transversal intersections. In Figure 5.5 we indicate the stable and unstable manifolds of p for the diffeomorphism Wµ o .a< 1, where 0 < >.< l and a > l are the eigenva lues of (dt.p)p- We say that >.is the dominating eigenvalue. Exactly as in the first example, we only modify t.pin a small neighbourhood of the curve C joining the points ruin wu(p) and Ts in W 8 (p) as in the figure. We define Wµ as before and '{)µ= Wµ o t.pso that we obtain a homoclinic orbit of tangency 0 for 0 in the above definition of UµNot e that the choice of a larger constant K leads to a slower rate of convergence of the limit.

§3 Homoclinic tangencies with bifurcation relative measure - idea of proof

set of small

In this section we want to outline the proof of the first theor em in the previous section. The proof of the more general and even global th eor em is the same except for a careful analysis of the consequences of the requir em ent that 0, we show that for µo sufficiently small, m(B( 0 and small. Then, to begin with. we must assume the transversality condition for all stable and unstable manifolds of cpo except along the (unique) orbit of tangency occurring say in wu(A;,) n ws(A;J. Besides, one has also to consider the unstabl e limit capacities of the basic sets Ak such that wu(Ak) n W 5 (A;,) =I-¢. Each such Ak is said to be positively involved with the tangency since wu(Ak) contains

5.4 Heteroclinic cycles

109

(a)

s1, s2: fixed saddles r1, r2: fixed sources a 1 , a2: fixed sinks (b)

s 1 : fixed saddle in a horseshoe s2: fixed saddle r 1 : fixed source a 1, a2: fixed sinks Figure 5.12 the orbit of tangency in its closure. Similarly, A.- iH negatively involved with the tangency if W·'(Ak) internects W"(A;,). We define the unstable limit capacity of the tangency d;' as the maximum of the unHtable limit capacities of the basic Hcts positively involved with the tangency. Similarly, we can define

llO

5 H omoclinic Bifurcations

the stable limit capacity of the tangency df. For a periodic orbit which is a basic set we define these limit capacities as zero even when they are sources or sinks. On the other hand for an attractor which is not periodic the stable limit capacity is 1; similarly for a repeller which is not periodic. as above we have the following. So, for a family O µo where B 9 (T,-1 ~

which were again discussed in the previous section. There, for ji, near - 2 and n big, we denoted by Pn(ji,) the fixed point of 'Pn,µ which is close to (x,y) = (2, 2). Corresponding to this fixed point, there is a periodic point with period n + N of ,Pµ which we also denote by Pn(µ). (Whenµ, ji,, and n appear simultaneously, it is understood that they are related by µ = Mn (ji,), see the theorem in Section 4, Chapter 3). In this section, which is entirely dedicated to the proof of the next propostion, we analyse the stable and unsta ble separatrices of Pn(µ) for µ near Mn(-2) . PRO POSITION l. For ji, near -2, there are compact arcs a;(ji,) and a~(ji,) in W 8 (Pn(ji,)) and wu(Pn(ji,)) containing Pn(ji,) and converging, for n --; '.X), to an arc in W 5 (po), respectively in wu(Po), containing at least one fundamental domain (Po is the saddle point of 1 and argui ng as in the proofs of the corollaries in Section 1 of this chapter, we obtain after thi s last perturbation a diffeomorphism which is in the boundary of an open set 2 UC Diff (M) with persistent homoclinic tangencies , as requir ed. REMARK 1: As a consequence of the main result in this chapter , we haw that the diffeomorphisms with a hyperbolic limit set are not C 2-dense in the space of all C 2 surface diffeomorphisms . (Similarly, the £-stable or n-st able

6. 6 Sensitive chaotic orbits near a homoclinic tangency

131

diffeomorphisms are not C 2-dense) . This fact, however is still not known in the C 1 topo logy, so we pose the following 1: Are the diffeomorphisms with a hyperbolic limit set C 1-dense in the space of all surface diffeomorphisms? PRO BLEM

§6

Sensitive

chaotic orbits near a homoclinic

tangency

In this final section we want to discuss the results that we have obtained so far in terms of occurrence of sensitive orbits and strange attractors. First , a homoclinic tangency, and its unfolding, give rise to hyperbolic basic sets of saddle type; in these basic sets, most orbits are sensitive. On the ot her hand , for a diffeomorphism in the plane near a homoclinic tangency , associa ted to a dissipative (area contracting) saddle point, there can be no nontri vial hyperbolic attractor near the orbit of tangency: such an attractor conta ins "holes" in its basin where the map must be expanding, see Plykin [P ,1974], which is impossible in the dissipative case. So, as long as th e dynamic s is hyperbolic , the chaotic orbits can only occupy a set of Lebesgue measure zero. Secondly, there is the phenomenon of the coexistence of infinitely many periodic attractors (sinks). Generically, the number of periodic attractors, with p eriod smaller than some constant, is finite. Thus, if there are infinitely many attractors, most should have very big period. Of course, periodic attrac tors are not sensitive, but in numerical experiments, where one can analyse only a finite part of an orbit, these periodic attractors of very high period may look chaotic . Finally, although we cannot expect nontrivial hyperbolic attractors, we do expect non-hyperbolic strange attractors. This is based on the fact that in the unfo lding of a generic homoclinic tangency there are Henon-like families of diffeomorph isms and for such families Mora and Viana [MV ,1991] proved the exis tence of "persistent strange attractors" . See the next chapter.

CHAPTER

7

OVERVIEW, CONJECTURES AND PROBLEMS A THEORY OF HOMOCLINIC BIFURCATIONS STRANGE ATTRACTORS

-

Based on recent developments , conveyed in the previous chapters and further discussed here, we now present some perspective, and indeed a program me. concerning homoclinic bifurcations and their relations to chaotic dynami cs. Actually, we consider homoclinic bifurcations as a main mechanism to unleash a string of complicated changes in the dynamics of a diffeomorphi sm (or, more generally, an endomorphism). Indeed, as we have seen, the one-parameter unfolding of a homo clinic tangency yields a striking number of dynamical phenomena: - hyperbolic Cantor sets (Chapter 2), - cascades of homoclinic tangencies (Chapter 3), and for locally dissipative surface diffeomorphisms · - cascades of period doubling bifurcations

(Chapter 3),

- relative prevalence of hyperbolicity of the limit set in a significan t number of cases (Chapter 5, Appendix 5) and its converse (Chapter 7), - infinitely many sinks (Chapter 6, Appendix 4), - Henon-like strange attractors, as proved in [MV,1991] extending the work in [BC,1991], to be discussed in this chapter. Thus, homoclinic bifurcations embody most of the known bifurcati ons of a nonlocal character, at least in the setting of surface diffeomorphis ms or three-dimensional flows without singularities. On the other hand , th ere is some evidence, still quite limited, that a nonhyperbolic diffeomorphi sm exhibiting (one of) the above complicated phenomena might be approxim ated by one exhibiting a homoclinic bifurcation. These considerations lead us to propose a number of related questions that when put together poin t toward a theory of homoclinic bifurcations . Our programme, as it will be discussed in this chapter, consists of (1) to determine a dense subset rt of all dynamical topology such that if cp E rt, then either -

n

Recall that if Po, ... , Pn are partitions, then

V Pi is the

partition whose

j =O

elements are sets of the form Pon•· -nPn for P1 E P1 and µ(Pon•· -nPn) > 0. Finally, we define hµ(cp) the (metric or measure theoretic) entropy of cp with respect toµ, as the supremum of hµ( cp,P) over all finite partitions of supp µ. \"otice that hµ(cp- 1) = hµ(cp).

This concept is also based on the intuitive notion of entropy that we have first presented. Instead of pursuing this point, which was quite clear with respect to the topological entropy , we now state the following important fact: the topological entropy of cp is the supremum of the metric entropies of cp with respect to all invariant probability (ergodic) measures. As a consequence, if h(;p)> 0 then there are invariant probability measures, even ergodic ones,

with respect to which the metric entropy is also positive. We refer the reader [B,1975a], [M,1987], [S,1976], [W,1982] for more references, details and some history. We now present a theorem of Oseledec [0,1968], the multiplicative ergodic theorem, that together with subsequent work of Pesin ([P,1976], [P,1977]) has very much influenced a basic line of research that allows one to extend results valid for hyperbolic diffeomorphisms, e.g. the stable manifold theory, to other classes of maps. We say that x EM is a regular point for cp if there are numbers >-1(x)> · · · > >-e(x) and a decomposition to

such tha t

for v E E j , v -=I-0, and 1 :::;j :::;£. The numbers ,\1(x) are called Lyapunov exponents of cp at x. The multiplicative ergodic theorem states that for every invariant probability measure µ, the set of regular points A is a Borel subset and µ(A) = l. Moreover, the maps x -> Aj(x) and x -> E 1(x) ar e measurable . If µ is ergodic, then the A1 's as well as the dimensions of the corresponding E/s arc constant a.e. (almost everywhere). We say that cp is nonuniformly hyperbolic if ,\1 (x) -=I-0, x a.e. in A and all IS j S £.

146

7 Overview, Conjectures and Problems

For nonuniformly hyperbolic C 2 (or even Cl+ °', a> 0) diffeomor phis ms, Pesin developed a "filtered" hyperbolic theory for the set A of regular poin ts. That is, ther e are closed, but in general not invariant, hyperbolic sets A 1 C - - - C Ak C · · ·, k E N and Ak = A. The main difference with the

LJ k

usual hyperbolic structure on A is that , as k increases, it may tak e long er and longer to detect the hyperbolic behaviour of dr.pon Eu(x) an d E 8 (x) for x E Ak, where Eu(x) and E 8 (x) denote the subspaces generat ed by the positive and negative Lyapunov exponents at x . Assuming that the measu re is ergodic and denoting by .X+ = min .X1, .X1 > 0, and by .X- = max .X1 , .X1 < 0. we have

ll(dr.pn)xvll ~ k - Ie>.+-t: llvll :S k e>.- +t: llwll ll(dr.pn)xwll for x E Ak, v E Eu(x), w E E 8 (x), n E N and E > 0 small . Moreov er, the stable and unstable sets of x E A are injectively immersed subm anifo lds of M, denoted , as in the hyperbolic case, by W 5 (x) and wu(x). (Noti ce, however, that in this case the angl es between Eu (x) and E 8 ( x) , x E Ak, m ay go to zero ask----->oo.) Thus, in the nonuniformly hyp erbolic case, many vectors will grow ex ponentially when it erating r.p.Vectors can be interpreted as "two infini tely close points" and, hence , we get exponential growth of their distanc es, whi ch implies, informally speaking , exponential sensitive dependence on the initial conditions . It seems reasonable to even expect that full sensi tivity is formally implied just by the existence of a positive Lyapunov expon ent. Based on the work of Pesin, Katok [K,1980] proved that the m axim al invariant set in each Ak is hyp erbolic with a dense subset of hyp erbol ic periodic points all of them exhibiting transversal homoclinic orbi ts. T his pretty result corroborates our pr evious comments about th e sens itivit y of the orbits. Moreover, since in general we have

known as Ruelle's inequality [R,1978], we conclude that if dim M = 2 and h( 0 then r.pis non uniformly hyperbolic with respect to som e ergo dic an d probability measureµ. In fact, there isµ such that h1,(r.p)is close to h(r.p) thus h1,(r.p)> 0. This implies the existence of a strictly positiv e Lyapun ov exponent by Ruclle 's inequality. Similarly, since h1,(r.p-1) = hµ('P), we get a strictly negative Lyapunov exponent , proving the assertion. Notice that. for invariant ergodic measures that arc absolutely continuous wit h rcsp 1. As we saw , residually in some intervals in the /t-paramctcr line, there arc infinitely many sinks for the corresponding maps cp1,. For the Henon-likF

150

7 Overview, Conjectures and Problems

strange attractors we can ask if it is possible that infinitely many of them occur simultaneously when a homoclinic tangency is unfolded. In any case. we have the following conjecture stating that the simultaneous exist ence of infinitely many sinks or Henon-like strange attractors might be a "rare'· phenomenon. CONJECTURE 1. The set ofµ 's for which 1, to ensure the existence of tangencies between leaves of F 5 (Aµ) and ?(Aµ) for every µ E (-£, 0) orµ E (0, c:), one can ask if the condition is necessary to achieve this persistence of tangencies. In work in development, Yoccoz and the first author of this book show that this is not the case for an open set of C2 diffeomorphisms. This leads to the following very interesting question . PROBLEM 3: Is there a necessary and sufficient condition, say involving fractal dimensions of the basic set A0 , to obtain an interval of the form (-c:,0) or (0,c) such that for eachµ E (-c:,0) orµ E (0, c:) th ere is a tangency between leaves of the stable and unstable foliations of Ao? Related to Problem 1 there is a relevant recent result of Diaz, Rocha and Viana [DRV,1991], stating that a saddle-node critical cycle correspond s to a point of positive density of Henon-like strange attractors for generi c oneparameter families of surface diffeomorphisms. That is, if SA( 0.

It is interesting to note that, by a remark of Mora, saddl e-node critical cycles are always present when generically unfolding a homoclinic tangc nc~·This can be seen as follows . As first pointed out in the last sect ion of Chapter 3 and much used in Chapter 6, we can study properties of cpl'that are "robust" under perturbations of the interval map y --+ l - ay 2 , since for some fixed integer N this is the limiting map as n--+ oo of 0. The key hypothesis in the results above is the hyperbolicity of th e fixed point p, i.e. the existence of a d,p-invariant splitting TpM = E• EBEu such that d 0 small enough in the construction of V (recall the outline of this construction in Theorem 5), then Vis We use this to justify our claims about the differentiability of the invariant sub manifolds introduced above. It is not difficult to check that V is rnor mally hyperbolic at p if and only if

er_

(1) when ever )q, >.2are eigenvalues of (dcp)p with l>. Let us 1 1 < a < i>21 consi der first the centre-stable manifold. Clearly, in this case we may take r

162

Appendix 1 Hyperbolicity: Stable Manifolds and Foliations

arbitrarily large satisfying (1). Hence, if cp is Ck, k finite, then w cs(p) may be chosen Ck, as we claimed . We point out that this is not true for k = oo: in general C 00 diffeomorphisms do not have C 00 centre-stable manifold s [S,1979], although they do have them of class Ck for any k ~ l. Thi s is related with the fact that wcs (p) is in general not unique. Analogo us comments hold for the centre-unstable manifold and the centre manifol d. Consider now the strong stable manifold. Again, we may take r as in (1) with arbitrarily large value, so W 88 (p) is Ck if cp is Ck. But now , due in part to the uniqueness of W 88 (p), this is still true for k = oo. The same holds for the strong unstable manifold, the stable manifold and the unstabl e manifold. We now want to use the theory of normal hyperbolicity to study the stable and unstable foliations, defined in a neighbourhood of a basic set of a surface diffeomorphism cp:M -+ M. Here, we are mostly concerned with the differentiability of these foliations. We first treat in some detail the fixed point case and then show how the conclusions can be extended to general basic sets. For diffeomorphisms cp: M -+ M as above, we construct a lift L(M)

1

M

.E£. L(M) _.£...

1

M

where L(M) = {(x, L) Ix EM and Lis a one-dimensional linear subsp ace of TxM} and Dcp is the diffeomorphism induced on L(M) by the deriv ative of cp. Clearly, if cp is Ck then Dcp is ck-l _ Let :F be a foliation of an open subset U of M by differentiable curves. Then U = {(x, L) Ix EU and Li s the tangent space of the :F-leaf through x} is a surface in L(M) and :F is a cp- invariant foliation if and only if U is a Dcp-invariant manifold . If U is then :Fis (i.e. there are local coordinates which trivialize :F) and we even have in this case that the tangent spaces Tx:F(x) to the leaves of :F depend on X. Let now cp be a C 2 diffeomorphism of M = IR2 and O E IR2 be a fixed

er

er

er

er

point of cp, with (dcp)0

= ( ~ ~),

0 < jaj < 1 < j>.j. If Eu denot es the

unstable eigenspace of (dcp0 ) then ou = (0, Eu) E L(IR2 ) is a fixed point of Dcp. The three eigenvalues of d(Dcp)ou are a>.-1 ..Hence , there is a two-dimensional locally invariant C 1 manifold U containing ou, such that TouU is the eigenspace of d(Dcp)ou corresponding to the eigenvalues a and >.. Note that [J projects diffeomorphically onto a neighbourhood U of 0 E IR2 . Then , there is a (unique) foliation :,:U such that TxP(x) = L for all (x, L) EU. Clearly, :,:U is a cp-invariant C 1 foliation and P(0) c wu(o)

Appendix 1 Hyperbolicity: Stable Manifolds and Foliations

163

(because :,:U(0)is an invariant submanifold tangent to Eu at the origin), so of 0 E IR2 • Moreover, one easily checks that U is r-normally hyperbolic at ou for all r < 1-log >./log a. Hence, if cp is C 3 and 0 < £ < - log>./ log a, £ :=:;1 then U is Cl+c, and so :,:U is a C 1+E foliation. In the same way we construct a C 1 stable foliation :,:snear zero and we show that :,:scan be taken of class c1+0 if cp is C3 . Before proceeding to the case of a general basic set, let us derive a corollary of the discussion above, which was much used in the text . :,:U is an unstable foliation on a neighbourhood

T HEOREM

7 [H,1964]. Let cp be a C 2 diffeomorphism of a surface Mand

p E M be a saddle point for cp. Then cp admits C 1 -linearizing coordinates near p. If cp is C 3 such coordinates can even be· taken to be C 1+c for some € > 0.

We detail the proof of the first part of the theorem; the C 3 case follows from the same argument. First we construct C 1-linearizing coordinates for cp j W 8 (p). Let a: IR -+ W 8 (p) be any C 2 parametrization of W 5 (p) with a( 0) = p, and define f: IR-+ JRby

Th en OE JRis an attracting fixed point for f with f'(0) = a, the contracting eigenvalue of (dcp)p- It is not difficult to check that h(x) = limn-+oo1:~x) is a well defined C 1 function in a neighbourhood of O ([S,1957]) . Clearly, we have h(f(x)) = a· h(x) for all x near 0. It follows that ~ = ho a - 1 (x) is a C 1-linearizing change of coordinates for cp I W 8 (p) near p, as we wished. Replacing cp by cp-1 , we get in the same way a C 1 coordinate, say 'r/, defined on wu(p) (near p), relative to which cp I wu(p)(ry) = >.rt,>.being the expanding eigenvalue of (dcp)p- We now define coordinates (~, rt) in a neighbo urhood of p as follows. Let :,:s,:F" be the stable and unstable foliations constructed above. For~ E W 5 (p) and 'r/ E wu(p) we take (~, ry) to be the int ersection point of P(~) and :F5 (ry). Since :,:sand :,:U are C 1 and transversal to each other this really defines C 1 coordinates near p. Moreover, due to t he invariance of the foliations we have

so the proof is complete. Now we indicate how to obtain stable and unstable foliations in a neighbo urhood of a basic set A of a two-dimensional diffeomorphism cp. One way that this can be done is by the limiting process described in Chapter 2 for the case of a horseshoe. A different but related proof involving the >.-lemma, wh ich we outlined above for the case of a hyperbolic fixed point, can also

164

Appendix 1 Hyperbolicity: Stable Manifolds and Foliations

be used to construct these foliations for any basic set of a surface diffeomorphism. Using this approach, we sketch the construction of the unstabl e foliation P and refer the reader to [M,1973a] for more details; of course , th e stable foliation is obtained in a similar way. After this is done, we discus s the differentiability of the foliations when the diffeomorphism is of class C 2 or C 3 . We observe that if the diffeomorphism is only C 1 , as considered in [M,1973a], then the foliations are in general not differentiable. To begin with define P in a fundamental neighbourhood of w s(A) that is in an open neighbourhood N of Wt (A) - 0, such that N n A = 0. That such a neighbourhood exists results from the fact that A is isolated [HPPS,1970] (see also [N,1980], [S,1978]). The definitio n of P on N must be done in such a way that 0, n big, cpn(x) is near the boundar y of cpn(U) and, hence, near wu(p) U W 8 (p). Since the part of the boundar y

Appendix 3

179

formed by W 8 (p) has decreasing length, cpn(x)is close to wu(p). This means □ that w(x) c wu(p) and so the proposition is proved. Note that this proposition explains that certain attractors are contained in wu(p), but not that they fill out all of it. In fact, there could be only one periodic attractor in wu(p). This is the reason why one is interested in nonexistence results for periodic attractors proved by Benedicks and Carleson [BC,1991). It is not hard to see that the above proposition applies to the Henon R.2 given by H(x, y) = (1 - tx 2 + y, fox); the existence of a map H: R 2 ------> homoclinic intersection was proved in [MS,1980]. In fact both Szewc and Tangerman proved more, namely that one can, in the case of the Henon map, choose U to be a neighbourhood of wu(p). Another interesting case can be obtained from the second example in Chapter 1, the pendulum, through small perturbations . With a first perturbation, we make the diffeomorphism, or even the differential equation attracting towards the {E = 1} level without perturbing the dynamics inside this level. With a second perturbation, we make transverse homoclinic intersections in all the branches of the separatrices of the saddle point (1r, 0) (and hence destroy {E = 1} as an invariant set). Although this example is defined not on the plane but on an annulus, the arguments of the proof of the proposition above still work.

APPENDIX

4

INFINITELY MANY SINKS IN ONE-PARAMETER FAMILIES OF DIFFEOMORPHISMS

In this appendix we shall prove a refinement of the main results of Chap ter 6. In that chapter we considered one diffeomorphism cp with a homo clinic tangency, and then proved that near this diffeomorphism there are man y diffeomorphisms with infinitely many sinks. Here, we consider a one-param eter family of diffeomorphisms cpµwith a homoclinic tangency (forµ= 0) , which satisfies certain generic properties , and then show that within this oneparameter family there are many diffeomorphisms cpµ, µ near zero, with infinitely many sinks (or sources). Such a result, based on Newhouse 's original work, was obtained in [R,1983) . We first point out that we can get this result by a rather formal argum ent in the following way. Let D = Difl'2( M) be the space of C 2 diffeomorphs ms on a 2-manifold M. Let :EC V be a codimension-1 submanifold consisting of diffeomorphisms with homoclinic tangency. We assume the usual generic properties (quadratic tangency and ldet dcpl at the saddle point different from 1) to be satisfied for all diffeomorphisms cp E :E. We shall consider one-parameter families µ 1---7 cpµ which intersect :E transversally for µ = 0. First, however, we reformulate the main result of Chapter 6. Let U C D be the (open) set of those diffeomorphisms cp which have a neighbourhood V such that, for each integer m, there is an open and dense subset V(m) c V so that each cp' E V(m) has at least m periodic sinks (or sources). From the results in Chapter 6, it follows that :E c U. Thi s means that, for any integer n, the following property is generic for one-param eter < 1/n , which families 0, the diffeomorphism 0,

r + t) n [- K r, K- r] : 0). If d(A) + d(B) < 1, then for each 8 > 0 there is an c(8) such that 1/r - m(Mre) < 8 wheneverE < c(8); m denotestheLebesguemeasure. Mr,

=

{t E

[0, r]IAE r fl (Be

Remark.One can considervalues t E Mr, -as the values for which the A in a (K r)-neighborhood translateby t of B intersects, or almostintersects, of 0. The constantK whichmayseemheresomewhatartificial is forcedupon us by the applications(see Chapter4) whichwe have in mind. Beforewe startthe proofof thispropositionwe firststate and prove the followinglemma. Let A be a scaled set withlimitcapacitysmallerthand'(A). Then thereis a constanta so that Ar can be coveredwith at most a (c/r) - d'(A) intervalsof lengthc wheneverE < r. LEMMA.

Proof Since the limitcapacity of A is smallerthan d'(A), there is a functiona(r) > 0 of r such that Ar can be coveredwithat most a(r) E-d'(A) intervalsof lengthc wheneverc < r. Let X be the scalingfactorof A. Because - d'(A) intervals of the scaling,AXr can be coveredby a(r) oflengthX e; on d'(A) the otherhand it can be coveredby a(X- r) (X e)such intervals.So we may assume that a(X- r) = Ad'(A) - a(r). From this,and the factthatwe may assume that a(r) = a(l) forr E (X, 1], we concludethatthereis a constanta suchthata(r) < a - rd'(A). Thisimpliesthelemma. Proofof theproposition.Accordingto thelemmathereis a constanta such that A(K+ 1) r can be coveredby at most

aa

(K

E )rr

(K +1)

r

d(A

intervalsof length c r (d'(A) is some numbergreaterthan d(A)). Hence, be coveredby the same numberof intervalsof length3 - r. In the same way, B.(r?1). r can be coveredby at most

A(K + 1) r can

b

(K E r )r

d(B

-r

i o(Kn+r1) intervalsof length3

r (also d'(B) > d(B),.but we take d'(A) and d'(B) so

This content downloaded from 131.156.157.31 on Sun, 23 Nov 2014 21:01:38 PM All use subject to JSTOR Terms and Conditions

J.PALIS

360

AND F. TAKENS

that d'(A) + d'(B) < 1). This meansthat Mr e can be coveredby at most a

b

r

E

(K +

.)

Ar)

d'(A) -d'(B)

intervalsof length6 Ec r. Fromthisit followsthat
0, the map u - A(y) n [- r, +r] is continuouswith respectto the Hausdorff metricforcompactsubsetsof R; (ii) The scalingfactorX(yu)of A(yu)is a Lipschitzfunctionof P. For such 1-parameter familiesA(y) we oftendenoteA(O) simplyby A. Let A(y) be a continuous1-parameter familyof scaled sets and let K > 0 be a fixedconstant.Then, for each c > 0 thereis an r(c) > 0 such that for each 0 < r < r(,), PROPOSITION.

Ar.F eD

(A(yu)n

[-

(K + 1) r, (K + 1) r])

for all 0 < P < r.

Proof:The proofof thispropositionis complicated,mainlybecause of the factthatthe scalingfactormaynotbe constant.If X(y) X(0),thentheproofis quite simple: we just take r(,c) so small that the distance,in the Hausdorff metric,betweenA n [-1, + 1] and A(y) n [-1, + 1] is less thanc/K + 1 for all 0 < p < r(c).

From now on we concentrateon the consequencesof the fact that the scalingfactormay not be constant.To be morespecific,we provethe proposition forthe case whereA(y) = {f (p))?ji 2 O} U {O}, thisbeing the simplest case withnon-constant scalingfactor.The generalcase can thenbe done in the same way. Let p(p) = (X(p))i and h = p(O). If AX is so small that 1/2 < Log(X(O) + AA)/Log X(O),thenfor IX(y) - X(O)I < AX, LEMMA.

h 1/2 Log h L IP(') - P(o)I X E 0, we have to choose r(E) so that 0 < r(E) < ro and so that E > L ((K + 1) * r( )) 1/3. For such a value of r(,c)the conclusionof the proposition holds. *

3. Diffeomorphic imagesof scaled setsand thefinalresult. PROPOSITION.

Let A(y) be a continuousI-parameter familyof scaled sets

and let : R -- R be a continuousI-parameter familyof Cl-diffeomorphisms with 'I1(O) = 0 and *'I(O) = 1. Let K > 0 be a constant. Then, for every E > 0 there is an r(,c) > 0 such that Ar-eO)

for all r,

L E

D

n [-

(*,u(A(p))

(K + 1)

.

r,(K + 1)

r)

[0, r(E)].

This is a simpleconsequenceof the factthatforL *,I(x)

E

[0, r(,c)],

- x

jxj

for jxj -. 0, uniformly in i; see also [11]. In the case that*I''(0) $ 1, one has to replace,in theconclusion,Ar.l(p) by the (r. E)-neighborhood of *'I(0) A(y). In Section 1, we investigatedthe set of values of t such that A has an almostintersection with B, shiftedovera distancet. In Section2 we analyzed 1-parameter familiesof scaled setsdependingcontinuously on a parameteri. In the presentsectionwe gave an estimateon the deformation of a scaled set by a Cl-diffeomorphism dependingon p. Combiningall thiswe obtainthe following theorem. .

THEOREM.Let A(1) and B(j) be two continuous1-parameter familiesof scaled sets so that d(A) + d(B) < 1. Let *,,,4(,,:R -- R be two continuous

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HOMOCLINIC

ORBITS

363

such that 'I'(O) = 0 and (F,,(O)= t. 1-parameter familiesof C'-diffeomorphisms Let K > 0 be a constantand letMr, ebe thesetof those i Ee [0, r] such thatthe distancebetween*I,(A(1)) n [- K r, K r] and 'Ti(B(i)) n [- K r, K r] is smallerthan E r. Then,foreach 8 > 0, thereis an E(8) such that m(Mr

e(8))


M, we recall that for Returningnow to the 1-parameter u= 0 we have: -

=

TO

= S(qpo); is hyperbolicon lim ,,AOQ2(Tp])

T has exactlyone orbitof tangencybetween a stable and an unstable separatrixofperiodicpoints,and thisorbitoftangencyis in factan orbitof homoclinictangency,i.e., of a tangencyof the stable and unstableseparatrixof the same saddlepoint p of p = TO; - the s-set of (p is theunionof Q2(TO)and the orbit( of tangency. hyperbolic,thereare basic sets Ai(A) Since, fort