Structural Stability and Hyperbolic Attractors
Artur Oscar Lopes
Transactions of the American Mathematical Society, Vol. 252. (Aug., 1979), pp. 205-219.
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http://www.jstor.org Wed Mar 26 16:36:54 2008 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 252, August 1979
STRUCTURAL STABILITY AND HYPERBOLIC A'ITRACTORS BY ARTLJR OSCAR LOPES ABSTRACT.A necessary condition for structural stability is presented that in the two dimensional case means that the system has a finite number of topological attractors. Introduction. One of the basic questions in Dynarnical System Theory is the characterization and the study of the properties of structurally stable systems (vector fields or diffeomorphisms). The standing conjecture about charac- terization, first formulated by Palis and Smale [2], is that a system is structurally stable if and only if it satisfies Axiom A and the strong transver- sality condition. That these conditions are sufficient was proved by Robbin [8] and Robinson [9], [lo] after a series of important results in this direction. It is also known that in the presence of Axiom A, strong transversality is a necessary condition for stability. Therefore the open question is whether stability implies Axiom A. Relevant partial results were obtained by Maiie [3], [4] and Pliss [5], [6]. Pliss [5] showed that C1-stability of a diffeomorphism implies that there is only a finite number of attracting periodic orbits (all hyperbolic). The purpose of this work is to show, generalizing this last result, that C1~structurallystability of a diffeomorphism implies that there is only a finite number of hyperbolic attractors. This result is a well known fact for Axiom A diffeomorphisms (Smale [ll]). We will now explain some facts about our result. Let M be a Cm-manifold without boundary and Diffr(M) be the space of Cr-diffeomorphisms of M with the Cr-topology, r > 1. A diffeomorphsm f E Diffr(M) is Cr-structurally stable if there exists a neighborhood U off in Diffr(M), such that if g E U there exist a homeomorphism h of M satisfying $4 = hg. Such a diffeomorphism f has the property that all diffeomorphisms sufficiently near to it have the same orbit structure. We will call a point x E M nonwandering for f when V U neighborhood of x in M there exist n E Z - {0), such that f"(U) n U f 0.We will denote such a set by Q(J. A closed subset A of the nonwandering set Q(f)is called an attractor when: (1) A is f-invariant (that is, f (A) = A).
Received by the editors April 4, 1978. AMS (MOS) subject classifications (1970). Primary 58F10; Secondary 58F15. Key worh andphruses. Structural stability, Axiom A, hyperbolic set, attractor, stable manifold.
O 1979 American Mathematical Society 0002-9947/79/0000-0359/$04.75 206 A. 0. LOPES (2) 3L neighborhood of A in M such that nn.,+f" (L) = A. (3)j is transitive. A subset A c M is called hyperbolic for a diffeomorphism f E Diffr(M) when (1) A is j-invariant. (2) There exist continuous bundles Es(x) and Eu(x) (x E A) that are invariant by Djand such that Es(x) @ Eu(x) = TM,, Vx E A. (3) 3K > 0, 3A,0 < A < 1, such that
Our purpose in this work is to prove the following: THEOREM.The number of hyperbolic attractors in a C1-structurally stable diffeomorphism is finite. In the study of Cr-structurally stable diffeomorphsms the following definitions were introduced (Smale [ll]): AXIOMA. We will say that a diffeomorphism j E Diffr(M) satisfies the Axiom A when: (1) the periodic points are dense in the nonwandering set, (2) the nonwandering set of the diffeomorphismj is hyperbolic. STABLE(RESPECTIVELYUNSTABLE) MANIFOLD. For a diffeomorphism j E Diffr(M) and a point x E M we will call stable (unstable) manifold of x the following set: ~'(x)= {y E Mld,,,(f"(x), f"(y))+O) (Wu(x) = {y E Mldn,,(f -"(x),f -"(Y)) -, 0)). It can be proved [I] that if j satisfies Axiom A then for any x E S2(j), Wyx) ( Wu(x)) is a 1- 1 immersed manifold in M and
STRONGTRANSVERSALITY.We will say that a diffeomorphism f E Diffr(M) satisfies the strong transversality condition when: (I)j satisfies Axiom A, (2) the intersections between stable and unstable manifolds of any pair of points in the nonwandering set are transversal. In the direction of the characterization of the Cr-structurally stable diffeomorphisms there exists the following conjecture: CONJECTURE1. A diffeornorphism j E Diffr(M) satisfies the Axlom A and the strong transversality condition if and only if it is Cr-structurally stable. It is a well known fact (Smale [ll]) that a diffeomorphism satisfying Axiom A and strong transversality conditon has a finite number of hyperbolic STRUCTURAL STABILITY AND HYPERBOLIC ATTRACTORS 207 attractors. A complete proof of this conjecture would therefore contain our result. We point out that from Maiie [4] it follows that if A is an attractor of a diffeomorphism C '-structurally stable in a manifold of dimension two, then A is hyperbolic. Therefore in dimension two our main result can be stated in the following way. The number of attractors in a C'-structurally stable diffeomorphism is finite. The proof of our result will be done in the following way: in $1 we relate and explain some known facts, mainly from [4] and [S]. In $2 we demonstrate some preliminary lemmas that we will lead in $3 to our objective. This paper is the author's doctoral thesis at IMPA under the guidance of J. Palis. I wish to thank him and R. Maiie for many helpful conversations. 1. For Axlom A diffeomorphisms there exists a generalization of stable (unstable) manifolds for periodic points to the points of the nonwandering set [I]. This generalization allows us to deal with a continuous family of discs that have some interesting properties. We will call this family local stable (unstable) manifolds and we will enumerate some of its properties in Lemmas 1.1 and 1.2. For C1-structurally stable systems we do not have "a prior?' ths nice structure of discs, that are defined by the hyperbolicity of the nonwandering set. However another continuous family of discs was introduced by Maiie [4] for C1-structurally stable diffeomorphisms, that has some but not all properties of the family of local stable (unstable) manifolds. We will call this new family of discs the family of pseudo-stable (pseudo-unstable) manifolds. We notice that these two kind of families stable and pseudo-stable (unstable and pseudo-unstable) coincide in the case where f satisfies Axiom A. In this section we will remember some of the properties of this two families. In Lemmas 1.1 and 1.2 we will talk about stable (unstable) manifolds, in the hyperbolic case. In Lemmas 1.3 through 1.8 we will be interested in the pseudo-stable (pseudo-unstable) manifolds of a C '-structurally stable diffeomorphism. We will just give the proofs of Lemmas 1.5 and 1.8 because the others are proved in [4]. In Lemma 1.9 we relate the two splittings that are associated with these two families in the case of a hyperbolic attractor of a c'-structurally stable diffeomorphism. Finally in Lemma 1.1 1 we will discuss the use of techniques introduced by Pliss in [S] applied to the pseudo-stable manifolds of a C1-structurally stable diffeomorphism. We will give before Lemma 1.11 an idea of where we will use all this machinery to prove the main result of this work. In the rest of this work we will consider a C" compact n-dimensional 208 A. 0. LOPES
manifold M without boundary and iff'(^) the set of diffeomorphisms of M with the C1-topology. We will consider d( , ) a fixed metric on M.
LEMMA1.1. For x in a hyperbolic set A with the periodic points dense in A, there exists an unstable (respectively stable) manifold of x denote by Wu(x) ( Wyx)) with the followingproperties: (a) WU(x)= {Y E Mldn,,(f-"(x), f-"(~))- 0) (WS(x) = {Y E M Idn,,( f"(x), f"(y)) -+ 0)) is a 1-1 immersed plane with the same dimension as EU(x)(EYx)), (b) Eu(x) (Eyx)) is the tangent plane of Wu(x)( Ws(x)) at x, (4f (WU(x)) = wu(f (x)) (f( ws(x)) = W"f (x))), (d) if A is a hyperbolic attractor then Wu(x) c A c Q(f) for all x E A.
PROOF.It appears in [I]. DEFINITION1.1. Let A be a hyperbolic set and x E A, we will define the u-metric in Wu(x) and the s-metric in Ws(x) in the following way: d,"(v, z) = inf(1engths in the metric d on M of the paths contained in Wu(x) that begins in v and ends in z) and d,"(v, z) = inf{lengths in the metric d on M of the paths contained in Ws (x) that begins in v and ends in z) . DEFINITION 1.2. Let A be a hyperbolic set and x E A, we will define Wf(x) the ball of center x and radius E in Wu(x) in metric d,". Similarly for W,"(x).
LEMMA1.2. Let A(f) be a hyperbolic set with the periodic points dense in A( f) and E > 0. Then for any 6 > 0 there exists an N > 0 such that f-"(Wf(x)) c W;(fp"(x)), Vn > N.
PROOF.Suppose K and A as in the definition of hyperbolicity in the introduction. Given 6 take n such that KA'E < 6. Now using (3) on the definition of hyperbolicity, Lemma 1.2 follows. In the next lemmas we will be interested in the family of pseudo-unstable (pseudo-stable) manifolds that exists for C'-structurally stable diffeomorphisms. We will call Per,(f) the set of peridoic points of the diffeomorphism f that has stable dimension j. Let A,(f) be the closure of the set Per,(f). We will use the notation %(M) to indicate the interior in Diff'(~)of the set of diffeomorphisms whose periodic points are all hyperbolic. It is easy to see that all C' structurally stable diffeomorphism is in 'F(M). We point out that Mafie [4] proved that iff E 'F(M) then it follows that Per(f) = Q(f). We will always suppose in this work the condition f E 'F(M)instead off C '-structurally stable. STRUCTURAL STABILITY AND HYPERBOLIC ATTITRACTORS 209
LEMMA1.3. Let f E %(M) with the periodic points dense in Aj # 0, for some 0 < j < dim M. Then there exists a continuous splitting TM/A, = ,??GI i?" and constants 0 < X < 1, K > 0 such that (a) (Tf)E" = E", dim E" = j, (TAP = fi (b) if x E Per,( f)and m is the period of x, then
(c)for all x E A,(f) and m E Z+
PROOF.It appears in [3]. Let f E 9(M). Given 0 < j $ dim M let A = Aj(j) and E, be the subbundles of TM/A given in Lemma 1.3. Denote by EmbA(DJ,M) the space of C' embeddings P: DJ + M, where DJ = {x E RJl llxll < l), such that p(0) E A. Consider EmbA(Dj,M) endowed with the C' topology. Then T: Emb,(Dj, M) + A defined by T(P) = P(0) is a fibration.
LEMMA1.4. There exist sections cp? A -+ EmbA(DJ,M), q ": A -+ Emb dim M-j A( , M) and E > 0 such that dejining we have the following properties: (a) T, W," (x) = E,", T, Wr(x) = E;. (b) If we dejine d,(a, b) = inf{length oj a(t), where a: [O, 11 + &(x) is such that a(0) = a, a(1) = b)for x E A, a, b E @(x), and dejine &(x) = {a E (x)ld,(a, x) < E),then @(x) = W,"(x). (c) There exists a power g = jNojj such that for all 0 < E, < 1 there exists 0 < E~ < 1 satisjjing for all x E A. REMARK.We call el(g(x))the set K1Jg(x)). PROOF.It appears in [4]. The corresponding properties are true for g,"(x) = c(x) and g = f -N, N > 0. These two families of discs will be called, respectively, pseudo-stable and pseudo-unstable manifolds for x E Aj(j). REMARK1. These concepts are "a priori" different from the stable and unstable manifolds defined in Lemma 1.1. These two lemmas (1.3, 1.4) allow us to use a continuous structure of discs that has some interesting properties. These properties will appear in Lemmas 1.5 through 1.8. This structure however does not have in general the uniform 210 A. 0. LOPES property that appears in the hyperbolic case of Lemma 1.2. As Lemma 1.3 shows, the @(x) correspond to the linear invariant part of f. REMARK2. We point out the following invariant properties of the pseudo- unstable family: for any E > 0 there exist a 6 > 0 such that Vx E A,, f - '(W,"(x))c @(f -'(x)). Suppose now that p is a periodic point, take 6 such that
f -J (@ (P)) c @ (f-j (P)), VO < j < T(P). where ~(p)is the period of p. Although in general f -"(p)(w~(~))g W,"(p), for such 6 > 0 we have f -"(P)(iz (p))c @'(p). We use here the uniformity of 2 of Lemma 1.3(b). Remark 2 corresponds to the corollary of Proposition 2.4 in [4].
LEMMA1.5. Let E > 0 be as in Lemma 1.4 and x E Per,(f) n A, hyperbolic, then there exist E, > 0 such that W:(x) = i:(x) (we recall that W:(x) is the unstable manifold of the hyperbolic periodic point x of size E,).
PROOF.Suppose x is fixed for f (otherwise take n such that f"(x) = x). Let U be the neighborhood of x of size E, given by Hartman's theorem (see [I]for reference); that is, j is conjugate to Tjx in U. By Remark 2 we can take 0 < el < E, such that f - '( w:(x)) c w:(x). By uniqueness of the W:(x), this property just happens for one disc of dimension j, the W{(x) disc. Then it follows that WJx) is equal to w:(x). We point out that this E, depends on x. DEFINITION1.3. For x,y E A, we define (x,y) = @:(x) n @:(y).
LEMMA1.6. There exists E > 0 such that if x E A,(f), y E Aj(f) and d(x,y) < E then (x,y) is onepoint and
PROOF.It appears in [4]. LEMMA1.7. > 0, m, > 0,0 < h < 1 and 2 > 0 such that: (a) If x E Per, (f)has period 2 m,,then &(f"(x),f" (y)) < E ,for all n $ 0 (d,(f"(x),f" (y))< for all n < 0) implies y = x. (b)Ifx,EAj(f),i = 1,2,y E M,N > Oandy = (xl,x2) d,(f"(~,),f"(~)) d,(f"(x,),f"(Y)) --n iu(f" (xz),~" (Y)) < KX ~,(x,?Y) d,(x,?Y) for all 0 < n < N. STRUCTURAL STABILITY AND HYPERBOLIC ATTRACTORS 211 PROOF.It appears in [4]. Lemma 1.8 relates the property that we will really use, and it is a consequence of Lemma 1.7(b). Lemma 1.8 says, informally speaking, that if the images of the positive iterates of e(x) are bounded, then the positive iterates of F(x) are limited too, for any x E A,. LEMMA1.8. Suppose that for E, > 0 small enough like in Lemma 1.7 and a, b E A, such that a E c(b), &(a, b) = E, and du*(f"(a), f"(b)) < El, Vn E Z,, then 3~~> 0 such that Vy E e(b), PROOF.Let no be such that Exno&(a, b)/d,(a, b) < 1 where K and i are the constants that appear in Lemma 1.7. Let E, be such that d,(f" (y),f" (b)) < el, Vn 0 < n < no, VY E & (b). Then by Lemma 1.7: Then Lemma 1.8 follows. The next lemma (Lemma 1.9) studies the relation, in the case of an hyperbolic attractor, between the splitting Eyx) @ Eu(x)(given by hyperbo- licity) and the splitting E" (x) $ E" (x) (from Lemma 1.3) for x in the attractor. For the last splitting to make sense of course we consider the attractor contained in A, for some 0 LEMMA1.10. Let A be a hyperbolic attractor and c E A such that f(c) = c. Then for all x, y E Wu(c), the sequence d,' (f" (x), f" (y)) + oo when n + oo. Therefore Wu(c) is not bounded for the metric d,'. PROOF.Let a: [0, 11 + WU(c)be such that a(0) =f"(x), a(1) = r(y).As j-"(a(t)) is in A, we have: where K and X are the hyperbolic constants for A. Since f is a diffeomorphism, all paths between f"(x) and f"(y) are f" of some path between x and y. Therefore: d,"(x, y)/ KXn < d,"( f"(x), f"(y)). Then Lemma 1.10 follows and consequently Wu(c)is not bounded for d,'. We can give now an idea of the proof of our main result. This program will be carried out in detail in 92 and $3. First of all, let us recall the work of Pliss about periodic atrattractors in [S]. (for a C1-structurally stable diffeomorphism there are just a finite number of periodic attractors points): he finds 6 > 0 as small as he wants, and attracting periodic points pi E An (with period ~(p,))such that there are jumps of constant ki-size such that fk(@ (f'kl(pi))) c 6(f ([+ ')kl(pi))is a uniform contraction by 1/4. Here 1 > 0 are integers such that lki < ~(p,).He can make ki and the biggest of such I arbitrarily big, taking ~(p,)big enough. Therefore if there exists an infinite number of attracting periodic points, we can take pi with a large number of elements in the above sequence of ki-size jumps (that is, the biggest of such 1, can be taken arbitrarily big). Since M is compact, two of such elements of a sequence must be at a distance less than 6/2; that is, there exist integers mi > ni > 0 such that d (f" k~ (pi), fn, (pi)) < 6/2 and m,k, - n,k, < ~(p,).Well jqk-qk is a contraction of W,"(j4Vp,)) in Fll4( j~k(~,));thus j(~k-%k)~(~)has just one fixed point in w:( j'+k(p,)) and different from pi because m,k, - n,k, < ~(p,).Thus he reaches a contradic- tion, and so it follows that the number of periodic attractors is finite. REMARK3. Pliss in [S] did not explicitly use pseudo-stable manifolds, but the neighborhoods of the points pi that he considered correspond to these manifolds. STRUCTURAL STABILITY AND HYPERBOLIC ATTRACTORS 213 Let us now see our program for proving that iff E F(M) then f has only a finite number of hyperbolic attractors. Suppose to the contrary that there exists an infinite number of hyperbolic attractors in Aj. Take pi a periodic point in each attractor, such that pi have minimum period in the attractor. First we notice that, with the same arguments as in [S], one can get the existence of ki-jump sequences as above for the periodic points pi with stable dimension j, 0 Thus this new periodic point has period less than period of pi. Therefore we will have a contradiction if we prove (and we will do it in $3) that this new periodic point is in the same attractor as pi. This is obviously not possible because pi has minimum period in its attractor. We will finish $1 with the new formulation of Pliss' lemma on jump sequences mentioned above. The proof is the same as in [S]. LEMMA1.1 1. Let f E %(M) and a sequence pi E Per,(p), that is, pi is a sequence of periodic points with stable dimension 0 PROOF.Similar to [5]. We point out that the corresponding result for is true, but unfortunately we cannot obtain simultaneously both results for the same periodic point pi. 2. We will consider in this section f E 9(M)with the periodic points dense in a(f).In the last section we saw that for every p E Per(fi there exists a 6 (p) > 0 such that Our main purpose in the present section is to obtain a univeral 6 > 0 so that @(x) = W$(x) for all x E A,, where A, denotes the union of all hyper- bolic attractors for f (Lemma 2.7).Moreover we will exhibit an integer N > 0 so that f -N is a uniform contraction of w$(x)in ~$(J-~(x))for a11 x E A, and the same S (Lemma 2.5). Of course these facts would be easily verified if we had only a finite number of attractors for f. The problem here is that with an arbitrary number of hyperbolic attractors we may not have uniformity in the constants of hyperbolicity. Let E > 0 be as in Lemma 1.4; such E will be used throughout this section in the following way: everywhere we mention V;(p) or W,"(p)for some r > 0, it is implicit that r < E. LEMMA2.1. Let p E AH n Per(f) n4 and E~ be such that Vq E W,"D(p), d(f-"(q),f -"(p)) < E, Vn 2 0; then W,"o(p)= W,"o(p). PROOF.Let ~(p)be the period of p, therefore by Remark 2 we have f -T(p)(@o(p)) c We know that there exists by Lemma 1.5 a 6 (p) > 0 such that e,,= W$(,,. - Suppose to the contrary W,"u(p)# W,"u(p),and take y on the boundary of n W,"o(p).Let V be a neighborhood of y in W,"u(p).There exist an n such that f -"T'P'(~)C = R(,)(P). As fm(,)(f -nT(p)(~))c e, because j"(~)q,,(~) c c(p),we have a con- tradiction with the way we took y. Therefore Po(p)= W,"o(p). REMARK4. The bundle E" is uniformly continuous, therefore 3.5, such that if d (x,y) < E, the angle between E" (x) and E" (y)is smaller than 1/4. We will consider in the rest of this work the E of Lemma 1.3 as being smaller than this E,. LEMMA2.2. Let p E Per(n n A, n A,, q E A,, E as in Remark 4 and suppose P(q)c A,, then there exists an e5 such that if d(p,q) < E, then WU(p)n E(x)# 0,Vx E e(q). STRUCTURAL STABILITY AND HYPERBOLIC A'ITRACTORS 215 PROOF.First of all note that there exists an E, < E such for any surface V of dimension j which is not limited has the following property: If the tangent planes of V and E"(y), Vy E W,"(q) are smaller than 1/4 and there exist z E V, d(z, q) < E,, then V is the graph of some function of W,"(q) to R. Therefore V n @(x) #0 for any x E W,"(q) such that d(x, q) < E. We just have to show that Wu(p), Vp E Perf n A, n A,, satisfies the conditions of V. Remember that by Lemma 1.lo, W" (p) is not limited and by Lemma l.l(d), Wu(p) c A,. Now by Lemma 1.9 E"(z) = EU(z),then for y E @(q) and z E W," (p) the angles between E" (y) and E" (z) are smaller than 1/4 by the way we took E (see Remark 4). Therefore Lemma 2.2 is true. REMARK5. Let E, be fixed satisfying the conditions of Lemma 2.2. Then there exists a C > 0 such that for p E Per, f n AH n A,, d(p, q) < E,, the distance between x and y in the same connected component of Wu(p) n (F(q) x W,"(q)) is uniformly bounded by C. The reason is that the angles between E"(x) and E"(z) are limited for all x E Wu(p) and z E W,"(q), d(p, x) < e. We will suppose in the rest of this work the E of Remark 4 smaller than this fixed E,. The following Lemmas 2.3 and 2.4 are just for periodic points but they are the central point of $2. The main purpose of this section, Lemmas 2.5 and 2.7, will follow by the continuity of W,"(x) in A, n A,. In the next lemm-a (Lemma 2.3) we will obtain a property that is very easy to prove for just one hyperbolic attractor (Lemma 1.2). The problem here is that we could have an infinite number of attractors and therefore the constants K can be arbitrarily close to 0. LEMMA2.3. 3~,> 0 such that VE,, 0 < e6 < E,, 36 > 0 such that f -"(@(p)) c e6(f -"(P)) for aNp E A, n Per,(fl n A,, Vn E Z,. PROOF.Let E, = e, as in Remark 5 after Lemma 2.2. Forp E A, we define Sp = {a E e~d(f-"(~),f-"(a)) < E,, Vn E Z,) and 6, = d((~,O), &(PI - S~), < Notice that by Lemma 1.3(b) we have that if ~(p)is big enough /-"(P)(F[(~)) c q(p), Vp E Per(fl n A,. Thus 6, > 0 always. Also if x E w[(p) then d( f -"(p), f -"(x)) < e6 for all n > 0. Suppose to the contrary there exist 6, = 6," +O,pn E Per,(fl n A, n A,. Let qn E A, n F[(pn), o < mn < ~(p,),d( f -%(qn),f -%(pn)) = e6. ~t is clear that mn + co because as f is C' and M is compact, the derivative of f is uniformly bounded. Let a = lim f -%(qn) E A,, b = limf -%(pn) E A,. We will show that d(f"(a), f"(b)) < e6, Vn E Z,. 2 16 A. 0. LOPES Let BA(x)= {y E Mld(x,y) < A). Suppose to the contrary that 3n0 > 0 such that d(f"o(a),f"o(b)) > E,. Let A, such that VxE BAl(yo(a)), Vy E BAl(P(b)) we have d(x,y) > E,. Let now A, be such that Vx E BA2(a),Vy E Bq(b) we have Let pn, qn be such that d(f -"(qn), a) < A,, d(f -"(pn), b) < A,, mn > no. Thus but this is a contradiction with the fact that qn E F<"(pn).Therefore Vn E z+,d(f"(47f"(b)) < &,. We point out that by the way we take a and b we have that a E q(b). Let now c E A, some periodic point of the sequence f -"(pn). By Lemma 2.2 if c is close enough of b then we have the following: there exists an a, in the intersection &(a) n Wu(c)and a b, in the intersection of e(b)n wu(c). Now by Lemma 1.8 if d(a,, a) < e, and d(b,, b) < E, for some E, > 0 then d(f"(a,),f"(b,))< E6, Vn E Z+. It is clear that we can take a, near a and b, near b by taking c near b. Thus d(f"(a,),f"(b,)) < E, for all n > 0 and a,, b, E WU(c). We will show finally that this will imply that di(f"(a,), f"(b,)) is limited, and therefore by Lemma 1.10 we will have a contradiction. The distance d,' (f" (a,),f"(b,))is limited by C (see Remark 5 of Lemma 2.2) because Therefore with this contradiction we conclude that Lemma 2.3 is true. LEMMA2.4. Let e0 be as in Lemma 2.3. There exists an 0 < E, < eo such that Fn (p)= diam f -" (c(p)) converges uniformly to zero for p E AH n Per(f) n A, when n+ a. PROOF.We have to prove the following: Ve, > 0, 3N > 0 such that for all p inAH n Per,(f) n A,, Vn > N,fpn(c(p))c F;(fpn(p)). Let E, > 0 be such that f-"(F;(p)) c e0(f-"(p)), Vn E Z+ as in Lemma 2.3. Given E, > 0 and E, < e,, again by Lemma 2.3 there exists 6 > 0 such that f -n(W,"(p))c K3(f-"(p)), Vn E Z+, Vp E (A, n Per,(f) n AH). Suppose to the contrary that In, sequence in Z+ nk +a such that Fnk(pk)> E, for some fixed E,. Let qk E e(p,) be such that It is clear that d(f -'(q,), f -'(pk)) > 6, Vj,0 integer larger than n,. Note that by the way we took e,, d(f-j(q,), f -'(P,)) < eo, VJ,0 < j < nk. Using the same arguments of Lemma 2.3 we have that 6 < d(f"(b), f"(a)) < Eo. Again (as in Lemma 2.3) by remarks of Lemma 2.2 we conclude that d,"(f"(a,), f"(b,)) is bounded for n E Z,, c as Lemma 2.3. As c E AH this last sentence is in contradiction with Lemma 1.10. Therefore our first assumption in the proof of this lemma is false, and consequently Lemma 2.4 is true. LEMMA2.5. Let eo, e, and F,,(p) as in Lemma 2.4, then F,,(p) converges uniform& to zero when n -+m for allp E AH n A,. PROOF.Suppose to the contrary that Lemma 2.5 is false, therefore 3e3 > 0, 3xk E A, n A,, 3yk E F:(x) and n, +m,such that For each k, take p, E A, n Per, f n A, and q, E e(pk)sufficiently close respectively to x and y, using the continuity of the pseudo-unstable families in A,. If these distances d(x, p,) and d(y, q,) are small, then d(f -,,k(p,), f -,,k(q,)) > E,. As n, + m this last sentence is in contradiction with Lemma 2.4. Therefore Lemma 2.5 is true. LEMMA2.6. There exists a constant S such that W,"(p) = F,"(p) for all p E AH n Per(f) n A,. PROOF.Let 6 be obtained as in Lemma 2.3 for the e, = eo < e. As we have seen in Lemma 2.3, f -"(F,"(p)) c W,"(f -n(p)), tln > 0. Therefore Vp E Perf n A, n A,, f -"(p)(W,"(p))c F," (p) by Remark 2 before- Definition 1.3. We conciude then by Lemma 2.1 that W,"(p) = w;(PI. LEMMA2.7. The constant 6 of Lemma 2.6 is such that W,"(p) = W(p), Vp E A, n A,. PROOF.The points p E Perf n A, n A, are dense in each attractor in AH n A,. Therefore by Lemma 2.6, and by the continuities of the pseudo- unstable manifold family in A, and unstable manifold family in an attractor in A,, we have that for any q E A, n A,, W,U(q)= @'(q). REMARK6. We point out that if we take e, < 6, E, < ~,/4and N > 0 as in Lemma 2.5, then we will have f -N a contraction in Wu(p)for all p E A, n A,. In this case the constant of contraction is smaller than 1/4. The reason is that any q E WU(p)is in A, n A,, and therefore W,"(q) = W,"(q). Thus as fpN(R7(q)c &(fN(q)), f-" is a contraction of Wu(p)in ~"(f-~(p)), Vp E A, n A,. 218 A. 0.LOPES THEOREM.If M is a compact Cm manifold without bounday and j is C '-structurally stable, then the number ojhyperbolic attractors is finite. PROOF.It is enough to prove that the number of hyperbolic attractors for f in A, is finite. We point out that each attractor is completely contained in some A,, and there is just a finite number of A,. All we will need for the proof is that j E S(M) and that the periodic points are dense in Q(j). For each hyperbolic attractor A in A, we will define i(A) = min{ per(x)l x E Per,( j) n A n A,}. Suppose to the contrary that there exists an infinite number of attractors, so that i(A) is not bounded as A varies in A, (all periodic points are hyperbolic). Takep, a periodic point with minimum period in each attractor in A,. It is clear that ~(p,)-+ co. Now using Lemma 1.1 1 for (p,), and 6/2 (6 as in Lemma 2.7 we will have the- following: 3p; in the sequence (p,) such that f("-q)k is a contraction of w," (f"kl(pi))in W," (fqk~(pi))with constant of contraction smaller than 1/4. As we have seen in Lemma 1.1 1, (m, - n,)ki can be taken as large as we want, d(fq4 (P;), Pk,(P;)) < 6/2 and 0 < (mi - n,)k, < ~(p,). If (mi - n,)k, is large enough then by Lemma 2.5, we will have that j-(''-q)k is a coritraction of W,",(jqk(pi))with constant of contraction smaller than 1/4. Let A be the hyperbolic attractor such that pi E A. Now let Now we will define the map g: S -+ S by This map is well defined by Lemma 2.7. We point out that S is a complete metric space for the induced metric of W,"(jqk(pi)).AS we have seen by Lemma 2.5, g is a contraction in S. Take x the fixed point of g and we will have that f-(~-q)~~is a contraction of W&(x) in W;(x). Therefore there exists a periodic point with period (mi - n,)ki in A, but this is impossible because (m, - n,)k, < m(p,). The conclusion is that there exists only a finite number of attractors in A,. 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