New Phenomena Associated with Homoclinic Tangencies

Transcription

1 New Phenomena Associated with Homoclinic Tangencies Sheldon E. Newhose In memory of Michel Herman Abstract We srvey some recently obtained generic conseqences of the existence of homoclinic tangencies in diffeomorphisms of srfaces. Among other things it has been shown that they give rise to invariant topologically transitive sets with maximal Hasdorff dimension, that they prohibit the existence of varios kinds of symbolic extensions, and that they form an impediment to the existence of SRB measres. The main new reslt described here, together with a positive answer to an as yet nproved conjectre of Palis, wold prove that generically on srfaces, SRB measres only exist on niformly hyperbolic attractors. 1 Introdction The stdy of homoclinic motions forms an important part of the modern theory of dynamical systems. Sch motions were first described by Poincare in his geometric stdies of the restricted three-body problem, and, since that time, they have been stdied by many athors. Transverse homoclinic motions prodce complicated invariant sets which can be completely described in terms of certain symbolic systems called sbshifts of finite type. Moreover the dynamical properties of these invariant sets persist nder small pertrbations of the nderlying dynamical system. On the other hand, homoclinic tangencies give rise to rich and varied dynamics nder small pertrbations. In this note we will srvey some of the phenomena associated to homoclinic tangencies. After reviewing some definitions and well-known facts, we describe some recent developments, and present a new reslt (Theorem 1.4) 1

2 which shows that generically homoclinic tangencies form an impediment to the existence of SRB measres. For historical information and varios interesting phenomena associated to homoclinic motions, we refer to the books by Palis and Takens [25] and Moser [19]. For general information on dynamical systems related to the concepts we stdy here, we refer to the books [30], [10], and [22]. While some of or discssion cold be extended to systems of arbitrary finite dimension, the strongest reslts occr in two dimensional systems. Hence, we will restrict orselves to dimension two. Let M be a compact C two dimensional Riemannian manifold, and, for r 1, let D r ( M) be the space of C r diffeomorphisms of M with the niform C r topology. It is known that D r ( M) is a Baire space: contable intersections of dense open sets are dense. For f D r ( M), a compact f invariant set Λ is a niformly hyperbolic set if there are constants C > 0, λ > 1 sch that for each x Λ there is a splitting T x M = Ex s Ex sch that, for n 0, max( Df n x E s x, Df n x E x ) Cλ n. (1) Here we let S E denote the norm of a linear map S restricted to a linear sbspace E of the domain of S. Let d denote the topological metric (distance fnction) on M indced by the Riemannian metric. It is well-known that if the conditions in (1) hold, then the sbspaces Ex, s Ex are niqe and depend continosly on x Λ. The invariant manifold theorem [11] garantees that there are C r injectively immersed manifolds W s ( x), W ( x) tangent at x to Ex, s Ex, respectively, which are defined by and W s ( x) = {y M : d( f n ( y), f n ( x)) 0 as n, } W ( x) = {y M : d( f n ( y), f n ( x)) 0 as n.} A niformly hyperbolic set Λ is called a hyperbolic basic set if it has a dense orbit and there is a neighborhood U of Λ in M, sch that n Z f n ( U) = Λ. In this case one calls the neighborhood U an isolating neighborhood of Λ. If Λ( f) is a hyperbolic basic set for f with an isolating neighborhood U, then there is a neighborhood N of f in D r ( M) sch that if g N, then Λ( g) def = n Z gn ( U) is a hyperbolic basic set for g, and there is a 2

3 homeomorphism h : Λ( g) Λ( f) sch that fh = hg. It is common to call the set Λ( g) the contination of Λ( f). A point q is a homoclinic point of f if there is a hyperbolic basic set Λ and points x Λ, y Λ sch that q ( W ( x) \ Λ) ( W s ( y) \ Λ). Since we are dealing with two-dimensional manifolds, sch a q can only exist if the sets W ( x), W s ( y) are one-dimensional manifolds; i.e., C r crves. The homoclinic point q is called a transverse homoclinic point if the crves W ( x), W s ( y) are not tangent at q. Otherwise, we call q a homoclinic tangency. Let B denote the collection of hyperbolic basic sets of a C r diffeomorphism. There is an eqivalence relation on B defined as follows. We say Λ 1 Λ 2 if either Λ 1 = Λ 2 or there are points x, y Λ 1 and x 1, y 1 Λ 2 sch that W ( x) \ Λ 1 has a non-empty transverse intersection with W s ( x 1 ) \ Λ 2 and W s ( y) \ Λ 1 has a non-empty transverse intersection with W ( y 1 ) \ Λ 2. This relation was considered in [22] for periodic points. If Λ 1 Λ 2, we will say that Λ 1 is homoclinically related (or h related) to Λ 2. We call an eqivalence class a homoclinic class (or h class). The closre of an h class will be called an h closre. An h closre containing more than a single periodic orbit will be called a homoclinic set. Let Λ be a homoclinic set which is the closre of a particlar h class C. We call Λ the homoclinic set of C. If p is a periodic point whose orbit is in C, we will say that p is associated to Λ and that Λ is the homoclinic set of p. We also say that Λ is the homoclinic closre of p. If a homoclinic set Λ contains a hyperbolic basic set Λ 1 with a homoclinic tangency, we will say that Λ contains a homoclinic tangency. The following reslt is proved in [22]. Every homoclinic set is an ncontable topologically transitive set containing a dense set of associated periodic points. Moreover, it coincides with the closre of the transverse homoclinic points of the orbits of any of its associated periodic points. We remark that homoclinic sets have a lowersemicontinity property in the following sense. Let Λ( f) be a homoclinic set for f, and let p = p( f) be an associated periodic point of minimal period. If g is C r close to f, then the contination p( g) of p has an h closre Λ( g) which we call the contination 3

4 of Λ( f). This is independent of the choice of p with minimal period provided that g is close enogh to f. The map g Λ( g) is lowersemicontinos as a map from a neighborhood of f in D r ( M) into the collection of compact sbsets of M with the Hasdorff metric. One can think of homoclinic sets as the basic bilding blocks of systems with chaotic motion. We are interested in detailed information abot homoclinic sets and the phenomena which are indced from them when the ambient diffeomorphism f varies with external parameters. We will see that, there are several interesting phenomena which appear after small pertrbations when a given map f has a homoclinic tangency. First, we recall a general sitation in which there is a whole open set in D r ( M) whose elements have no homoclinic tangencies. An ɛ chain is a seqence x 0, x 1,..., x n in M sch that d( fx i, x i+1 ) < ɛ for 0 i < n. A periodic ɛ chain is an ɛ chain x 0, x 1,..., x n sch that x 0 = x n. A point x is chain recrrent if, for every ɛ > 0 there is a periodic ɛ chain passing throgh x. The collection of chain recrrent points is a non-empty, compact, f invariant set called the chain recrrent set and is denoted R( f). We say that the diffeomorphism f is hyperbolic if R( f) is a hyperbolic set. It is well-known that a diffeomorphism f is hyperbolic if and only if it satisfies Smale s Axiom A and No Cycle properties in which case the chain recrrent set eqals the non-wandering set. Hyperbolic diffeomorphisms are chain stable. This means that there is a neighborhood N of f in D r ( M) sch that if g N, then there is a homeomorphism h : R( g) R( f) sch that fh = hg. A famos reslt de to Palis [23] following work of Mane [16] states that in D 1 ( M) hyperbolicity is eqivalent to chain stability. A diffeomorphism is called Anosov if the whole manifold M is a hyperbolic set. In dimension two, M mst be a tors and R( f) = M. It is easy to see that the existence of a homoclinic tangency is an obstrction to hyperbolicity. Indeed, it can be shown that sch a tangency is in the chain recrrent set, and there can be no splitting as reqired for hyperbolicity. We say that f has persistent homoclinic tangencies if there is a hyperbolic set Λ( f) with a homoclinic tangency and there is a neighborhood N ( f) sch that if g N ( f), then the contination Λ( g) of Λ( f) is defined and also has some homoclinic tangency. Ths diffeomorphisms with persistent homoclinic tangencies have neighborhood none of whose elements are hyperbolic. 4

5 The following reslt [21] shows that on srfaces any diffeomorphism with a homoclinic tangency can be pertrbed to create persistent homoclinic tangencies. The book of Palis and Takens [25] contains a nice proof of this reslt. Theorem 1.1 Let M be a compact C manifold, let r 2, and let f D r ( M) have a homoclinic tangency. Then, given any neighborhood N of f in D r ( M) there is a g N which has persistent homoclinic tangencies. Homoclinic tangencies freqently give rise to infinitely many sinks. In fact, C r generically with r 2, a homoclinic set is in the closre of the periodic sinks provided that it contains a tangency and a dissipative periodic orbit. A periodic point p with f τ ( p) = p is called dissipative if detdf τ ( p) < 1. It is known that if f is a hyperbolic diffeomorphism which is not Anosov, then the Hasdorff dimension of the chain recrrent set is strictly less than two. The next theorem, which was recently obtained in [8], shows that generically with r 2, each homoclinic set which contains a homoclinic tangency has maximal Hasdorff dimension. Recall that a sbset B of a topological is called residal if it contains a contable intersection of open dense sets. In D r ( M) sch sets are, of corse, dense. Theorem 1.2 Let r 2. There is a residal sbset B of D r ( M) sch that if f B, and Λ is a hyperbolic basic set for f with a homoclinic tangency, then the homoclinic set of Λ has Hasdorff dimension two. We sketch the idea of the proof of Theorem 1.2 referring the reader to [8] for the details. The idea of the proof comes from that of a similar reslt for C 1 area preserving diffeomorphisms which goes back to ([20]). A given f with a homoclinic tangency is C r pertrbed to get persistent homoclinic tangencies. Then, one makes se of a reslt of Gonshenko, Traev, and Shilikov to make a frther C r pertrbation to obtain a periodic point p whose stable and nstable manifolds have a whole interval of homoclinic tangencies. Having this, a frther pertrbation gives special zero dimensional hyperbolic basic sets Λ 1, Λ 2 homoclinically related to p sch that the nstable Hasdorff dimension of Λ 1 is close to one and the stable Hasdorff dimension of Λ 2 is close to one. Since Λ 1 is homoclinically related to Λ 2, Lemma 8 in [21] gives another hyperbolic basic set Λ in the same h class which contains them both. C r 5

6 The set Λ will have Hasdorff dimension close to two. This proves that, given n > 0, there is a dense set B n of diffeomorphisms with homoclinic tangencies having homoclinic sets of Hasdorff dimension greater than 2 is also open since the Hasdorff dimension of hyperbolic basic sets varies continosly ([26]). Theorem 1.2 follows taking the intersection B = Dring the past fifteen years or so there has been mch interest in the stdy of so-called SRB measres in area decreasing planar diffeomorphims. An f invariant probability measre ν of a diffeomorphism will be called an SRB measre if it is ergodic, has compact spport, and has absoltely conditional measres on nstable manifolds. A family ( a, b) f a,b of diffeomorphisms which is near the Henon family H a,b ( x, y) = (1 + y ax 2, bx) is called a Henon-like family. Diffeomorphisms f a,b in Henon-like families are called Henon-like diffeomorphisms. It is a conseqence of the work of many athors (e.g., [2], [3], [18], [31]) that for 1. This set n Henon-like families with b small there is a positive Lebesge measre set of parameters A( b) sch that f a,b has an SRB measre for a A( b). Ths, sch measres exist with positive probability in parameters in Henon-like families. At the present time all known SRB measres in Henon-like diffeomorphisms are spported on homoclinic sets with tangencies. n B n. To be more precise, in a Henon-like family f a,b let s call a pair of parameters ( a, b) good if f a,b has a hyperbolic periodic saddle point p a,b and an SRB measre ν a,b spported on the homoclinic set of p a,b. All presently known good pairs ( a 0, b 0 ) have the following property. There is a seqence a 1, a 2,... of real parameters converging to a 0 sch that each f ai,b 0 contains persistent homoclinic tangencies. It may be that techniqes of the proof of Theorem 1.4 below can be extended to show that for any good pair ( a 0, b 0 ) one can find a seqence a 1, a 2,... converging to a 0 sch that there is no SRB measre spported on the homoclinic set of the following conjectre. p a i,b 0. This prompts s to make Conjectre 1.3 Let H a,b be the Henon family of planar maps. Then, for each parameter b, there is a residal set of parameters a sch that H a,b has no SRB measre. For b = 0 this conjectre is tre since the maps essentially redce to the familiar logistic family a f a ( x) = ax(1 x) and one may apply the well-known reslts of Graczyk and Swiatek [9] and Ljbich [15]. These imply that there is a dense open set of parameters a sch that f a has an attracting 6

7 periodic orbit whose basin contains a set of fll measre. For b = 1 the map H a,b is area preserving, and the conjectre implies that, for a residal set of parameters a, one has that H a,b has zero measre-theoretic entropy on any invariant set of finite area, We s proceed to state the main new reslt in the present note. If ν is an invariant probability measre for f, we say that a sbset Λ carries the measre ν if ν(λ) = 1. Theorem 1.4 There is a residal sbset U in D r ( M) for r 2 sch that if f U and Λ is a homoclinic set for f which contains a tangency and has an associated dissipative periodic point, then Λ does not carry an SRB measre. Theorem 1.4 has interesting relations to a conjectre of Palis [24]. This conjectre states that there is a dense sbset A in the space D r ( M) of C r diffeomorphisms of a compact srface M sch that if f A, then either f is hyperbolic or f has a homoclinic tangency. For r = 1, this conjectre has recently been proved by Pjals and Samborino [28]. It is still nproved for r 2. It follows from Theorem 1.1 that if the conjectre is tre for r 2, then the set A can actally be chosen to be dense and open and the homoclinic tangency occrs in some homoclinic set. If one wants information abot all homoclinic sets, one has to pass from dense and open to residal. Ths, we state the Conjectre 1.5 (Weak Palis Homoclinic Conjectre) Let r 1. There is a residal sbset A in D r ( M) sch that for f A, each homoclinic set for f is either niformly hyperbolic or has a homoclinic tangency. In view of Theorem 1.4 we have The Weak Palis Homoclinic Conjectre implies that there is a residal sbset A of D r ( M) sch that if f A, then any SRB measre whose spport contains a dissipative periodic point mst be spported on a niformly hyperbolic attractor. Next, we wish to srvey some connections between homoclinic tangencies and the existence of symbolic extensions. Let Z denote the set of integers, N 2 be a positive integer, J = {1,..., N},and let Σ N = J Z be the set of dobly infinite seqences of symbols in the alphabet J. Elements a in Σ N are denoted 7

8 We give Σ N the standard metric d( a, b) = a = ( a i ), i Z. i Z a i b i 2 i, making (Σ N, d) into a compact metric space. The shift atomorphism is the map σ = σ N : Σ N Σ N defined by σ N ( a) i = a i+1 for i Z. A sbshift or symbolic system is a pair ( S, X) where X is a closed σ N invariant sbset of σ N and S is the restriction of σ N to X for some N. Sbshifts ( S, X) are known to have many special dynamical properties. Let s list some of these. 1. They are expansive. That is, there is a positive real nmber ɛ > 0 sch that if x y in X, then there is an integer n sch that d( S n x, S n y) > ɛ. This implies, for instance, that if ν is an S invariant probability measre, then its metric entropy can be compted as the mean entropy h ν ( α) for any finite partition α whose elements have diameter less than ɛ. 2. The metric entropy fnction ν h ν ( S) is ppersemicontinos as a fnction of ν. Hence, any sbshift has measres of maximal entropy. 3. The topological entropy h top ( S) can be compted as h top ( S) = lim n 1 n log card B n where card B n is the nmber of n blocks appearing in X. If a given system ( f, M) cold be modelled in some sense by a sbshift, then one wold have mch information abot the dynamics of most sefl notion of modelling is that of topological conjcacy. However, since sbshift spaces have topological dimension zero, most systems cannot be topologically conjgate to sbshifts. Ths, one wold like to weaken the notion of topologically conjgacy somewhat to se symbolic systems as models. f. The 8

9 Let f : X X and g : Y Y be homeomorphisms of the compact metric spaces X and Y, respectively. We say that ( g, Y ) is an extension of ( f, X) if there is a continos srjection π : Y X sch that πg = fπ. In this case, we call the triple ( g, Y, π) an extension triple of ( f, X). It is natral to ask when a given system ( f, X) has a symbolic extension. That is, when can we find an extension ( g, Y ) which is a symbolic system. An obvios necessary condition is that h top ( f) <. Arond 1988, J. Aslander asked the converse qestion: Does every homeomorphism of a compact metric space with finite topological entropy have a symbolic extension? This qestion trned ot to be highly non-trivial. In 1990, Mike Boyle fond a conter-example sing an inverse limit constrction which led to a certain zero dimensional system with no symbolic extension. One can ask for more precise information regarding the kinds of extensions a system may or may not have. One says that an extension triple ( g, Y, π) is a principal extension of ( f, X) if the extension map π simltaneosly preserves entropies of all invariant probability measres. That is, for every g invariant probability measre ν, we have h π ν( f) = h ν ( g) (here, π ν = ν π 1 ). In the sense of information theory, a principal extension of a given system is indistingishable from that system. Arond the time of the discovery of Boyle s example mentioned above, several athors proceeded to stdy the existence of symbolic and principal symbolic extensions. Tomasz Downarowicz [7] gave a characterization of finite entropy zero dimensional systems ( f, X) which have principal symbolic extensions in terms of certain ppersemicontinos fnctions on the space of invariant probability measres. It trned ot that a necessary and sfficient condition was that f be asymptotically h expansive in the sense of Misirewicz [17]. Later, Boyle, D. Fiebig, and U. Fiebig [5] extended this to the general case; i.e. a finite entropy system has a principal symbolic extension if and only if it is asymptotically h expansive. On the other hand J. Bzzi [6] proved that every C map is asymptotically h expansive. So, these reslts together yield the following striking reslt. Theorem 1.6 Every C diffeomorphism of a compact manifold has a principal symbolic extension. 9

10 It is natral to ask abot systems with a finite amont of smoothness. Since every C 1 self map of a compact manifold has finite topological entropy, the qestion simply becomes when does a C r map on a compact manifold with 1 r < have a symbolic or principal symbolic extension. In this connection, recently T. Downarowicz and the athor [8] have proved the following reslts. Theorem Given any compact C manifold M, there is a C 1 diffeomorphism f on M with no symbolic extension at all. 2. There is a residal sbset A in the space of C 1 symplectic diffeomorphisms on a compact orientable two dimensional manifold sch that if f A, then either f is Anosov or f has no symbolic extension at all. 3. For any r 2 and any compact two dimensional manifold M 2, there exists a residal sbset A of an open sbset U D r ( M 2 ) sch that if f A, then f has no principal symbolic extension. Regarding C r maps with 2 r <, we have the following Problem. Let 2 r <. Does every C r self-map of a compact C manifold have at least one symbolic extension? It trns ot that intervals of homoclinic tangencies again play a fndamental role in the proof of Theorem 1.7. After making certain pertrbations, one gets diffeomorphisms sch that the entropy fnctions of invariant invariant probability measres exhibit pathological continity properties. Then one applies reslts of Boyle and Downarowicz [4] concerning the existence of varios types of symbolic extensions. Dring or lectre at the conference, Dennis Sllivan made the following interesting observation. Note that if a system ( f, X) possesses a (principal) symbolic extension, then so does any system ( f 1, X 1 ) which is topologically conjgate to ( f, X). Ths, the above reslts have some interesting new corollaries. For instance, a generic C 1 non-anosov area preserving diffeomorphism on a srface is not topologically conjgate to any C diffeomorphism. 2 Proof of Theorem 1.4 We begin with a proposition which says that if a homoclinic set carries an SRB measre ν, then the spport ν contains the fll nstable manifold of the orbit of any associated periodic point. 10

11 Proposition 2.1 Sppose that Λ is a homoclinic set which carries an SRB measre ν. Then, the spport of ν consists of the closre of the orbit of the nstable manifold of every periodic point associated to Λ. Proof. Since ν is ergodic, Λ is f invariant, and ν(λ) > 0, we have, in fact, that ν(λ) = 1. Also, since ν has non-zero Lyapnov exponents, there is an invariant set Λ 1 Λ of fll ν-measre sch that each x Λ 1 has a C r nstable manifold W ( x). This is the set of points y sch that d( f n x, f n y) approaches 0 exponentially fast as n. Since ν has absoltely continos conditional measres along nstable manifolds, it is known that one can choose Λ 1 so that if x Λ 1, then W ( x) is completely contained in the spport of ν. We refer to standard places (e.g. [1], [27], [14]) for proofs of these facts. Ths, µ is spported on a set which contains many fll nstable manifolds. We wish to show that every nstable manifold of a periodic orbit associated to Λ is also in the spport of ν. Note that, obviosly, Λ 1 contains no periodic orbits. Let p be a hyperbolic periodic point associated to Λ. For any point x, let O( x) = {f n ( x) : n Z} denote the orbit of x. Let x Λ 1, and let y Λ be an ω limit point of x. Ths, there are three distinct iterates x 1 = f n 1 x, x 2 = f n 2 x, x 3 = f n 3 x near y so that 1. there is a small crvilinear rectangle D sch that f( D) D =, 2. the bondary of D consists of pieces γ 1 W ( x 1 ), γ s 1 W s ( x 1 ), γ 2 W ( x 2 ), γ s x W s ( x 2 ), 3. x 3 is contained in the interior of D, and 4. O( p) D =. Let s recall a reslt de to Katok in [13]. He proves that there is a seqence µ 1, µ 2,... of measres spported on hyperbolic periodic orbits which converges weakly to ν. Indeed his proof shows that we may find three hyperbolic periodic points p i arbitrarily close to x i, sch that there are crves ηi W ( p i ), ηi s W s ( p i ) sch that 5. η i is C r close to γ i and η s i is C r close to γ s i for i = 1, 2, 3, 6. there is a rectangle D close to D and bonded by the crves η 1, η1, s η2, η2 s sch that p 3 interior( D), and 11

12 7. O( p) D =. These properties are illstrated in Figre 1. γ 2 η 2 γ 3 s x 2 η 1 s γ 3 η 3 x 3 p 3 s η 3 η 2 s p 2 γ 1 s x 1 γ 1 p 1 η 1 γ 2 s 4 Figre 1: Solid lines correspond to x i, dashed lines correspond to p i It now follows that each W s ( p i ) has some non-empty transverse intersections with W ( O( x)), so the Inclination Lemma (Theorem 2, page 155, in [25]) implies that W ( O( p i )) Closre( W ( O( x))) Λ (2) for each i. Frther, since Λ is the closre of the set of transverse homoclinic points of p, we have that W s ( O( p)) accmlates on x 3. Since O( p) D =, we may find a q O( p) sch that W s ( q) meets both the interior and exterior of D. Hence, it mst also cross either η 1 or η2. In the first case, an elementary two dimensional argment sing the Inclination Lemma shows that W ( O( p)) Closre( W ( O( p 1 )), and a similar argment in the second case gives W ( O( p)) Closre( W ( O( p 2 ))). Now (2) implies the reqired statement that W ( O( p)) Λ. QED. The next Lemma, which gives new information abot the typical way in which sinks approach a homoclinic set, is the key new ingredient of the proof of Theorem

13 Let p be a hyperbolic saddle periodic point, and q be a hyperbolic sink (attracting periodic point). We say that q is related to p (short for nstably related) if W ( O( p)) W s ( O( q)). We also say that q is related to a homoclinic set Λ if there is a periodic saddle point p associated to Λ sch that q is related to p. Lemma 2.2 Let r 1, and let f be a C r diffeomorphism of a compact srface, and let p( f) be a dissipative hyperbolic periodic point of f whose homoclinic closre contains a tangency. Then, arbitrarily C r close to f one can find a g sch that p( g) has a related sink. Proof. Observe that the orbit of a homoclinic tangency consists of homoclinic tangencies, and the stable and nstable manifolds of p accmlate near the orbit of the homoclinic tangency in Λ( f). Ths, the stable and nstable manifolds of p accmlate on some homoclinic tangency in Λ( f). After a small pertrbation, we may assme that W ( p) has a point of tangency with W s ( p). Let τ > 0 be the period of p. Next, we can find a g 1 C r near f sch that 1. g 1 is C, 2. the contination p( g 1 ) is defined and hyperbolic, 3. there is a C 2 linearization of g τ 1 near p( g 1 ), and 4. W ( p( g 1 )) and W s ( p( g 1 )) have a qadratic tangency at a point q 1. Replacing g 1 by g τ 1, we may assme that p( g 1 ) is a fixed saddle point of g 1. Let λ, σ denote the eigenvales of Dg τ 1 ( p( g 1 ) with 0 λ < 1 < σ. Replacing g 1 by g1, 2 we may assme that λ and σ are positive. Next, we embed g 1 in a one parameter family µ f µ of diffeomorphisms defined for µ near 0 so that the family creates a non-degenerate tangency of W ( p µ ) and W s ( p µ ) at q 1. Here, f 0 = g 1 and p µ is the contination of p( g 1 ). 13

14 Now, we se the techniqes of the proof of Theorem 1 on page 47 of the book [25] of Palis and Takens, to show that, for some µ s near 0, there are crvilinear rectangles D µ near q 1 whose first retrn maps F µ of f µ are smoothly conjgate to small pertrbations of certain Henon maps H µ,b defined on the sqare Q = [ 1, 1] [ 1, 1] where µ = µ( µ) depends smoothly on µ.. We will see that the maps H µ,b can be chosen to have fixed sinks z 0 ( µ) whose stable sets contain a fll-width strip of fixed height in Q. Ths, the corresponding maps f µ will have periodic sinks w 0 ( µ) whose stable manifolds contain fll-width sbrectangles of D µ. The retrn maps F µ will contract area very sharply for small µ on a larger rectangle D µ D µ which has a part of W ( p 1 ) as a fll-height crve. This will imply that p 1 is -related to the sinks w 0 ( µ). Let s proceed to the details. We find a one-parameter family µ f µ of C r diffeomorphisms on M sch that the following conditions hold. 1. f 0 = g 1 2. the family {f µ } creates a non-degenerate homoclinic tangency between W ( g 1 ) and W s ( g 1 ) at q 1 3. there is a µ-dependent C 2 linearization ψ µ of f µ in a neighborhood of a compact crve γ s ( µ) W s ( p µ ) sch that γ s (0) contains q 1 and p( g 1 ) 4. there is a seqence J n of parameter intervals converging to 0 as n so that if µ J n, then one can find a rectangle D µ near q 1 and a C r diffeomorphism Φ µ : D µ Q sch that, letting F µ be the first retrn map of f µ to D µ and λ = λ µ, σ = σ µ be the eigenvales of Df µ at p µ, we have (a) Φ µf µ Φ 1 µ def = G n,µ is defined on Q and has the form G n,µ ( x, y) = ( y, µ + y 2 λ n µσ n µx) + S( µ, x, y) where S( µ, x, y) 0 as n and µ µ σ 2n µ, (b) µ crosses all of [0, 1 2 ] as µ crosses J µ, and (c) denoting the height of D µ by height( D µ ) and the width of D µ by width( D µ ), we have height( D µ ) σ 2n µ, width( D µ ) height( D µ )

15 Ths, we have that the first retrn map F µ on the rectangle D µ is conjgate to a small pertrbation of the Henon map H µ,b ( x, y) = ( y, µ + y 2 bx) on the rectangle Q with b = λ n µσ n µ and n large. We claim that, for small b, and µ [ H µ,b has a niqe attracting fixed point z 0 = z 0 ( µ, b) sch that W s ( z 0 ) contains the rectangle R = [ 1, 1] [ 7, 7]. 8 8 To see this, first consider the singlar family H µ,0. The calclations redce to analogos ones in the one dimensional logistic family y µ + y 2. One easily comptes that, for µ [ H µ,0 has the attracting fixed point 1 z 0 ( µ, 0) = ( 1 4 µ, µ ), and its stable manifold contains R. Now, for 2 2 small b, the fixed point z 0 ( µ, 0) contines to a nearby one z 0 ( µ, b) with the reqired properties. Here we se the fact that stable manifolds of hyperbolic fixed points depend continosly on compact sets even for singlar maps. This follows from the proofs of the stable manifold theorem in [12] or [29]. Now, retrn to the map F µ above. In the domain of the linearizing coordinates ψ µ, enlarge the rectangle D µ horizontally to a rectangle Dµ which projects vertically onto all of γ s and has D µ as a fll-height sbrectangle. Ths, Dµ will contain both D µ and a fll-height crve γ which is contained in W ( p µ ). Since λ µ σ µ < 1, for n large, the image F µ ( D µ ) is a slight thickening of F µ ( γ ) which is mch narrower than the width of D µ as in Figre 2. This implies that the stable manifold of the attracting fixed point of F µ will meet F µ ( γ ), proving Lemma 2.2. QED. We can now complete the proof of Theorem 1.4. For each positive integer n, let U n be the sbset of D r ( M) sch that if f U n, then all periodic points of f of period less than or eqal to n are hyperbolic with characteristic exponent sm different from zero. As is wellknown, argments as in the proof of the Kpka-Smale theorem give that U n is a dense open sbset of D r ( M). For f U n, let Pn( t f) denote the periodic points p of f of period less than or eqal to n sch that 1. p is dissipative, and 2. there is a seqence of diffeomorphisms g 1, g 2,... converging to f in D r ( M) sch that, for each i, the h closre of p( g i ) contains a homoclinic tangency. 15

16 W (p) ~ F (D ) µ µ W (p) γ p ~ Dµ q 1 D µ Figre 2: Dµ is a rectangle containing D µ which is wide enogh to project onto a line segment in W s (p) which contains both p and q 1 ; F µ ( D µ ) is near q 1 and is a slight thickening of the crve F µ (γ ). Let V n be the set of f s in U n sch that P t n( f) is empty. Then, V n is an open sbset of U n. Letting F n = U n \ V n, we have that f F n implies that P t n( f) is a finite non-empty set of periodic points. Label these as with s n = s n ( f). For m > 0, let and let P t n( f) = {p 1 ( f), p 2 ( f),..., p sn ( f) } F n,m = {f F n : s n ( f) = m}, Then, T n = {m : F n,m }. T n is a finite set of positive integers and we have the disjoint nion F n = F n,m. m T n 16

17 For m T n and 1 j m, let F n,m,j be the set of f s in F n,m sch that the h closre of p j ( f) has persistent tangencies and a related sink. By Theorem 1.1 and Lemma 2.2, each F n,m,j is dense and open in F n,m. Hence, the set def E n = m T n [ 1 j m F n,m,j ] is also dense and open in F n. Let G n = V n En. This set is dense and open in D r ( M) for each n. If f G n, then any dissipative periodic point p of f with period less than or eqal to n whose h closre contains a tangency mst have a related sink. By Proposition 2.1, no sch h closre can carry an SRB measre. Now, set U = n G n. This is the residal sbset of D r ( M) reqired in Theorem 1.4. QED. References [1] M. Herman A. Fathi and J.C. Yoccoz. A proof of pesin s stable manifold theorem. In J. Palis, editor, Geometric Dynamics, volme 1007 of Lectre Notes in Math., pages Springer-Verlag, [2] M. Benedicks and L. Carleson. The dynamics of the Henon map. Annals of Math., 133:73 169, [3] M. Benedicks and L-S. Yong. SBR measres for certain Henon maps. Inventiones Math., 112: , [4] M. Boyle and T. Downarowicz. The entropy theory of symbolic extensions. preprint, mmb/papers.html. [5] M. Boyle, D. Fiebig, and U. Fiebig. Residal entropy, conditional entropy, and sbshift covers. preprint, mmb/papers.html. [6] J. Bzzi. Intrinsic ergodicity for smooth interval map. Israel J. Math, 100: ,

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