NancyGuelman OntheapproximationoftimeonemapsofAnosovflowsbyAxiomAdiffeomorphisms

MATHEMATICAL

SOCIETY Bull Braz Math Soc, New Series 33(1), 75-97

© 2002, Sociedade Brasileira de Matemática

On the approximation of time one maps of Anosov flows by Axiom A diffeomorphisms

Nancy Guelman

Abstract. We prove that iff₁is the time one map of a transitive and codimension one Anosov flowφand it isC¹-approximated by Axiom A diffeomorphisms satisfying a property calledP, then the flow is topologically conjugated to the suspension of a codimension one Anosov diffeomorphism.

A diffeomorphismf satisfies property_Pif for every periodic point inMthe number of periodic points in a fundamental domain of its central manifold is constant.

Keywords: Anosov flow, suspension of Anosov diffeomorphisms, time one map, Axiom A.

Mathematical subject classification: 37D20, 37D30, 37D05.

Introduction

Throughout this paperMdenotes a smooth compact Riemannian manifold with- out boundary, andφ :M×R→MaC^r flow, withr ≥1.

Recall that the suspension of an Anosov diffeomorphism is an Anosov flow in the corresponding manifold. Let us consider a transitive Anosov vector field Xand letfτ = Xτ be the flow of X at timeτ. Althoughfτ is not an Anosov diffeomorphism, there exists aDfτ-invariant splitting ofT M

T M=E^s ⊕E^c⊕E^u,

such thatDfτ|E^sis uniformly contracting,Dfτ|E^uis uniformly expanding, and E^cis a nonhyperbolic central direction.

The object of our study are transitive Anosov flows ( i.e. the case when the non-wandering set is the whole manifold).

A codimension one Anosov flow defined on ann-manifoldMis an Anosov flow

Received 15 March 2001.

such that for allx ∈M,dimE^s(x)=1 anddimE^u(x)=n−2 ordimE^s(x)= n−2 anddimE^u(x)=1.

An interesting question is what kind of dynamical system can appear under perturbations of a time one map of a transitive Anosov flow.

Palis and Pugh (see [9]) wondered whether the time one map of a transitive Anosov flow could be approximated by hyperbolic or Axiom A diffeomorphisms.

It is a well known fact that in the case when the flow arises from the suspension of an Anosov diffeomorphismg:N →Nsuch an approximation can be carried out with Axiom A diffeomorphisms.

The suspension manifoldNg is obtained from the direct productN × [0,1] by identifying pairs of points of the form(x,0)and(g(x),1)forx ∈ N. The suspension flowϕ(x, t )is determined by the vector field _∂t^∂. The manifoldNg

is fibered over S¹ and the projection of ϕ(x,1) onto S¹ is the identity map.

Letf be a diffeomorphism preserving fibers,C¹- close toϕ(x,1)such that the projection off overS¹is a Morse-Smale map. We have thatf is an Axiom A diffeomorphism.

In this spirit, Bonatti and Díaz ( see [2]) proved that ifτ is a period of a periodic orbit of a transitive Anosov flow, then there exist an open set U of nonhyperbolic and transitive diffeomorphisms, and a sequence(gn)n∈N,gn∈U such thatgn→fτ in theC¹- topology.

Throughout this paperτ will be 1.

Our aim is to give a partial answer to the Palis-Pugh question. We will say that a diffeomorphismf satisfies propertyPif for any periodic pointx the number of periodic points betweenxandf (x)in the connected component of its central manifold is constant (see Section 1).

This property is not so strange. It is, for instance, verified in the case whenf is a convenientC¹-perturbation of the time one map of a transitive Anosov flow arising from the suspension of an Anosov diffeomorphism. In fact, the above example verifies that the number of periodic points betweenx andf (x)in the connected component of its central manifold is constant, ifx is a periodic point off.

We will show that, in the general case there is an open dense setV ⊂Msuch that the number of periodic points betweenxandf (x)is constant for all thef- periodic points inV. Here, as before,f is aC¹-perturbation of the time one map of a transitive Anosov flow.

We will prove the following:

Theorem 1. LetMa smooth compact riemannian manifold without boundary, dim(M)≥3. If the time one map of a transitive codimension one Anosov flow isC¹-approximated by Axiom A diffeomorphisms satisfying propertyP, then the flow is topologically conjugated to a suspension of a codimension one Anosov diffeomorphism.

Perhaps it is worthwhile to note that Verjovsky ( see [10]) proved that ifn >3 any codimension one Anosov flow is transitive (see [5] for a counterexample in dimension 3). Then the hypothesis of transitivity can be omitted if the dimension is higher than 3.

From Theorem 1 follows the next corollary.

Corollary 1. LetMbe a negative curvature closed surface. The time one map of the geodesic flow can not beC¹-approximated by Axiom A diffeomorphisms verifying propertyP.

In Section 1 we prove that PropertyPis a “reasonable property” and we study some properties of attractors of Axiom A diffeomorphism close to the time one map of a transitive Anosov flow. In Section 2 we introduce maps which will play an important role in the proofs of the theorems and we examine some basic facts about them. Section 3 deals with the continuity of the above mentioned maps.

In Section 4 we prove that there is a repeller basic set which is a hypersurface and we complete the proofs of the Theorem in Section 5.

1 Properties of basic sets

We begin recalling some basic definitions about flows and diffeomorphisms.

Definition 1. A compactφt−invariant set,⊂M, is called a hyperbolic set for the flowφ if there exist a Riemannian metric on an open neighborhood_U of, andλ <1< µsuch that for allx ∈there is a decomposition

Tx(M)=E^s_x⊕E_x^u⊕E_x⁰

such that∂tφ (x, t )|t=0 ∈ E⁰_x− {0},dim(E⁰(x)) =1,Dxφt(x)(E_xⁱ) ⊂E_{φ (x,t )}ⁱ , withi=s, u, and

Dxφ (x, t )|E^s(x) ≤λ^t witht ≥0 Dxφ (x, t )|E^u(x) ≤µ^t witht≤0.

AC^r flowφ :M×R→M, is called an Anosov flow ifMis a hyperbolic set forφ.

Letf :M→Mbe aC^r diffeomorphism .

Definition 2. Anf-invariant setis called hyperbolic if there exists aDf- invariant decomposition ofTMsuch that

TM =E^s⊕E^u

andDf|E^s is uniformly contracting andDf|E^uis uniformly expanding. More precisely, there arec >0 ,λ, with 0< λ <1 such that for allx ∈

Dxfⁿ|E^s(x)< cλⁿ and

Dxf⁻ⁿ|E^u(x)< cλⁿ.

A diffeomorphismf :M →Mis called an Anosov diffeomorphism ifMis a hyperbolic set forf.

Letf1:M →M,the time one diffeomorphism ofφdefined as f1(x)=φ (x,1),∀x∈M,

whereφ :M×R→Mis a codimension one Anosov flow ifdim(M) >3 (In the case thatdim(M)=3, codimension one property is replaced by transitivity.) Without loss of generality we may assumedimE^s(x)=n−2 anddimE^u(x)= 1 for allx ∈M.

Sinceφ has no singularities, it follows that there existf1-invariant foliations F^cs,F^cu,F^ss,F^uuandF^c. Notice that the leaf ofF^cthroughxis the same as theφ-orbit ofx, and we denote it byF^c(x)orW_φ^c(x).

By well known properties of transitive Anosov flows, we have that {F^c(x)|F^c(x)is a closed set}is dense inM.

{F^c(x)|F^c(x)is dense inM}is a residual set.

If_Ois a periodic orbit of φ, thenW^s(O)consists of all points whose foward φorbits never stay far fromOandW^u(O)of all points whose reverseφ orbits never stay far fromO. Both of them are dense in M, and so areF^cs(x) and F^cu(x)∀x ∈O.

Since f1 is C^r, we have that the leaves of F^cs, F^cu and F^c are C^r. Let f : M → M be a diffeomorphism C¹-close to f1. The map f is plaque expansive (see [7] ), there exist_Ff^cs,_Ff^cuand_Ff^c and there is a homeomorphism h:M →Mclose to the identity such that ifh(x)=x, thenF_f^c(x´)isC¹-close

toF_f^c

1(x)in compact sets and the manifoldsF_f^cs(x)andF_f^cs

1(x)areC¹-close in compact sets. In addition,

hof1(F_f^c₁(x))=f oh(F_f^c₁(x)).

The mapf is normally hyperbolic atF_f^c, therefore every leaf ofF_f^c is invariant and every periodic point off is in a closed leaf ofF_f^c.

According to what was mentioned above we have that {F_f^c(x)|F_f^c(x)is a closed set}is dense inM and

{F_f^c(x)|F_f^c(x)is dense inM}is a residual set.

Let us denote byF_f^c(x)or byW^c(x)the leaf of the central foliation through the pointx.

We recall that a diffeomorphismf :M → M satisfies Axiom A if the non- wandering set(f )is hyperbolic and the set of periodic points is dense in(f ).

From now on we will assume that f is an Axiom A diffeomorphism C¹− close tof1. Moreover, we will make the following assumption: the number of periodic points in the connected component ofW^c(x), betweenx andf (x)is constant, for allf-periodic point inM. We will consider the number of periodic points inW^c(x), betweenxandf (x), in such a way that the length of this curve is almost of the same length of the trajectoryφ (x, t )ˆ of the Anosov flow, witht varying between 0 and 1, andxˆ being af1periodic point nearx. Sometimes we have to consider the number of periodic points when the segment of the curve betweenx andf (x)winds around itself more than once. The last property will be called propertyP. We will prove that this property is verified in an open and dense set of the manifold.

LetO=F_f^c(x)whereF_f^c(x)is a closed curve.

The rotation number off must be rational, because if it were irrational, there would be an hyperbolic minimal setI ⊂Oand it would be included in a basic set.

IfO⊂(f )thenOwould be in a basic set andf|Owould be expansive which leads to a contradiction with the nonexistence of one dimensional expansive diffeomorphism. Lety ∈Othenα(y)=ω(y)=I, hence

y ∈W^s(I )∩W^u(I )⊂W^s()∩W^u(), thereforey∈(f )which is a contradiction.

Then, there exist at least two periodic points in_Obecausef is an Axiom A diffeomorphism. All the points in(f )∩Omust be periodic because if there were a nonperiodic point,x ∈(f )∩Othen the invariance of(f )∩Oimplies thatα(x)andω(x)would be periodic points of different indices so they would be in different basic sets.

From now on, we choose an orientation forF^c, and denote C_b^a the curve included in a central foliation leaf, between a and b. We will consider the connected component ofF^c(a)betweena andbin the positive direction from a, in the case thatF^c(a)is a closed curve.

Proposition 1.1. There exists an open and dense setV ⊂Msuch that property P is verified for f|V i.e. all periodic points in V have the same number of periodic points in the connected component ofW^c(x), betweenxandf (x).

Proof. The metric induced by the Riemannian metric on the leaves ofF^cwill be denotedd^c.

The lengths of the curvesC_{f (x)}^x are bounded away from 0, and asf is Axiom A there existsκ such thatd^c(p, q) > κ, ifpandq are periodic points in the same leaf ofF^c. Then, there existsm∈Nsuch that

m=min{n∈N:W^c(x)has exactlynperiodic points inC_{f (x)}^x }. Letpa periodic point verifying that the number of periodic points inC_{f (p)}^p ism.

We claim that there is an open neighborhood U of C_{f (p)}^p such that for all periodic pointxinU the number of periodic points inC_{f (x)}^x ism.

If not, there exists a sequence of periodic pointspn→psuch that the number of periodic points inC_{f (p}^pn

n) is greater thanm, so there exist more thanmlimit points inC_{f (p)}^p . Since these limit points must be periodic, this contradicts our assumption.

Therefore, there exists a curve included in a dense leaf of central foliation in U. So, if we saturateUby the central foliation we have an open and dense set such that any periodic pointq in it has exactlymperiodic points inC_{f (q)}^q . Let us recall that there exists a finite number of attractors (repellers) whose basin of attraction (repulsion) are open sincef is Axiom A.

Here are some elementary properties of attractor basic sets.

LetAdenote an attractor basic set of the spectral decomposition off. Notice thatA=Mbecausef can not be an Anosov diffeomorphism. There is no loss of generality if we consider that_Ais connected.

Lemma 1.1. Dim(W^s(x))=n−1,∀x∈A.

Proof. We have assumed thatdim(E_φ^s) =n−2, then asf isC¹-close tof1

we have thatdim(W^s(x))=n−1 ordim(W^s(x))=n−2 for allx ∈(f ).

Letx ∈A∩per(f ), whereper(f )is the set off-periodic points.

Suppose thatdim(W^s(x))=n−2.

SinceAis an attractor,W^u(x)⊂A; henceF_loc^c (x)⊂W^u(x)⊂A. The set Ais closed andf-invariant so there existsx ∈F^c(x)∩A∩per(f ).

Butdim(W^s(x))=n−1 sincedim(W^s(x))=n−2. It follows that there exist two periodic points of different indices inA, which is impossible.

Lemma 1.2. For every closed curve O in F^c there exists a periodic point p∈A∩O.

Proof. SinceO is closed, W^s(O)is dense inM andW^s(A)is an open set, there existyinW^s(O)∩W^s(A)andy ∈W^ss(y)∩Osuch thaty ∈W^s(A).

Asy ∈O,y ∈W^s(p)for a periodic pointp∈O. Thenp∈A∩O. LetK = maxx∈Mlengt h(C_{f (x)}^x ). K is finite becauseM is compact and the mapg: M →Rsuch that everyx ∈ M is mapped into the length ofC_{f (x)}^x is continuous.

The previous lemma asserts that in every segmentγ of central closed curve withlengt h(γ )≥K, there exists a periodic pointp∈γ ∩A.

Corollary 1.1. Every leaf ofF^cintersectsA.

Proof. Letγ ⊂F^cwithlengt h(γ )≥K. Since

{F_f^c(x)|F_f^c(x)is a closed set}is dense inM,

we can choose arcsγnsuch thatγnare included in closed leaves ofF^c,γn →γ, andlengt h(γn)≥K. Then, there exists a sequence(pn)such thatpn ∈A∩γn,

and any of its limit pointsp∈γ ∩A.

Lemma 1.3. In every leaf ofF_f^cthere exists at least one point outside ofW^s(A).

Proof. IfF_f^c(x)is closed, by Lemma 1.2 we have that there exists a periodic pointp∈A∩F_f^c(x)and by Lemma 1.1dim(W^s(p))=n−1. The hyperbolicity offimplies that there exists a periodic pointq ∈F_f^c(x)such thatdim(W^s(q))= n−2, henceq ∈ where is a basic set off, = A. So we proved the claim in the case thatF_f^c(x)is closed.

In the case thatC0 = F_f^c(x) is a future-dense curve, this is, iff (x) > x in the chosen orientation, thenW^c+(x) = {y ∈ W^c(x)/y ≥ x} is dense, and if f (x) < xthenW^c−(x), with the obvious definition, is dense.

Clearly we have thatC0∩W^s(A)= ∅.

We only need to show that_C0is not included inW^s(_A), i.e._C0∩∂(W^s(_A))=

∅.

Suppose that for everyy inC_{f (x)}^x , we have thaty ∈W^s(A). There exists an open and nondense setU, such that_A⊂U,f (U ) ⊂U andC_{f (x)}^x ⊂U; then ifC0intersectsU,C0would be included inUin the future. This contradicts the nondensity ofU, so there existsy ∈C_{f (x)}^x such thaty /∈W^s(A).

It still remains to prove the claim in the case that_C =F_f^c(x)is any curve.

Recall thatK =max_x∈Mlengt h(C_{f (x)}^x ).

Suppose that there exists a curve γ ⊂ F_f^c(x) such that γ ⊂ W^s(A) and lengt h(γ )≥K+1.

Then there exists an open setV,V ⊂W^s(A)andγ ⊂V. There existsy ∈V such thatW^c(y)is dense inM, andW^c(y)∩V has length greater or equal than K. This gives the existence of a fundamental domain inW^c(y)∩V, and then in W^s(A). This contradicts the previous case.

Note that we have proved that every leaf of the central foliation “goes away”

from the basin of attraction of any attractor.

Lemma 1.4. No curveγ,γ included inF_f^c(x)for anyx, satisfiesγ ⊂A.

Proof. Suppose the statement is false, i.e. there existsγ ⊂W_loc^c (x)such that γ ⊂A. Sinceγ ⊂A⊂W^s(A), then the negative iterates ofγ are included in_Aand the length of them grow exponentially.

Letz∈α(x)thenz∈Aand by the proof of Lemma 1.3W^c(z)has to intersect

∂(W^s(A)), butW^c(z)⊂A⊂W^s(A), which yields a contradiction.

All the above lemmas admit versions for repeller basic sets and the proofs are analogous. In fact, ifis a repeller basic set, then forx ∈,Dim(W^s(x)) = n−2, every leaf of F_f^c intersects, in every leaf of F_f^c there exists a point outside ofW^u(), and noγ included inF_f^c(x)satisfiesγ ⊂.

2 Properties of the projection along the central foliation

In this section, we will introduce some maps which are important from the technical point of view.

Definition 2.1. LetSA :W^s(_A)→∂W^s(_A)be a map such that, for everyx in the basin of the attractorA,SA(x)is the nearest point in its central leaf in the positive direction verifying that it is not in the basin of attraction ofA.

Definition 2.2. LetS˜A:W^s(_A)→∂W^s(_A)be the map analogous toSA, but in the negative direction of the central foliation .

Definition 2.3. Let S : A → ∂W^s(A) be the restriction of SA to A and S˜ :A→∂W^s(A)the restriction ofS˜AtoA.

Lemma (1.3) makes the preceding definitions possible.

LetW^c(x)denote the connected component ofW^c(x)∩W^s(A)which contains x.

Letl:A→R,l(x)=lengt h(C_S(x)^x ).

Lemma 2.1. lis lower semicontinuous.

Proof. SinceC_S(x)^x − {S(x)} ⊂W^s(A)andW^s(A)is an open set, there exists a neighborhoodV such thatC_S(x)^x − {S(x)} ⊂V ⊂W^s(_A).

The central foliation is aC¹- lamination becausefisC¹-close to the time one map of an Anosov flow (see [7]), hence for all >0 there exists a neighborhoodUx

ofxsuch that ify ∈Uxthen the curveC_y^y included in_F^c(y)withlengt h(C_y^y)= l(x)− is included inV, and hence inW^s(A). Thenl(y)≥ l(x)− which

proves thatlis a semicontinuous map.

Sincel:A→Ris semicontinuous, the setRof points of continuity oflis a residual set. Let:M×R_≥0→M such that(x, l)=z, ifz∈W^c(x),zis in the positive direction ofW^c(x)andlengt h(C_z^x) =l. is a continuous map then

S(x)=(x, l(x)) is continuous overR.

Without loss of generality we can assume thatRis a residual set of continuity for bothSandS.˜

Analogously there exists a residual set Q in W^s(_A) such that Q is a set of continuity forSAandS˜A.

Following, we prove some properties of the mapS. They are verified byS˜and the proofs are analogous.

Lemma 2.2. S(R)isf-invariant.

Proof. Letx∈ R,y =S(x). For allz∈C_y^x− {y}, we have thatz∈W^s(A), f (z) ∈ W^c(f (x))andf (z) ∈ W^s(A). Fromf (y) ∈ ∂W^s(A)it follows that f (y)=S(f (x)). Replacingf byf⁻¹we conclude that

f (S(R))=S(R).

Lemma 2.3. For ally ∈S(R),dim(W^s(y))=n−2.

Proof. Let y = S(x) with x ∈ A; since dim(W^ss(y)) = n − 2 and dim(W^uu(y)) = 1, dim(W^s(y)) = n−1 or n−2, but by Lemma (1.1) if z∈C_y^x− {y}thenz∈W^s(x). Then

W^c(y)= {z∈W^c(y)such thatd^c(z, y) < }

can not be included inW^s(y)and we can assert thatdim(W^s(y))=n−2.

Lemma 2.4. The set of periodic points inA\Ris nowhere dense inA.

Proof. In order to prove the lemma it is enough to prove:

Let(pn)n∈Nbe a sequence of periodic points such thatSis not continuous at pnandpn →x. ThenSis not continuous atx.

Letqn=S(pn).

Sincepnis a point of discontinuity, there existα >0 and(rnk)⊂Asuch that limk→∞rnk =pnand

lengt h(C_S(r^rnk

nk)) > lengt h(C_S(p^pn_n))+α

and for any with 0< < ^α₂ there exist(snk)⊂R such that limk→∞snk =pn

and

lengt h(C_S(s^snk

nk))≥lengt h(C_S(r^rnk

nk))− > lengt h(C_S(p^pn

n)).

It follows that there exists a periodic limit point ofS(sn_k),q_n, inW^c(pn).

Bothqnandq_nare inW^c(pn)∩S(R), are periodic and dim(W^s(qn))=dim(W^s(q_n))=n−2.

Sinceqnandq_nare in the same closed leaf ofF^c, it follows that there exists a periodic pointp_n, such thatp_n ∈C_q^qn

n anddim(W^s(p_n))=n−1.

Suppose, contrary to our claim, thatSis continuous atx.

Frompn →xwe conclude thatqn→S(x)by the continuity ofSatx.

Besidesq_n → S(x)because there exist(sn_k) ⊂R such that limk→∞sn_k =pn

and limk→∞S(snk)=q_n. Letting a convenient subsequencek(n), we can assert that

nlim→∞sn_k(n) =x and lim

n→∞S(sn_k(n))=S(x) by the continuity ofSatx. This givesq_n →S(x).

Thendist (qn, q_n)→0 whenn→ ∞andd^c(qn, q_n)→0 whenn→ ∞. Butd^c(qn, q_n) > min{d^c(p_n, q_n), d^c(pn, q_n)}and this leads to a contradiction becausep_n andq_n (orpn andq_n ) are in different basic sets because they have different indices.

We have proved thatSis not continuous atx. p

q

p’

q’

n

n

n

n

x

S(x)

Figure 1

Observe that as a consequence we have that for allx ∈ A there exists a sequence of periodic points(pn)n∈N⊂Rsuch thatpn→x.

Lemma 2.5. S(R)is transitive andS(R)⊆(f ).

Proof. Since F^c is continuous, the set of periodic points is dense in R and S(p)is periodic ifpis periodic, then the set of periodic points is dense inS(R), hence

S(R)⊆(f ).

Analogously the image of a dense orbit is dense inS(R).

Corollary 2.1. From the above properties we conclude thatS(R)is included in, a basic set of the spectral decomposition off.

Lemma 2.6. S(W^s(x))⊂W^s(S(x)).

Proof. Let x ∈ A, y ∈ W^s(x) ∩ A. Suppose that S(y) /∈ W^s(S(x)).

Since S(y) ∈ F^cs(x) there exists z = W^s(S(y)) ∩W^c(x). We have that

∀w∈∂(W^s(A)),W^s(w)⊂∂(W^s(A)), thenW^s(S(x))⊂∂(W^s(A))∀x ∈A, andz∈∂(W^s(A)), but this contradicts the definition ofS.

Lemma 2.7. Ifxis a point of continuity ofS, then all the points inW^s(x)∩A are continuity points ofS.

Proof. Letx be a point of continuity ofS,y ∈ W_loc^s (x)∩A. We first prove thatyis a continuity point ofS.

Let{yn}n∈N⊂A, such that limn→∞yn=y. There existsxn=W_loc^s (yn)∩W^u(x) andyn∈W^s(xn). By continuity of the stable foliation, we have limn→∞xn =x, and by continuity ofSatxwe conclude that limn→∞S(xn)=S(x).

Fromyn ∈ W^s(xn), and the above lemma, it follows thatS(yn) ∈ W^s(S(xn)), henceS(yn)=W_loc^s (S(xn))∩W^c(yn).

By the continuity ofW^s andW^cwe have that:

nlim→∞W_loc^s S(xn)=W_loc^s S(x)and lim

n→∞W^c(yn)=W^c(y); hence

nlim→∞S(yn)=W_loc^s S(x)∩W^c(y)=S(y).

We have proved that∀y ∈W_loc^s (x)∩A,Sis continuous aty i.e. S|W_loc^s (y)∩Ais continuous.

Now, ifz ∈ W^s(x)∩Athere isN > 0 such thatf^N(z)∈ W_loc^s (f^N(x))∩A

and the previous argument still applies.

Remark. Note that Lemmas (2.6) and (2.7) are verified not only by SandS˜ but also bySAandS˜A. The proofs are analogous.

Lemma 2.8. Ifx ∈A, thenxis a point of continuity ofSif and only ifxis a point of continuity ofSA.

Proof. We only have to prove that ifx ∈ Ais a point of continuity ofSthen it is a continuity point ofSA.

Letybe a point close tox, theny =W_loc^u (x)∩W_loc^s (y)is a point in_Asuch that S(y)is close toS(x)and

SA(y)=W^s(S(y))∩W^c(y)is close toSA(y)=S(y).

HenceSA(y)is close toSA(x)=S(x).

Proposition 2.1. Iff satisfies propertyPthen for every periodic pointp,Sis continuous atp.

Proof. Letk denote the number of periodic points in C_{f (x)}^x , for all periodic pointx ∈ A. Supposex is a periodic discontinuity point ofS, then we have a sequence(xn)n∈Nof periodic points of continuity such that limn→∞xn=x and lengt h(C_S(x^xn

n)) > lengt h(C_S(x)^x )+α, withα >0.

For everyxn, there existkperiodic pointsx_n¹ < . . . < x_n^k inC_{f (x}^xn

n), ordered by the chosen orientation.

Since limn→∞W^c(xn) = W^c(x)in compact sets, there existxⁱ, limit point of x_nⁱ inW^c(x), andxⁱ must be periodic. Since the number of periodic points in C_{f (x}^xn _n) and inC_{f (x)}^x is the same, then there exists only a limit point ofx_nⁱ, i.e.

limn→∞x_nⁱ =xⁱ.

In particular limn→∞x_n¹ = x¹, and this gives limn→∞S(xn) = S(x); so lengt h(C_S(x^xn

n)) < lengt h(C_S(x)^x )+α ifnis big enough, which is absurd.

We have proved thatSis continuous at every periodic point.

Lemma 2.9. Letbe a basic set andx∈.

1. Ifdim(W^s(x))=n−1 then there is a finite number of points ofin the connected component ofW^c(x)∩W^s()that containsx.

2. Ifdim(W^s(x))=n−2 then there is a finite number of points ofin the connected component ofW^c(x)∩W^u()that containsx.

Proof. We will prove just the first statement.

Suppose that it is false. Then we can choose{xi}in∩W^s()∩W^c(x), such thatx1< x2< . . . < xl < . . . in the given orientation ofW^c(x). There exists k > 0 such that f⁻¹|W_k^c(x) "expands", ∀x ∈ A. Then there exists n1 ∈ N verifying thatlengt h(f⁻ⁿ¹(C_x^x₁)) > k, for alln≥n1. There existsn2∈Nsuch thatlengt h(f⁻ⁿ²(C_x^x₂¹)) > k, for alln≥n2. Letl0such thatkl0> K+1, where K = maxx∈Mlengt h(C_{f (x)}^x ). We continue in this way obtainingn3, . . . , nl0. LetN =max{n1, . . . , nl0}, then

lengt h(f^−N(C_x^x_l

0)) > kl0 > K+1

Hence, as in the proof of Lemma 1.3 we conclude that there existsp∈f^−N(C_x^x

l0) such thatp ∈ ∂W^s() and thereforef^N(p) ∈ ∂W^s()andf^N(p) ∈ C_x^x_l

0 ⊆

W^s(); which is a contradiction.

We have actually proved that there are no more than[^K+_k¹]points ofin the

connected component ofW^s()∩W^c(x).

3 Continuity of the map S.

Let us first prove the next lemma.

Lemma 3.1. Letxbe a continuity point ofSAandS˜A, (i.e. x ∈Q) then for all y∈W^s(x),W^c(y)∩A= ∅.

Proof. Let >0 be such that∪x∈AW^s(x)⊂W^s(A).

Letx ∈ QandUx be a neighborhood of x such that for all y ∈ Ux we have thatlengt h(W^c(y))is close enough tolengt h(W^c(x)), and lety ∈Ux∩W^s(x).

Since W^c(y) ⊂ W^s(A) and W^s(A) is open, there exists a neighborhood of W^c(y),V, such thatV ⊆W^s(A)andV ⊂ ∪z∈Ux(W^c(z)), in such a way that ifz∈V ∩Athenlengt h(W^c(z))is close enough tolengt h(W^c(y)).

By the density of the closed leaves in the central foliation, there exists a curveζin V, included in a closed leaf of the central foliation,Osuch thatζ =O∩W^s(A).

There exists a periodic pointpsuch thatp∈ζ∩A,ζ =W^c(p)and sinceSA

andS˜Aare continuous atyby the remark of Lemma 2.7, the lengths ofW^c(y) andζ are close; and the lengths of the curvesC_S^p

A(p), andCp^S˜A(p)are greater than thepreviously defined.

Then, considering open setsVn such thatVn → W^c(y), we can assert that there exist curvesζn ⊂Vnand periodic pointspn∈ζn∩Asuch that the lengths ofW^c(y)andζnare close; and the lengths of the curvesC_S^pn

A(p_n), andCp^S˜An^(pn)are greater than.

Sinceζnconverges toW^c(y)and the distance ofpnto∂(W^s(A))is bounded away from 0, there exists a limit pointpofpnsuch thatp∈A∩W^c(y).

We have proved that ifx∈Qthen

∀y ∈W_loc^s (x),∃p∈W^c(y)∩A.

Successive applications of this proceeding enables us to conclude that ifx ∈Q

∀y ∈W^s(x),∃p∈W^c(y)∩A.

Corollary 3.1. =S(R)is a repeller set.

Proof. Letx ∈ Q∩A, z ∈ W^s(S(x)) and z = W^c(z)∩W^ss(x). Since z ∈W^s(x)withx∈Q, then by Lemma 3.1 there existsq ∈W^c(z)∩A; hence S(q)=zandz∈S(R). Then

∀x ∈Q∩A, W^s(S(x))⊆S(R).

We have proved thatS(R)is included in a basic set. Now, ify =S(x)with x∈A∩Qthen

W^s(y)⊆S(R)⊆S(R)⊆⊆W^s(y).

It follows thatS(R)is a basic set, and since it contains a stable manifold we have

that=S(R)is a repeller set.

Let us consider the following maps.

Definition 3.1. Let:W^u()→∂W^u()be a map such that, for everyx in the basin of repulsion of,(x)is the nearest point in its central leaf in the positive direction verifying that it is not in the basin of repulsion of.

Definition 3.2. Let˜ : W^u() → ∂W^u() be the map analogous to, but in the negative direction of the central foliation.

Definition 3.3. Let : → ∂W^u() be the restriction of to and ˜ :→∂W^u()the restriction of˜to.

The version of Lemma (1.3) for repeller sets makes the preceding definitions possible.

As done after Definition 2.3 we defineW^c(x)as the connected component of W^c(x)∩W^u()which containsx, ifx ∈W^u().

All the properties verified byS,S,˜ SA andS˜A are verified by,,˜ and ˜ with the obvious modifications. In particular, there exists a residual set ⊂W^u()such thatand˜are continuous in. Besides, ifx ∈then for ally ∈ W^u(x)we have thatW^c(y)∩ = ∅. Once again, if property_Pis verified, all the periodic points ofare continuity points for all these maps.

Lemma 3.2. Letx ∈. Suppose thaty ∈W^u(x). Then W^c(y)∩= ∅.

Proof. By the version of Lemma 3.1 for repeller sets and the continuity of

and˜restricted to, we have that for all pointx ∈there is a neighborhood Uxsuch that ify ∈Ux andz∈W_loc^u (y)thenW^c(z)∩= ∅.

Let

U= ∪x∈Ux. Uis an open and dense set inW^u().

Letx ∈and suppose by contradiction that there existsy0∈ W^u(x)such that W^c(y0)∩= ∅. In addition, there exists a neighborhoodVy0 ofy0such that if z∈Vy0 ∩W^u(y0)thenW^c(z)∩= ∅.

Since W^u(x) is dense in , there exists v ∈ W^u(x)∩U, hence there exists

˜

v∈W^uu(x)∩U.

Let C ⊆ W^uu(x) an arc such that is maximal with respect to the following property: ify∈C,W^c(y)∩= ∅.

Letr˜be an extreme of C andr =W^c(r)˜ ∩.

Ifw∈C∩W_loc^uu(˜r)thenw=W^c(w)∩W_loc^uu(r)exists and verifies thatW^c(w)∩ = ∅; so we can define

W^u+(r)= connected component of{y ∈W^u(r)|W^c(y)∩= ∅}

such thatW^u+(r)∩W^u(r)= ∅for any >0.

For alln∈N,fⁿ(r)∈andW^u+(fⁿ(r))contains an arcDn⊂W^uu(fⁿ(r)) whose length grows exponentially and it has an extreme infⁿ(r).

Letq ∈ ω(r), then W^u+(q)contains a “half plane” of W^u(q), i.e. with an adequate orientationonW^uu(q), we have

W^u+(q)= { v ∈W^u(q)|W^c(v)∩W^uu(q)q}

We may also assume thatfⁿ(r)→q. Takingnandmbig enough we obtain thatfⁿ(r)andf^m(r)are as close as we wish, then there is no possibility that W^s(fⁿ(r))intersectsW^u(f^m(r))inW^u+(f^m(r))because this point would be inW^u+(f^m(r))∩.

In the same way there is no possibility thatW^s(f^m(r))intersectsW^u(fⁿ(r))in W^u+(fⁿ(r)).

It follows that ifnandmare big enough thenW^s(fⁿ(r))intersectsW^u(f^m(r)) inW^c(f^m(r))because the central-stable foliation locally separatesM.

Then there are two possibilities:

1. There exist infinite many stable manifolds off^j(r), withj ∈ N. In this case, there exist infinite many points in∩W^c(f^m(r)), but this contradicts Lemma 2.9.

2. There exists a finite number of different stable manifolds off^j(r), with j ∈N.

We can suppose thatW^s(fⁿ(r))is the same for alln∈N. Sincefⁿ(r)→ q, we have thatq is periodic point; and sinceW^u+(q)∩ = ∅,q is not a continuity point ofand˜, because it would contradict the version of Lemma 3.1 for repeller sets.

On the other hand, the version of proposition 2.1 for repeller sets asserts that all periodic points inare continuity points of, and, and hence˜ of, and˜, which yields a contradiction.

We notice that it is at this point where PropertyPis used.

We have proved that for allx ∈, and for ally ∈W^u(x)

W^c(y)∩= ∅.