An Open Dense Set of Stably Ergodic Di!Eomorphisms in a Neighborhood of a Non-Ergodic One

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Topology 40 (2001) 259}278

An open dense set of stably ergodic di!eomorphisms in a neighborhood of a non-ergodic one

Viorel Nit7 icay * , Andrei ToK roK k Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700 Bucharest, Romania Received 16 February 1999; received in revised form 27 April 1999; accepted 14 June 1999

Abstract

As a special case of our results we prove the following. Let A3Di! P(M) be an Anosov di!eomorphism. P ; Then there is a C -neighborhood of A Id1 that contains an open dense set of partially hyperbolic di!eomorphisms that have the accessibility property. If, in addition, A preserves a smooth volume l and j is ; ; the Lebesgue measure on S , then in a neighborhood of A Id1 in Di! J"H(M S ) there is an open dense set of (stably) ergodic di!eomorphisms. Similar results are true for a neighborhood of the time-1 map of a topologically transitive (respectively volume preserving) Anosov #ow. These partially answer a question posed by C. Pugh and M. Shub. We also describe an example of an accessible partially hyperbolic di!eomorphism that is not topologically transitive. This answers a question posed by M. Brin. ( 2000 Elsevier Science Ltd. All rights reserved.

MSC: 58F

Keywords: Stably ergodic; Accessibility property

1. Introduction

Let X be a compact smooth manifold without boundary endowed with a smooth volume k. P P P Denote by Di! (X) the set of C -di!eomorphisms of X, and by Di! I(X) the set of volume P 3 P preserving C -di!eomorphisms of X.Adi!eomorphism f Di! I(X) is called ergodic if for any

* Corresponding author. Department of Mathematics, University of Notre-Dame, Notre Dame, IN 46556-5683, USA. Tel.: #219- 631-8848; fax: #219-631-6579. Current address for both authors: Department of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA. E-mail addresses: [email protected] (V. Nit7 icay ), [email protected] (A. ToK roK k)

L k " k ! " 3 P measurable f-invariant set A X either (A) 0or (X A) 0. A di!eomorphism f Di!I(X)is P called stably ergodic if there is a neighborhood of f in Di! I(X) consisting only of ergodic di!eomorphisms.

Remark. If no other mention is made, the notations CP and Di! P are used for the case r*1.

Classical results of Kolmogorov, Arnold and Moser show that stable ergodicity is not always P a generic property in Di! I(X). Indeed, by the KAM theory, near a non-degenerate integrable Hamiltonian system there is a set of positive measure of invariant tori. In contrast, Anosov showed in [1] that a volume preserving uniformly hyperbolic di!eomorphism is stably ergodic. Grayson et al. [7] found the "rst example of a non-hyperbolic stably ergodic di!eomorphism, the time-1 map of the geodesic #ow of a surface S of constant negative curvature. Wilkinson [15] generalized this result to the case when S has variable negative curvature. Pugh and Shub [13] later proved it for higher-dimensional manifolds of (almost) constant negative curvature. These results were improved in [14]. Results about genericity of ergodic skew-products over Anosov di!eomorphisms with a compact connected Lie group as "ber were found by Brin [3]. These rely on earlier work of Brin and Pesin [4] about ergodic partially hyperbolic di!eomorphisms. Recently, Burns and Wilkinson [5] proved results about the stable ergodicity of ergodic skew-products with the "ber a compact connected Lie group. Recall that a partially hyperbolic di!eomorphism has stable and unstable foliations, =Q and =S. In addition, an r-normally hyperbolic di!eomorphism has also an invariant center `leaf-immer- siona with CP leaves, tangent to the center distribution. (For the precise de"nitions see Section 2.) Given a partially hyperbolic di!eomorphism f3Di! (X), we say that two points x,y3X are accessible (or, more precisely, (u,s)-accessible) if they can be joined by a piecewise di!erentiable piecewise nonsingular path consisting of segments tangent to either ES or EQ.(Essential) accessibility of f means that (almost) each pair of points x,y3X is accessible. We say that a set ALX is accessible if any two points x,y3A are accessible. It was conjectured by Pugh and Shub that the set of stably ergodic di!eomorphisms is open and dense among the partially hyperbolic C volume-preserving di!eomorphisms of a compact manifold X. They conjectured also that stable accessibility is an open and dense property among C partially hyperbolic di!eomorphisms, volume preserving or not. (Note that in both conjectures the openness follows from the de"nitions.) Accessibility is relevant due to the following remarkable result of Pugh and Shub [14]:

3 Theorem 1.1 (Pugh, Shub). If f Di!I(X) is a center bunched and dynamically coherent partially hyperbolic diweomorphism with the essential accessibility property then f is ergodic.

This paper provides some evidence for a positive answer for the latter conjecture. Our main result shows that in an open class of partially hyperbolic di!eomorphisms, there is an open and dense set consisting of di!eomorphisms that have the accessibility property:

P P Theorem 1.2. In both Di! (X) and Di! I(X) the stably accessible partially hyperbolic diweomorphisms form a C-open and CP-dense set among those r-normally hyperbolic diweomorphisms with a V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278 261 one-dimensional center distribution that have two compact periodic leaves whose Hausdorw distance is small enough. (By Theorem 6.1 of Hirsch et al. [9], the latter condition is open in CP.) More precisely, we need the following (CP-open) conditions on the center leaves: E there is a periodic compact one-dimensional center leaf C, and E each point of C is close enough to a periodic compact center leaf that is disjoint from C (by compactness, xnitely many such leaves fulxll this condition for all the points of C). The precise meaning of `nearbya and `close enougha is given by Dexnition 2.1.

Theorem 1.2 follows from Theorems 3.2 and 3.3. In the former we exhibit a C-open set of partially hyperbolic di!eomorphism that are accessible. In the latter we show that some partially hyperbolic di!eomorphisms can be made to belong to this set by an arbitrarily CP-small perturbation. If a smooth volume is given, the perturbation can be made volume preserving. In some cases the existence of periodic center leaves follows from the PoincareH Recurrence Theorem and a Shadowing Lemma in [9]:

Lemma 1.3. Assume X is a compact manifold endowed with a probability measure k that is positiveon 3 open sets. Let f Di!I(X) be a partially hyperbolic diweomorphism that is dynamically coherent and whose center lamination is plaque expansive. Then the periodic center leaves of f are dense in X.

A proof of this fact will be sketched in Appendix B. Therefore, Theorems 1.1 and 1.2 imply that the set of stably ergodic partially hyperbolic di!eomorphisms is open and dense in certain open sets of Di!I(X):

Theorem 1.4. In Di!I(X) the stably ergodic partially hyperbolic diweomorphisms form an open and dense set among those diweomorphisms that are 2-normally hyperbolic, center bunched, dynamically coherent, plaque expansive, with one-dimensional center distribution, and have compact center leaves. (By Theorem 6.1 of Hirsch et al. [9] and Proposition 2.3 of Pugh and Shub [13], the latter set is open in C.)

The main examples consist of time-1 maps of Anosov #ows, and of Di!(S)-valued skew- products over Anosov di!eomorphisms:

P + , Theorem 1.5. Let be the time-1 map of a C Anosov yow R of a compact manifold M. Then the following are true: + , P (a) Assume that the yow R is topologically transitive. Then there is a C -neighborhood of in Di! P(M) that contains an open and dense set of diweomorphisms with the accessibility property. + , k (b) Assume that there is a R -invariant volume on M. Then there is a C -neighborhood of in Di! I(M) that contains an open and dense set of stably ergodic diweomorphisms. Indeed, for time-1 maps of Anosov #ows that are topologically transitive or volume preserving, the compact center leaves (i.e., the closed trajectories of the #ow) are dense. This follows from 262 V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278

Anosov's Closing Lemma. Note that, in general, the time-1 map of an Anosov #ow is not accessible. For example, in the case of a suspension #ow over an Anosov di!eomorphism, the strong stable and strong unstable distributions are jointly integrable.

Theorem 1.6. Let A be a CP Anosov diweomorphism of a compact manifold M and b : MPDi! P(S) a CP map (i.e.,(x,y)3M;SCb(x)(y)3S is CP). Assume that

""¹QA""(inf m(¹b(x)) and sup ""¹b(x)""(m(¹SA). (1.1) VZ+ VZ+ Let f : M;SPM;S be given by the skew-product f (x, y):"(Ax, b(x)(y)). Then the following is true: (a) There is a CP-neighborhood of f in Di! P(M;S) that contains an open and dense set of diweomorphisms with the accessibility property. (b) Assume moreover that f is center bunched and there is a smooth f-invariant volume k on M;S. ; Then there is a C -neighborhood of f in Di!I(M S ) that contains an open and dense set of stably ergodic diweomorphisms. (Assumption (1.1) implies that the skew-product f is a partially hyperbolic diweomorphism. For the dexnition of the conorm m( ) ) see Section 2.)

l j 3 Corollary 1.7. Let be a smooth volume form on M and the Lebesgue measure on S . If A Di! J(M) ; 3 ; is hyperbolic then in a C -neighborhood of A Id1 Di! J"H(M S ) there is an open dense set of ergodic diweomorphisms.

A related question is the topological transitivity of partially hyperbolic di!eomorphisms. Recall that a homeomorphism f of a topological space X is topologically transitive if for some point x3X the orbit + f L(x) " n39, is dense. If X is a compact metrizable perfect space then topological transitivity is equivalent to the existence of a point y3X whose positive semi-orbit +f L(y) " n"0,1,22, is dense (see Exercise 1.4.2 in [10]). Adi!eomorphism of a compact Riemannian manifold that has an ergodic invariant volume is topologically transitive. Therefore, by (b) of Theorem 1.6, there is an open and dense set of ; ; transitive di!eomorphisms in a C -neighborhood of A Id1 in Di!I(M S ). Brin proved in [3] that a partially hyperbolic di!eomorphism that has the accessibility property and is recurrent is also topologically transitive. He asked if accessibility of a partially hyperbolic di!eomorphism is su$cient for topological transitivity. Our next corollary answers this question in the negative due to the fact that, if A is an Anosov di!eomorphism, the set of topologically ; transitive di!eomorphisms is not dense in any C -neighborhood of A Id1. We are grateful to Burns for pointing out this consequence of our results and to the referee for suggesting that we state it in this generality.

Corollary 1.8. Let A be a C Anosov diweomorphism of a compact manifold M and R a rotation of S. Then there exists arbitrarily C-close to A;R a C-open set of partially hyperbolic diweomorphisms that are accessible, but not topologically transitive. V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278 263

Proof. By (a) of Theorem 1.6 it is enough to "nd arbitrarily close to A;R an open set of non-transitive di!eomorphisms. " 3 For simplicity, we consider "rst the case R Id1. Pick Di! (S ) as close to Id1 as desired such that has a "xed attracting point. Hence, there are open sets ;,, where i /(n ) for some n. One can choose now arbitrarily close to R a circle di!eomorphism for which there exist + , O M ' M L " ; n disjoint open intervals JG G such that JG IG and (IG) JG>. Let f A . Then ; M L ; f (M IG) M JG>, and any map that is C -close to f has the same property. As above, such a map cannot be topologically transitive. ᮀ

On the other hand, Bonatti and DmH az [2] proved, among other striking results, that if M and N are compact boundaryless manifolds and A is a transitive C Anosov di!eomorphism of M, then 3 ; 3 " ; there is a C -arc FR Di! (M N), t [0,1], such that F A Id, and there is a C -neighbor- + , hood of FR RZ( consisting of non-hyperbolic transitive di!eomorphisms. Let us also mention that according to Grayson et al. [7], "nding a C-neighborhood of the time-1 map of the geodesic #ow of a surface of constant negative curvature consisting of topologically transitive di!eomorphisms is an open question. As we pointed out earlier, one can study accessibility and ergodicity in thinner classes of transformations, e.g., skew-products over Anosov di!eomorphisms. The proof of Theorem 1.2 can be easily adapted to show that accessibility is generic in a class of HoK lder extensions of Anosov di!eomorphisms (see Theorem A.1). We also extend Brin genericity result about topological transitivity to the class of HoK lder cocycles (Theorem A.2). Note that the corresponding result for ergodicity is proved in [6]. We discuss these results in Appendix A. The existence of the stable and unstable foliations for such transformations does not follow in a straightforward way from the general theory of partially hyperbolic di!eomorphisms, so we prove their existence as well.

2. De5nitions

We recall "rst several standard facts about partially hyperbolic di!eomorphisms. Given a linear transformation ¸ between two normed linear spaces, the norm and the conorm of ¸ are de"ned by

""¸""": sup+""¸v"" " ""v"""1, and m(¸):"inf+""¸v"" " ""v"""1,"""¸\""\.

Remark. In the sequel by a CP-lamination, r'0, we mean a C-foliation whose leaves are immersed CP-submanifolds that vary continuously in the CP-topology. A foliation stands for a C-foliation. 264 V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278

Let X be a compact, connected, boundaryless manifold. Denote by ¹X the tangent bundle of X. A C-di!eomorphism f : XPX is called partially hyperbolic if the derivative ¹f : ¹XP¹X leaves invariant a continuous splitting ¹X"EQEAES, EQO0OES, such that, with respect to a "xed Riemannian metric on ¹X:

""¹Sf \""(1, ""¹Qf ""(1, (2.1) ""¹Q ""( ¹A ""¹A ""( ¹S 3 N f m( N f ), N f m( N f ) for all p X. (This is a slightly weaker condition than the one used in [14].) EQ, EA and ES are called the stable, center, respectively, unstable distributions. If the center distribution EA"0, then f is called an Anosov (or hyperbolic)di!eomorphism. The distributions EQ and ES are tangent to unique HoK lder laminations =Q and =S which have C leaves. These are called the stable and unstable foliations. [To be precise, we could call these laminations, but that is not the standard terminology.] If ESEA, EA, and EAES are also tangent to continuous foliations with C leaves =AS, =A, respectively, =AQ, and if =A and =S subfoliate =AS, while =A and =Q subfoliate =AQ, then f is said to be dynamically coherent. Let f : XPX be a partially hyperbolic di!eomorphism. f is r-normally hyperbolic if the center distribution EA is integrable to a CP-boundaryless leaf immersion (see [9, Section 6]) and ¹S *""¹A ""I ""¹Q "") ¹ A I " m( N f ) N f , N f m( N f ) , k 0,2,r. Roughly speaking, the center distribution integrates to a `laminationa that can have self-intersections; its leaves are CP. This set-up is necessary in order to assure that r-normal hyperbolicity is a C-open condition. The center bolicity of f is the ratio

""¹Af "" b" . m(¹Af )

The map f is said to be center bunched if b is close to 1. More precisely (see Section 4 of [14]; our notation is di!erent): let 0(c,l,j(1bede"ned by c ": min+m(¹A f ),""¹A f ""\,, l" + ¹Q ""¹S ""\, j " +""¹S \"" ""¹Q "", " l* : min m( f ), f , : max f , f and set q : logH 1; then f is center bunched if

log c 3#2q!((3#2q)!8 ( . log j 4

In particular, l(j(c and the spectra of ¹Qf, ¹Af and ¹Sf are contained in the annuli corresponding to the intervals [l, j], [c, c\], respectively, [j\, l\]. This is a stricter condition than the `relativea partial hyperbolicity introduced in (2.1). By Theorem 6.1 in [9], r-normal hyperbolicity is an open property in Di! P(X). The property of being center bunched is preserved by C-small perturbations. By Proposition 2.3 of Pugh and Shub [13], dynamical coherence is stable under C-small perturbations, provided the center lamination is plaque expansive. Plaque expansiveness is a technical condition on an f-invariant lamination. Intuitively, it means that separate leaves of the lamination eventually diverge to a "xed distance V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278 265 under either forward or backward f-iterations (see Section 7 of Hirsch et al. [9] or Appendix B for more details). For Anosov di!eomorphisms, i.e., when the center lamination consists of points, this property reduces to expansivity. By Theorem 7.2 of Hirsch et al. [9], the center lamination is plaque expansive if it is a C-foliation.

De5nition. If F is a foliation of a Riemannian manifold M and F(x) is the leaf passing through 3 x M, denote by FB(x) the connected component of the intersection of F(x) and the open ball of radius d around x.

The following local product structure property follows immediately from compactness and transversality for any partially hyperbolic map that has the distributions EQEA and ESEA tangent to the continuous foliation =AQ and =AS with C-leaves: There exist constants e'0, d'0 and K'1 such that for any x, y3X with dist(x, y)(e, there is Q 5 AS S 5 AQ a unique point z in =B(x) =B (y), and a unique point z in =B(x) =B (y). Moreover + ,( max dist(x, z), dist(y, z) K dist(x, y), + ,( max dist(x, z), dist(y, z) K dist(x, y), E AQ AS AQ AS the submanifold pairs =B (x) and =B (y), respectively, =B (y) and =B (x), intersect transversally; E AQ 5 AS " A AQ 5 AS " A =B (x) =B (x) =B(x) and =B (y) =B (y) =B(y).

By Theorem 6.1 of [9], the quantities e, d and K are lower-semicontinuous with respect to the C-topology on Di!(X).

De5nition 2.1. Two neutral leaves C,C are called nearby if the Hausdor! distance between them is smaller than d"d( f ):"e/(5K). A point x3X is close enough to a center leaf C if dist(x,C)(d.

3 3 (d De5nition 2.2. Given a center leaf C and points xH C , x X such that dist(xH, x) ,wede"ne the holonomy H : (x )P by H (x )"x , where x 3 (x ) and x is constructed as VH V CBY H C VH V CBY H follows:

" Q 5 AS x : =B(x) =B (x), " S 5 AQ L S 5 x : =B(x) =B (x) =B(x) C , " Q 5 AS x : =B(x) =B (x), " S 5 AQ L S 5 x : =B(x) =B (x) =B(x) C .

If necessary, we will refer to the points x in the above construction by H(I) (x ), 1)k)4. Here I VH V d C is the center leaf containing x. The choice of assures that the intersections de"ning x,2,x 266 V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278 have exactly one element.

Note that the pair (x,x)is(u,s)-accessible.

Remark 2.3. Assume that in the above de"nition C is connected and one-dimensional, and O x x. Then there is an open neighborhood of x in C whose points are accessible from x. 3 " " Indeed, by choosing a small continuous path x(t), t [0,1] that connects x x(0) to x x(1) and constructing the (u, s)-loops given by H (x ), t3[0,1], we can access from x at least V V(R) Q S a closed interval I of C ending at x. Since the foliations = and = are continuous, there is a very 3 short (u, s) loop starting from a point x C close to x but outside I and ending in I. Contract this loop to x with the same procedure as above to cover the half-neighborhood of x not contained in I.

3. Proof of the main results

Remark. By a CI-small di!eomorphism we mean a di!eomorphism that is CI-close to the identity. The support of a di!eomorphism f : XPX is the closure of the set +x3X " f (x)Ox,.

Lemma 3.1. Assume that f is a normally hyperbolic diweomorphism of a compact manifold X (hence, its center distribution is integrable). Then, given any point x3X and any center leaf C, x can be (u,s)-connected to C. In particular, if f has a center leaf C such that any two points of C are accessible, then the whole manifold is accessible.

Proof. Denote by CV the set of center leaves that can be accessed from x, and let DV be the set of points in X that lie on one of the center leafs in CV. We will show that DV is both open and closed, hence it is equal to X. V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278 267

, Openness follows from local product structure property. To show that it is also closed, let C CV. Then, again by the local product structure, there is an e-neighborhood of C that does not intersect DV. This shows that the complement of DV is open. 3 3 For the last statement, notice that given x,y X there are x ,y C such that the pairs (x,x ) and (y,y) are accessible. By the hypothesis, the pair (x,y) is also accessible. ᮀ

Theorem 3.2. Let f be a 1-normally hyperbolic diweomorphism whose central leaves are one-dimen- N " " N sional. Let C be a compact periodic central leaf of f, f (C) C. Set F f . Assume that: (1) F C has rational rotation number; (2) all periodic points of F are transverse (hence xnitely many only; denote them by y , i"1,2,K); C G (d (3) there are points yG close to yG (i.e., dist(yG,yG) ( f )) such that

H (y )Oy for all i"1,2,K. WG WG G G

Then any two points of C are (u, s)-accessible. Hence, by Lemma 3.1, f is accessible.

Remark. Condition (3) can be replaced by the fact that the local (u,s)-holonomy is non-trivial at 3 O each point yG: there is a point yG C, yG yG, which is (u, s)-accessible from yG through a path that is null-homotopic (mod C).

Proof. If the map f satis"es (3), then it follows from Remark 2.3 that there are open connected L neighborhoods ;G C of the points yG's such that each ;G is accessible. Denote L " 8 U ;G. G Since the periodic points of F are transverse, there is a positive integer k such that C

I 8 G " F (U) C. G\I Because the stable and unstable foliations are invariant under f, it follows that if a set A is accessible then the set f (A) is accessible as well. Moreover, if the sets A and B are accessible and A5BO, 6 ᮀ then A B is accessible. Therefore C is accessible.

Properties (1)}(3) that appear in the statement of Theorem 3.2 are stable under small C perturbations, hence, in order to prove Theorem 1.2 it is enough to show the following:

Theorem 3.3. Fix r*1. Let f be an r-normally hyperbolic diweomorphism with a one-dimensional center distribution that has two nearby compact periodic center leaves. Then there are arbitrarily CP-small diweomorphisms g3Di! P(X) such that g " f satisxes the conditions of Theorem 3.2. If a smooth volume form k is given, one can choose g to preserve k. 268 V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278

More precisely, we need the following conditions on the center leaves of f : E there is a periodic compact one-dimensional center leaf C, and E each point of C is close enough to a periodic compact center leaf that is disjoint from C (by compactness, xnitely many such leaves fulxll this condition for all the points of C).

We prove Theorem 3.3 after a few preliminary lemmas. We will assume that a smooth volume form k is given. The strategy is as follows: we do all the perturbations by composing the original map by time-t maps of certain volume preserving CP-#ows. These #ows are constructed in Lemma 3.4. The e!ect of these perturbations on the center leaves is discussed in Lemma 3.5. Lemmas 3.6 and 3.7 show how to obtain conditions (1) and (2) of the Theorem 3.2. After presenting these lemmas we describe the remaining part of the proof.

Lemma 3.4. Let M be a C manifold and k a smooth volume form on M. Assume NLM is a compact CP submanifold of codimension at least one, possible with boundary *N. Let ; be an open neighborhood of N. Let X be a CP\ vector xeld on N which vanishes in a neighborhood of *N. Fix an open set XLN such that supp(X)LXLXM LN!*N and a neighborhood V of X in XP\(N), the space of CP\ vector xelds on N. 1P P Then there is a continuous path of volume-preserving diweomorphisms : Di! I(M) given by a non-autonomous yow, such that E " Id+; E L " 31 supp( R) ; and R(N) N for t ; E 3 LX " X , V and supp(X ,) , where X : d R/dt R.

Remark. In some cases (e.g., in the situation described by Corollary 1.7), we can realize as the #ow of an autonomous vector "eld.

Proof. Denote dim(N)"n and dim(M)"m. For convenience we assume that M is endowed with a smooth metric. Due to the boundary of N, we have to consider open half-balls as well, i.e. sets + " 31L """ ""( * , x (x,2,xL) x r, x 0 . + , 5 M L Choose a "nite cover of N by balls ;? in M such that ;? N are open (half-) balls, ;? ;? s P L1K and ? : ;?

E s k s L1K ( ?)H is the Lebesgue measure on ?(;?) ; E s 5 " ;+ ,L ; K\L Ls L1K L1L ?(N ;?) D ? 0 D? BallC (0) ?(;?) with D? an open ball, where D? is equal to either D? or a half-ball of it. +t , + , Choose on M a smooth partition of the unity ? subordinated to ;? and let "t 3 P\ " s 3 P\ * X? : ?X X (N), v? : ( ?)H(X?). Then v? X (D?) because X vanished near N. By standard P 3 P approximation arguments (see, e.g., [8]), one can "nd C vector "elds v? X (D?) such that supp(v?) + sH 3 + sH LX is compact in D?, ? ? (v?) V and supp ( ? ? (v?)) . 3 P ; K\L We extend each v? to a gradient-free compactly supported vector "eld w? X (D? BallC (0)) as 1L 1K\L follows: denote the coordinates of by x,2,xL, those of by y,2,yK\L, and the compo- L nents of v? by v?,2,v?.De"ne " t ) w?(x,y): ( (y) v?(x),>(x,y),0,2,0), where E t3 !e e 1 CA (( , ), ); E t(0)"1; E !Ct " (s)ds 0; *v G E > "!+L ? ) Wt (x,y): G* (s)ds. xG ? 1P P s 1P P Let : Di! *%"( ?(;?)) be the #ow generated by w?. Then : Di! I(M) given by " s\ " ? " s R : ( ? R ?) ? satis"es the conditions of the Lemma (for the above composition "x any order of the indexes a). ᮀ

Lemma 3.5. Let f be an r-normally hyperbolic diweomorphism on the compact manifold X and C be 3 P L " L 39 a center leaf of f. If g Di! (X) is C -close to f and f (C) g (C) for all n then C is a center leaf of g as well.

Proof. This follows from the characterization of the center leafs of g given in Theorem 6.8 of Hirsch et al. [9]: it roughly says that to each center leaf C of f there corresponds a center leaf C of g, uniquely determined by the fact that f L(C) and gL(C) stay close to each other for all n39. ᮀ

We say that the rotation number of a di!eomorphism 3Di!(S) is stable if all di!eomorphisms C-close to have the same rotation number.

Lemma 3.6. Let X be a compact manifold with a smooth volume k. Given any r-normally hyperbolic 3 P N " map f Di! (X) with a p-periodic one-dimensional compact center leaf C, f (C) C, there are 3 P " arbitrarily small diweomorphisms g Di! I(X) such that g(C) C, C is a p-periodic center leaf of " " N g f and the rotation number of (g f ) on C is rational and stable. P U P Proof. Fix a C -di!eomorphism : S C and pick a tubular neighborhood ; of C such that + I " " ! , ; does not intersect f (C) k 1,2,p 1 . 270 V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278

3 P " L In view of Lemma 3.5, if g Di! I(X) is small enough, g(C) C and supp(g) ; then C is a center leaf of g " f (and, obviously, p-periodic). N P Notice "rst that if f C has irrational rotation number then by an arbitrarily small C - " N perturbation g as above we can achieve that (g f ) have rational rotation number on C. Indeed, P + , by Lemma 3.4, there is a volume preserving C -#ow R supported in ; and preserving C with the property that it does not have any "xed point on C. Then for small t the circle di!eomorphisms U\ " NU U\ NU ` a 1 ( R f ) and f are ordered in the sense that their compatible lifts to do not intersect U\ " NU"U\ " N U (see [10, De"nition 11.1.7]; note that ( R f ) ( R f ) ). Therefore, by Proposition 11.1.8 of Katok and Hasselblat [10], o U\ " NU Oo U\ NU ( ( R f ) ) ( f ) for any small tO0. In particular, the rotation number has to take rational values for arbitrarily small values of tO0. o "o It remains to show that the rational rotation number can be made stable. Assume that (F) , o 30 "U\ " NU with , where F : ( R f ) .IfF is not a rotation then it has at least one semistable ` a " periodic point, hence pushing R f in the right direction with a #ow around C produces a stable periodic point (see Proposition 11.1.10 in [10]). If F is a rotation, let (xH,yH) be an interval of S that contains no element of the F-orbit of xH. Take a #ow +t , preserving such that Osupp(t )L(x ,y ). Then (t " " f )N has rotation Q C Q C H H Q R o ᮀ number but is not a rotation, hence we can apply the previous argument.

Lemma 3.7. Let f3Di! P(X) be an r-normally hyperbolic diweomorphism that has a p-periodic one-dimensional compact leaf . Assume that f N has stable rational rotation number o . Then there C C 3 P " are arbitrarily small diweomorphisms g Di! I(X), such that C is a p-periodic leaf of g f, and (g " f )N has rotation number o and only transverse periodic points. C

+ I " " Proof. Fix a tubular neighborhood ; of C such that ; does not intersect f (C) k ! , + , 1,2,p 1 . Use Lemma 3.4 to construct a volume preserving #ow R that preserves C,is supported in ;, and is non-singular for "t"(e. We will obtain the desired perturbation by " considering R f for small values of t. In view of Lemma 3.5 and our hypothesis, the only part of the conclusion that has to be checked is that one can make all periodic points of ( " f )N transverse. For this we use the following R C consequence of Sard's Theorem (see [8, Theorem 2.7]):

Parametric Transversality Theorem. Let <,M,N be CP-manifolds without boundary and ALN a CP-submanifold. Let F :

" " ; "+ " 3 , " !e e Let M C, N C C, A (x,x) x C , < ( , ) and " 3 C " N O 3 F(t) [x M (x,( R f ) (x)) N] (note that ( " f )N "( " f N ) ). R C R C ' Condition (i) is obvious and condition (iii) holds for r 0. Note that R is a di!eomorphism for any t and coincides with the identity for t"0, hence the derivative D ( ) is positive for V R C 3 !e e * " N O * O 3 3 t ( , ). A straightforward computation shows that ( R f ) / t V 0 for any t < and x C, because the partial derivative is the sum of q terms, each having the same sign as the vector "eld. Hence the vector *FCT/*t and the tangent space of A span the tangent space of N at each point in the intersection of the image of FCT with A. Therefore, condition (ii) is true as well. It follows from the Transversality Theorem that there are values of t3(!e,e) arbitrarily close to zero for which ( " f )N has only transverse periodic points. ᮀ R C

The last ingredient is provided by the following:

Lemma 3.8. Let f3Di! P(X) be a partially hyperbolic diweomorphism as in Theorem 3.3, having two disjoint periodic compact center leaves C and C of period dividing p. Let K be a compact subset of X that is f-invariant and does not intersect either C or C. Given two close enough points 3 3 P 3 P I " " xH C, x C, there is an arbitrarily C -small diweomorphism g Di! I(X) such that f : g f has and as periodic center leaves, fI N "f N , fI N "f N and C C C C ) )

H (y )OHI (y ), WH W H WH W H where H is the f-holonomy and HI is the fI -holonomy. WH W WH W

Remark. In our case K will consist of the f-trajectories of a "nite number of periodic compact center leaves. By Lemma 3.5, center leaves of f that are contained in K remain center leaves for fI as well.

Proof. Let H :

Let g3Di! P(X) be a small di!eomorphism such that g(NO)"NO and supp(g)L;. Let fI ": g " f and denote the stable and unstable foliations of fI by =I Q, respectively =I S. I I By Lemma 3.5, C and C are center leafs of f as well, hence we can consider the f-holonomy HI :

HI "f O " U " g\ " U\ " f \O " H , (3.1) WH W WH W where U : NOPf \O(H (<))L is the holonomy along the unstable foliation of f within =AS( ). WH W C C Indeed, pick y3< and let y (I) ": HI (I) (y) for 1)k)4 (the notations correspond to those of WH W I L I L AQ AQ () * De"nition 2.2, applied for f ). Since f coincides with f on =B (y) and =B (y ) for all n 0, I Q " Q I Q () " Q () I S () " S () (I)" (I) =B(y) =B(y) and =B(y ) =B(y ). Similarly, =B(y ) =B(y ). We conclude that y y for y3< and 1)k)3. I It remains to consider the transition from N to C. Denote by H, respectively H the holonomy S AS I S I AS from N to C along = within = (C), respectively along = within = (C). The functions \LN I \LN O>N * U f and f coincide on N for all n 0, therefore the holonomy along the unstable leaves O>N I 3 from N to C is the same for both f and f (recall that the points of the local unstable leaf of x X are characterized by the fact their backward trajectory does not stray away from that of x). By the invariance of the unstable foliations we obtain that O>N " U " " O>N I O>N " U " I " I O>N f H f , respectively f H f , O>N I O>N " O " " N both relations being considered on N . Since f ,O>N f g f ,O>N, we obtain that I " O>N " U " \N " \ " N " U\ " \O\N " H , f f g f f H ,. N " U " \N"U By the same invariance, f f , which implies (3.1). 3 P O By Lemma 3.4, there is a #ow R Di! I(X) that is supported in ;, preserves N and does not "x U\ " \O () O ' the point f (yH ) for small t 0. Therefore, there is t 0 such that

HI (y )OH (y ), WH W H WH W H " (" "( ᮀ provided g R and 0 t t.

3 P Proof of Theorem 3.3. Let f Di! (X) be as given in the theorem, with C a compact periodic center leaf. 3 P By the Lemmas 3.6 and 3.7, we can "nd arbitrarily small di!eomorphisms g Di! I(X) such that g " f satis"es conditions (1) and (2) of Theorem 3.2. Thus, we may assume without loss of generality that f itself satis"es conditions (1) and (2). We will construct a small perturbation fI of f which induces on C the same map as f and satis"es condition (3) as well. ) ) N Let p be the period of C. Denote by yG,1 i K, the periodic points of f on C.LetyG be points close enough to y situated on periodic compact leaves and H ": H : < P the G CG G WG WG G C corresponding f-holonomies (we are not going to write explicitly the dependence of the holonomy ` a ` a on the function unless it is not clear from the context). We label the points yG as bad or good , V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278 273 depending on whether they satisfy condition (3): "+ " " ) ) , "+ " O ) ) , B : yG HG(yG) yG,1 i K , G : yG HG(yG) yG,1 i K . We use Lemma 3.8 to do successive perturbations which move points from B to G. During these perturbations and remain periodic center leaves and fN is not a!ected, hence the set of C CG C periodic points on C does not change and conditions (1) and (2) are still ful"lled. Due to the continuity of the stable and unstable foliations with respect to changes in the di!eomorphism, small perturbations will never take a point from G to B. Therefore, after a "nite number of ᮀ perturbations all points yG will be in G, as desired.

Appendix A

Let M and N be compact manifolds. " 3 Consider on Homeo(N) the distance d,(g, h): supWZ, dist,(g(y), h(y)), where g, h Homeo(N). It has the following properties: if g, h, u, v3Homeo(N) then " d,(gu, hu) d,(g, h), )"" "" d,(ug, uh) u **d,(g, h), hence ) # ) #"" "" d,(gh, uv) d,(gh, uh) d,(uh, uv) d,(g, u) u **d,(h, v), (A.1) \ \ " \ \ " \ \ )"" \"" d,(g , h ) d,(g h, h h) d,(g h, g g) g **d,(g, h). (A.2) Fix a C Anosov di!eomorphism A : MPM. There is j3(0,1), and a splitting ¹M"EQES such that with respect to an adapted Riemannian metric on M ""¹Av"")j""v"", v3EQ, ""¹A\v"")j""v"", v3ES.

De5nition. Amapb : 9;MPDi!(N) is called a cocycle over A if it satis"es b(m#n, x)"b(m, ALx) " b(n, x) (A.3) for m, n39, and x3M. h3 b h ' Given (0,1], the cocycle is said to be -HoK lder if there is a constant C& 0 such that b b ) F 3 d,( (1, x), (1, y)) C&dist+(x, y) , x, y M. (A.4) h ( b The smallest value C& that can be used in the above formula is called the -Holder (semi) norm of , ""b ) "" denoted by (1, ) F. The cocyle b is said to be h-close to the identity if ""¹b(1, ) )!""jF(1. (A.5) In view of (A.2), conditions (A.4) and (A.5) imply that b(1, ) )\ is h-HoK lder as well, "" b ) \ "" )j\F""b ) "" (1, ) F (1, ) F. 274 V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278

b 9; P ; P ; A cocycle : M Di! (N) determines a skew-product f@ : M N M N given by " b f@(x, y): (Ax, (1, x)(y)). (d( ( (R For 0 1, 0 H denote by Sk& B(A,N) the set of skew-products over A given by b 9; P h h3 ""b ) "" ( cocycles : M Di! (N) that are -HoK lder for some (0,1] and satisfy (1, ) F H, ""¹b(1, ) )!""jF(d. It follows from formula (A.3) that a cocycle b is determined by the map b ) P (1, ):M Di! (N), hence we can see Sk& B(A,N) as a subset of Maps(M,Di! (N)). Endow the space of maps from M to Di!(N) with the topology of uniform convergence and consider on Sk& B(A,N) the induced topology. For maps in Sk& B(A, N) one can construct the equivalent of the stable and unstable foliations of partially hyperbolic C-maps. Since the existence of these foliations for HoK lder skew-products does not follow from the standard theory of partially hyperbolic di!eomorphisms, we include a proof of this fact in Theorem A.3. Using these foliations, one can speak of accessibility. The stable and unstable foliations of the skew-product are contracting, respectively expanding, and the leaves of the foliations depend continuously on the cocycle b. Therefore, the proof of Theorem 1.2 can be applied to yield:

Theorem A.1. In Sk& B(A, S ) the accessible maps contain an open and dense set.

Another motivation for Theorem A.3 is a generalization of a result of Brin (see [3]) about topologically transitive C skew-products. Indeed, if G is a compact connected Lie group, then the set of translations by elements in G can be embedded in Di!(G) as Haar measure preserving % transformations. Denote by Sk (A) the set of skew-products f@ over A given by HoK lder cocycles b 9; P % d : M G. Note that any element f@ in Sk (A) belongs to Sk& B(A,G), for some H, . Denote % " 5 % % Sk& B(A) Sk& B(A,G) Sk (A). Any element in Sk& B(A) has stable and unstable foliations satisfying all the properties in Theorem A.3. Using this fact, one can follow Brin [3] to prove:

Theorem A.2. Let A : MPM be a topologically transitive C Anosov diweomorphism and G a com- % pact connected Lie group. Then Sk& B(A) contains an open dense set of topologically transitive transformations. Indeed, according to Brin [3], one has to check that a generic map is recurrent, and has the accessibility property. Since A is topologically transitive, it has a dense set of periodic points. From % this follows that any skew-product in Sk& B(A) is recurrent. Generic accessibility can be proved as in [3], using the properties of the stable and unstable foliations listed in Theorem A.3. We will discuss in Theorem A.3 the stable foliation only; for the unstable foliation consider f \ instead of f. In order to simplify the notation, denote b(1,x)byb(x). In the sequel =Q stands for P the stable foliation of the Anosov di!eomorphism A : M M and distQ for the distance induced by the metric of M on a leaf of =Q.

Theorem A.3. Let A : MPM be the C Anosov diweomorphism considered above and b 9; P h ( h c Q P : M Di! (N) a -Holder cocycle which is -close to the identity. Dexne V L : = (x) Di!(N)LHomeo(N) by c "b \b V L(t): (n, t) (n, x). V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278 275

Then: +c , (a) The sequence V L LV is pointwise convergent in Homeo(N). (b) The limit

c " c Q P V : lim V L : = (x) Homeo(N) L h ( is uniformly -Holder from distQ to d,. c " (c) V(x) Id,. Q "+ c " 3 Q , 3 3 (d) The family of graphs W (x,y): (t, V(t)y) t = (x) , x M, y N, gives an f-invariant foliation of M;N, where f3Homeo(M;N) is the skew-product f (x,y):"(Ax,b(x)y), x3M, y3N. (e) If t3=Q(x) and l'jF then

l\L L L c " lim dist+",( f (x,y), f (t, V(t)(y)) 0, L

3 ; + 3 Q " ) , ' uniformly in (x,y) M N and t = (x) distQ(x,t) K , for any K 0. +c Q P " 3 , (f) The family of maps V : = (x) Homeo(N) x M is uniquely determined by either of the following conditions: (i) properties (b)}(d); (ii) property (e) for a value l satisfying jF(l(m(¹b)"""¹b\""\. (g) The application b P 3+b " 3 ,C+c Q P , : M Di! (N) f@ Sk& B(A, N) V : = (x) Homeo(N) VZ+ constructed in (a) is continuous from the topology of uniform convergence of maps from M to Homeo(N) to the topology of uniform convergence on compact sets of maps from the leafs of =Q to Homeo(N).

Proof. We will use (A.1) repeatedly to estimate the distance between two products in Homeo(N). ' Recall (see [12]) that there is a constant CQ 1 such that L L ) jL distQ(A x, A t) CQ distQ(x, t) (A.6) for n*0 and t3=Q(x). +c , * (a) It is enough to show that the sequence V L(t) L is Cauchy. Let m n be positive integers: c c " b \ " 2 " b L\ \ " b L \ " 2 " b K\ \ " b d,( V L(t), V K(t)) d,( (t) (A t) (A x) (A x) (m, x), b(t)\ " 2 " b(AL\t)\ " b(ALt)\ " 2 " b(AK\t)\ " b(m, x)) K\ ) + ""b \ " 2 " b I\ \"" b I \ b I \ (t) (A t) **d,( (A x) , (A t) ) IL ) ""¹b\""LjLF F C distQ(x,t) , (A.7) 276 V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278 where ""b( ) )\"" C " F ) CF. 1!""¹b\""jF Q (b) Let s,t3=Q(x), and n be a positive integer. Then c c " b \ " 2 " b L\ \b b \" 2 " b L\ \b d,( V L(s), V L(t)) d,( (s) (A s) (n,x), (t) (A t) (n,x)) L\ ) + ""¹b\""I jIF F) F C ** distQ(s,t) C distQ(s,t) , I b where the constants C,C are independent of s, t, m, n, x, . (c) This is obvious. (d) The invariance is equivalent to b c "c L b 3 Q (n, t) V(t) LV(A t) (n, x), t = (x) and this follows from the identity b c "c L b (n, t) V K(t) LV K\L(A t) (n, x). (e) One has f L(x,y)"(ALx,b(n,x)(y)), L c " L b c f (t, V(t)(y)) (A , (n,t) V(t)(y)). In view of (A.6) it only remains to prove that

l\L b b c " lim d,( (n,x), (n,t) V(t)) 0. L By (d), (c) and (b) b b c " b c L b d,( (n,x), (n,t) V(t)) d,( (n,x), LV(A t) (n,x)) " c L ) jLF F d,(Id,, LV(A t)) C distQ(x,t) .

+u Q P " 3 , (f ) Let V : = (x) Homeo(N) x M be another family of functions. +u , u "b \u L b (i) Assume that V satis"es (b)}(d). From (d) we conclude that V(t) (n,t) LV(A t) (n,x), t3=Q(x). But then, using also (c) and (b): u c " b \u L b b \b d,( V(t), V L(t)) d,( (n,t) LV(A t) (n,x), (n,t) (n,x)) )""b \"" u L ) ""¹b\""LjLF F (n,t) **d,( LV(A t),Id,) C distQ(x,t) , which implies the desired conclusion. +u , (ii) Assume that V satis"es (e). Then l\L b c b u " 3 Q lim d,( (n,x) V(t), (n,t) V(t)) 0 for t = (x). L V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278 277

But l\L b c b u *l\L""b \""\ c u d,( (n, x) V(t), (n, t) V(t)) (n, x) ** d,( V(t), V(t)) jF(l(""¹b\""\ c u " and the right-hand side converges to zero for if and only if d,( V(t), V(t)) 0. bCc ` (g) This is a consequence of the fact that the application V is the uniform limit of a sequence of continuous functionsa. b b c c Indeed, let and be two cocycles whose skew-products are in Sk& B(A,N). Denote by V L, V and c c V L, V the corresponding families considered in (a). Let t3=Q(x). Estimate (A.7) implies that

Hj\ d (c (t), c (t))) C dL max+dist (t, x),1, , V L V 1!d Q Q c c and a similar inequality for V L and V. On the other hand, in view of (A.1), for a "xed n one can c c b b ᮀ make d,( V L(t), V L(t)) as small as desired by taking and close enough to each other.

Appendix B

We prove here Lemma 1.3. Recall its statement:

For the sake of completeness we recall a few de"nitions from Hirsch et al. [9, Section 7]. o ! P Let C be a C -lamination of X with leaves of dimension c.Aplaque is a C -embedding : X ! L1A +o, of the closed unit ball into one of the leaves of C.Aplaquation P is a family of plaques ! such that each leaf of C is the union of the images of the interior of through some plaques and +o, ! is precompact in Emb ( , X). By Theorem 6.2 of Hirsch et al. [9], each C -lamination admits a plaquation. + , Let f be a di!eomorphism that preserves C. A pseudo-orbit xL LZ9 preserves P if for each n, f (xL) and xL> lie in a common plaque of P. e' e + , The di!eomorphism f is plaque expansive if there is an 0 such that if two -pseudo-orbits pL + , (e and qL both respect P and satisfy dist6(pL, qL) for all n then for each n, pL and qL lie in a common plaque of P. We will use the following Shadowing Lemma ([9, Lemma 7A.2 and its proof]):

Lemma B.1 (Hirsch, Pugh, Shub). Assume that the center distribution of the partially hyperbolic diweomorphism f3Di!(X) integrates to a C-lamination C of X. If ( f,C) has local product structure and l, g'0 are given, then there exists d'0 such that any d-pseudo-orbit for f can be l-shadowed by an g-pseudo-orbit for f which respects C. 278 V. Nit7 icay ,A.To( ro( k / Topology 40 (2001) 259}278

Dynamical coherence of f implies that ( f,C) has local product structure (see Section 7A of Hirsch et al. [9] for the de"nition of the latter).

Proof of Lemma 1.3. Let e be the constant given by the plaque expansivity of =A. Pick 0(l(e/2, g"e and let 0(d be given by Lemma B.1. Let BLX be an open set of diameter at most d. By the PoincareH Recurrence Theorem, there * 5 , O 3 , (d exists N 1 such that B f (B) , hence there is x B such that dist6(x,f (x)) . Consider the N-periodic d-pseudo-orbit 2, f ,\(x), x, f (x), f (x),2, f ,\(x), x, f (x), f (x), 2 + , g l A and let yL be the -pseudo-orbit that -shadows it and respects = . g + , + , " ) l Since the -pseudo-orbits yL and zL , zL : yL>,, satisfy dist6(yL,zL) 2 for all n and both A A " A respect = , the plaque expansivity of = implies that yL and zL yL>, lie in the same leaf of = . + , A , A , But yL respects = , hence f (yL) and yL>, also lie in the same leaf of = . Therefore, yL and f (yL) lie in the same center leaf; we obtain an N-periodic center leaf at a distance at most l from B. ᮀ

Acknowledgements

We would like to thank Keith Burns, Clark Robinson, and Amie Wilkinson from Northwestern University for helpful conversations and remarks about this paper. We would also like to thank the referee for the suggestions made.

References

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