Density of Hyperbolicity and Tangencies in Sectional Dissipative Regions

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www.elsevier.com/locate/anihpc

Density of hyperbolicity and tangencies in sectional dissipative regions

Enrique R. Pujals

^a,^∗

, Martin Sambarino

^b

aInstituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, CEP: 22460-320, Rio de Janeiro, RJ, Brazil bCentro de Matemática, Universidad de la República, Iguá 4225 esq. Mataojo, CP:11400, Montevideo, Uruguay

Received 16 March 2008; received in revised form 16 April 2009; accepted 26 April 2009 Available online 27 May 2009

Abstract

In this paper we extend the notion of sectionally dissipative periodic points to arbitrarily compact invariant sets. We show that given a sectionally dissipative and attracting region for a diffeomorphismsf, there is a neighborhood off and a dense subset of it such that any diffeomorphismgin this dense subset either exhibits a sectional dissipative homoclinic tangency or the part of the limit set ofgin this attracting region is a hyperbolic compact set. The proof goes extending some results on dominated splitting obtained for compact surfaces maps.

Résumé

Dans cet article nous étendons la notion de points périodiques sectionnellement dissipatifs à des ensembles compacts invariants quelconques. Nous montrons qu’ayant une région sectionnellement dissipative et attrayante pour un difféomorphismef, il y a un voisinage def et un sous-ensemble dense de celui-ci tels que tout difféomorphismegdans ce sous-ensemble a une tangence homoclinique sectionnellement dissipative oú la partie de l’ensemble limite degdans la région attrayante est un ensembe compact hyperbolique. La preuve est une géneralisation des résultats obtenus pour des difféomorphismes de surfaces.

Keywords:Generic dynamics; Partial hyperbolicity; Dominated splitting; Homoclinic bifurcations; Homoclinic tangencies

1. Introduction

During the early times of non-conservative dynamics was a common sense that “non-pathological” systems behave in a very simple form such as the nonwandering set consisting of finitely many periodic elements. The achievement of Peixoto that an open and dense subset ofC¹vector fields on surfaces consist of the now-called Morse–Smale systems is paradigmatic of this view. However, in the early sixties (by Anosov and Smale following Birkhoff, Cartwright and Littlewood, and others) it was shown that “chaotic behavior” may exist within stable systems and this was the starting point of the hyperbolic theory and the modern non-conservative dynamical systems theory. A major result

* Corresponding author.

E-mail addresses:enrique@impa.br (E.R. Pujals), samba@cmat.edu.uy (M. Sambarino).

doi:10.1016/j.anihpc.2009.04.003

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in hyperbolic theory is the so-calledΩ spectral decomposition theorem for Axiom A systems. This means that for these systems, the nonwandering set can be decomposed into finitely many compact, disjoint and transitive pieces.

Although this pieces could exhibit a chaotic behavior (and nowadays well understood) there are just finitely many of them and this recovers the old vision by replacing finitely many periodic elements by these finitely many “dynamically irreducible” pieces.

It was soon realized that hyperbolic systems were not as universal as initially thought: there were given examples of open sets of diffeomorphism were none of them are hyperbolic. Nevertheless in all these new examples the nonwandering set still decomposes into finitely many compact, disjoint and transitive pieces. It was through the seminal work of Newhouse (see [10–12]) where a new phenomena was shown: the existence of infinitely many periodic attractors (today called Newhouse’s phenomena) for residual subsets in the space ofC^r diffeomorphisms (r2) of compact surfaces. The underlying mechanism here was the presence of ahomoclinic tangency: non-transversal intersection of the stable and unstable manifold of a periodic point.

In the late eighties, Palis conjectured (see [14,15,21]) that for surface diffeomorphisms, homoclinic tangencies are the solely mechanisms that leads to the explosion of the limit set into an infinite number of transitive isolated sets:

Any C^r-diffeomorphism on a surface can beC^r-approximated by one which is hyperbolic or by one exhibiting a homoclinic tangency.

The above conjecture was proved to be true for the case of surfaces and the C¹topology (see [17]). Moreover, in [20], it was proved that anyC²-diffeomorphisms having infinitely many periodic attracting points with unbounded period, can beC¹-approximated by another diffeomorphisms exhibiting a homoclinic tangency.

One may think that in higher dimensions the unfolding of a homoclinic tangency may lead to the breakdown of a finite decomposition of the nonwandering set. However, there are examples of robust transitive diffeomorphisms that coexist with the presences of a homoclinic tangency (see for instance [2]).

Nevertheless, it was shown in [22] that for smooth diffeomorphisms on manifold with dimension larger than two, the unfold oftangencies associated to sectional dissipative periodic points(tangencies associated to a periodic point such that the modulus of the product of any pair of eigenvalues is smaller than one) leads to the same phenomena that holds in dimension two: residual subsets of diffeomorphisms exhibiting infinitely many periodic attractors.

Regarding the previous comments and following the conjecture formulated by Palis, it is naturally to ask if is true that any diffeomorphisms on a finite-dimensional manifold can be eitherC^r-approximated by another one such its dynamic is hyperbolic restricted to a“sectionally dissipative regions of the limit set”, or it isC^r-approximated by a system exhibiting a sectional dissipative homoclinic tangency.In few words, any result in this direction, would be a converse to the one proved in [22] and mentioned above. This is one of the aims of this paper (see Corollary 1.1).

1.1. Definitions and statements

Letf:M→Mbe a diffeomorphism of a compact Riemannian manifold without boundary. We denote byΩ(f ) the nonwandering set of f and byL(f )its limit set which is defined as the closure of the forward and backward accumulation points of all orbits, i.e.

L(f )=

x∈M

ω(x)∪α(x).

A setΛis called hyperbolic forf if it is compact,f-invariant and the tangent bundleT_ΛMcan be decomposed as T_ΛM=E^s⊕E^uinvariant underDf and there existC >0 and 0< λ <1 such that

Df_/Eⁿs(x)Cλⁿ and Df⁻ⁿ

/E^u(x)Cλⁿ for allx∈Λand for every positive integern.

We say thatf is a hyperbolic diffeomorphism ifL(f )is hyperbolic.

We recall that the stable and unstable sets W^s(p)=

y∈M: dist

fⁿ(y), fⁿ(p)

→0 asn→ ∞ , W^u(p)=

y∈M: dist

fⁿ(y), fⁿ(p)

→0 asn→ −∞

areC^r-injectively immersed submanifold whenpis a hyperbolic periodic point off. Theindexofpis the dimension ofW^s(p). IfW^s(p)andW^u(p)has a nontransverse intersection we say thatphas ahomoclinic tangency.

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A periodic pointpof periodmis calledsectionally dissipativeif the modulus of the product of any two distinct eigenvalues ofDf_p^mis smaller than one. A homoclinic tangency associated to a sectionally dissipative periodic point is called asectionally dissipative tangency.

We wish to “extend” the notion of sectional dissipativeness to non-periodic points. This is done as follows.

Definition 1.1(Two-dimensional determinant).Let, be the Riemannian metric ofM. LetG2(M)be the Grassman- nian space of all two-dimensional subspaces inT M. Observe that given(x, L)∈G2(M)we can consider the metric , x restricted to L and in particular it induces a two dimension volume formwL onL. The derivative off acts naturally overG2(M), i.e.:Df (x, L)=(f (x), Df (L)). The determinant ofDf at(x, L)is defined as the unique real number det(Dfx|L)such that

f^∗(wDf (L))=det(Dfx|L)wL, wheref^∗is the pull back associated tof.

Definition 1.2(Sectionally dissipative compact sets).Let f :M→M be a C¹-diffeomorphism and Λa compact invariant set. We say thatf is sectionally dissipative onΛ(orΛis a sectionally dissipative set forf) if for any point x∈Λand for any two-dimensional subspaceLholds that

det(Dfx|L)<1.

We remark that ifp is a periodic point and the orbitO(p)is a sectionally dissipative set thenp is a sectionally dissipative periodic point, i.e, the modulus of the product of any two eigenvalues ofDf_p^mis less than one. The converse is not true even ifpif fixed.

More generally, givenλ >0, we denote withSDf(λ)the set SDf(λ):=

x∈M: det(Dfx|L)< λfor any two-dimensional subspaceL⊂T_xM . We define

SDf(λ):=

x: O(x, f )⊂SDf(λ) ,

whereO(x, f )is the orbit ofxbyf. Notice that ifΛis a sectionally dissipative set thenΛ⊂SDf(1).

We denote by

L(f,1):=L(f )∩SDf(1).

Finally, given two compact invariant setsA⊂B we say thatAis isolated withinB if there is a neighborhoodU ofAsuch thatB∩U=A. We say thatUis an attracting region iff (U )⊂Uand observe thatL(f )∩Uis isolated withinL(f ).

Now we can formulate our main theorem that relates tangencies and hyperbolicity in the sectionally dissipative regions of the limit set:

Theorem A.Letf :M→Mbe aC²-diffeomorphism of a finite-dimensional compact Riemannian manifoldM. Let Λ⊂L(f,1)be a compact invariant isolated set inL(f )and such that the periodic points inΛare hyperbolic. Then, one of the following statements holds:

1. For any neighborhoodU(f )and a neighborhoodUofΛthere exitsg∈U(f )exhibiting a sectionally dissipative tangency associated to a(sectionally dissipative)periodic pointpsuch thatO(p, g)⊂U.

2. Λ=Λ₁∪Λ₂whereΛ₁ is a hyperbolic set andΛ₂ consists of a finite union of periodic simple closed curves C1, . . . ,Cn, normally hyperbolic and such thatf^mⁱ:Ci→Ci is conjugated to an irrational rotation(mi denotes the period ofCi).

The following corollary is an immediate consequence of Theorem A and represents a weak converse of the main result in [22]. Before to state it, we introduce the set

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Λ₀=P₀(f )

the closure of the periodic attractors. Observe thatΛ₀is a compact invariant set inL(f ).

Corollary 1.1.Letf∈Diff²(M)be a diffeomorphism exhibiting infinitely many attracting periodic points and let us assume thatΛ₀is an isolated compact set inL(f,1)such that all the periodic points are hyperbolic. Then,f can be C¹-approximated by a diffeomorphismghaving a sectionally dissipative tangency in a neighborhood ofΛ0.

The previous result follows immediately sinceΛ₀cannot verify the second option of Theorem A. Another version of this corollary is the following:

Corollary 1.2. Letf ∈Diff²(M)be a Kupka–Smale diffeomorphism and let U be an attracting region such that U⊂SDf(μ), μ <1. Assume thatf has infinitely many periodic attractors inU. Then,f can beC¹-approximated by a diffeomorphismghaving a sectionally dissipative tangency inU.

Indeed, observe thatΛ=L(f )∩Uis isolated withinL(f )sinceUis an attracting region andΛ⊂SDf(1). Since the second option of Theorem A cannot happen sincef has infinitely many periodic attractors then the first one must occur.

An important consequence of Theorem A is also the following result which extends in some sense a bidimensional result in [9]:

Corollary 1.3.Letf ∈Diff¹(M)and letUbe an attracting region such thatU⊂SDf(μ),μ <1. Then, there exist a neighborhoodU(f )and a residual subsetR⊂U such that for anyg∈Rone of the following statements holds:

1. ghas infinitely many periodic attractors inU. 2. L(g)∩Uis hyperbolic.

Another straightforward important consequence is also the following result which extends in some sense a bidimensional result in [17]:

Corollary 1.4. Let f ∈Diff¹(M)and let U be an attracting region such thatU⊂SDf(1). Then, there exists a neighborhood U(f ) such that any g∈U(f )can be C¹-approximated by a diffeomorphismg either exhibiting a sectionally dissipative tangency inUor such thatL(g)∩Uis hyperbolic.

The proof of these two last corollaries are given in the next section.

In the direction to prove Theorem A, we shall extend some results on dominated splitting that we have obtained for compact surfaces. Letf :M→Mbe aC¹diffeomorphism of a compact Riemannian manifoldM. Anf-invariant setΛis said to have dominated splitting if we can decompose its tangent bundle in two invariant subbundleTΛM= E⊕F, such that:

Df_/E(x)ⁿ Df⁻ⁿ

/F (fⁿ(x))Cλⁿ, for allx∈Λ, n0, withC >0 and 0< λ <1.

We say that the dominated splitting is a codimension one dominated splittingif dimension(F )=1. We say that a codimension one dominated splitting is acontractive codimension one dominated splittingif the directionEis a contractive direction, i.e.: there existC >0 and 0< λ1<1 such that for anyxand any positive integernholds that

Df_/E(x)ⁿ < C λⁿ₁.

In this case we denote the directionEasE^s.

The next result establishes the relation between contractive codimension one dominated splitting and not being approximated by sectionally dissipative tangency. Let us state first a definition.

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Definition 1.3.Given a compact invariant setΛwe say thatf_/ΛisC¹-far from sectionally dissipative homoclinic tangencies if there is a neighborhoodU⊂Diff¹(M)off and a neighborhoodUofΛsuch that anyg∈U does not exhibit a sectionally dissipative tangency associated to a periodic pointpofgwithO(p, g)⊂U.

Theorem B.LetΛbe a compact invariant set inL(f,1)and isolated withinL(f ). Let us assume thatf_/ΛisC¹-far from sectionally dissipative tangencies and all the periodic points in Λare hyperbolic. Then,Λ\P₀(f_/Λ)(where P₀(f_/Λ)is the set of periodic attractors off inΛ)has a contractive codimension one dominated splitting.

Now we prove, that under certain conditions, contractive codimension one dominated splitting are actually hyperbolic.

Theorem C.Let f :M→M be a C²-diffeomorphism. Let Λbe a compact invariant set contained in L(f )and exhibiting a contractive codimension one dominated splitting. Let also assume thatΛis isolated withinL(f )and all the periodic points inΛare hyperbolic. Then,

Λ=Λ₁∪Λ₂

whereΛ1is a hyperbolic set andΛ2consists of a finite union of periodic simple closed curvesC1, . . .Cn, normally hyperbolic and such thatf^mⁱ:Ci→Ci is conjugated to an irrational rotation(mi denotes the period ofCi).

Remark 1.1.Observe that in Theorem C we are not assuming that the setΛis contained inL(f,1).

We will prove also the next corollary from Theorem C.

Corollary 1.5.Letf :M→Mbe aC²-diffeomorphism. Assume thatM has a contractive codimension one dom- inated splitting and all the hyperbolic periodic points are of saddle type. Thenf is an Anosov diffeomorphism and M=Tⁿ.

The paper is organized as follows: In Section 2 we give the proofs of Theorem A, Corollaries 1.3, 1.4 and 1.5 assuming that Theorems B and C hold. In Section 3 we give the proof of Theorem B. The proof of Theorem C is given in Section 5. To perform the proof, we need a series of results about the dynamical geometry of sets having a contractive codimension one dominated splitting. In this direction, in Section 4, under the hypothesis of contractive codimension one dominated splitting, we show the existence of Markov partition for a general class of sets that include the homoclinic classes. This result is a fundamental tool in the proof of the rest of Theorem C.

2. Proof of Theorem A and Corollaries 1.3, 1.4 and 1.5

Through this section we assume that Theorems B and C hold.

Proof of Theorem A. Let f andΛbe as in the statement and assume that the first option does not happen, i.e.

f_/ΛisC¹-far from sectionally dissipative tangencies. By Theorem B follows thatf_/Λ_\_P₀_(f_/Λ₎ exhibits a contractive codimension one dominated splitting. Given a neighborhoodV ofΛ\P₀(f_/Λ)we have thatP₀(f_/Λ)∩V^c<∞and setP₀(f_/Λ)∩V^c= {p₁, . . . , p_n}. If the neighborhoodV has been appropriately chosen, we have that any compact invariant set inV has contractive codimension one dominated splitting. ThereforeΛ˜=Λ\ {O(p1), . . . ,O(pn)}has contractive codimension one dominated splitting. On the other hand, ifΛis isolated withinL(f )so it isΛ. Now˜ applying Theorem C toΛ˜ we have the desired decomposition of it as a union of a hyperbolic set and finitely many periodic curves “supporting an irrational rotation”. SinceΛ= ˜Λ∪ {O(p1), . . . ,O(pn)}andp1, . . . , pnare hyperbolic (periodic attractors) we have the desired decomposition ofΛas required in the second option of Theorem A. 2 Proof of Corollary 1.3. LetU be such that for anyg∈U we have thatg(U )⊂U andL(g)∩U⊂SDg(1). For g∈U consider the map Γ that Γ (g)=P₀(g, U )whereP₀(g, U )is the set of attracting periodic point ofg inU. This map is lower semicontinuous and there is a residual subset R₁⊂U(f )of continuity points of Γ. Let R₂=

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{g∈R₁: P₀(g, U )= ∞}and considerV=U\R₂. It follows that ifg∈R₁∩V thenghas finitely many periodic attractors and sinceR₁is formed by continuity points ofΓ we have that there isV1open and dense inV such that anyg∈V1has finitely many periodic attractors. The set ofC²Kupka–Smale diffeomorphismginV1is dense inV1

and by Theorem A all of them satisfyL(g)∩Uis hyperbolic. Since there cannot exist a cycle among the basic pieces of L(g)(sincegis Kupka–Smale and the basic pieces have index n−1 or are periodic attractors) it follows by a straightforward adaptation of theΩ-stability theorem that there isV2open and dense inV1such that for anyg∈V2

holds thatL(g)∩Uis hyperbolic. Hence R=V2∪R2

satisfies the conclusion of Corollary 1.3. 2

Proof of Corollary 1.4. In the same way as in the proof of Corollary 1.3, letU(f )be such that anyg∈U(f )satisfies g(U )⊂U andL(g)∩U⊂L(g,1). Recalling again that the set ofC²Kupka–Smale diffeomorphism inU(f )are dense and arguing again as in Corollary 1.3 the result follows by a direct application of Theorem A. 2

Proof of Corollary 1.5. By Theorem B holds thatL(f )is the union of a hyperbolic set and a finite number of periodic simple closed curves normally hyperbolic (attracting) “supporting an irrational rotation”. It follows that there must be a hyperbolic repeller inL(f ). In other words, there existsΛ⊂L(f )such thatΛis maximal invariant with local product structure and it is a repeller. Moreover it has stable indexn−1. On the other hand, by [19] follows thatF is uniquely integrable. Now, the exact same proof of the main theorem in [13] where it is proved that a repeller in a codimension one Anosov diffeomorphism is also an open set applies here toΛ. ThusM=Λand hencef is Anosov.

By a result in [5] follows thatM=Tⁿ. 2

3. Proof of Theorem B: Dominated splitting for systems far from sectionally dissipative homoclinic tangencies LetΛbe as in Theorem B, that is,Λ⊂L(f,1)and it is isolated withinL(f ). Recall thatf_/Λis far from sectionally dissipative homoclinic tangencies and hence there exist a neighborhoodU of ΛandU(f )such that anyg∈U(f ) does not exhibit a homoclinic tangency associated to a sectionally dissipative periodic point of gwhose orbit lies entirely inU. From now on and through this section,UandU(f )will be as above.

We denote by Per^SD_n₋₁(g, U )

the set of sectionally dissipative periodic points (i.e the product of any two distinct eigenvalues is less than one) ofg having indexn−1 and whose orbit lies entirely inU.

We shall split the proof of Theorem B in the following sequence of propositions.

Proposition 3.1.LetΛbe as in Theorem B and fix anyη >0. Letx∈Λand assume thatxis not a periodic attractor.

Then, there exist sequences of diffeomorphisms{gn}and periodic points{qn}such thatgn→f,qn→x such that qn∈Per^SD_n₋₁(g, U )∩SDg_n(1+η).

For the next proposition we need the definition of angle between subspaces. Let E andF be two subspaces of finite-dimensional vector spaceV with an inner product and assume thatE⊕F=V. Hence dim(F )=dim(E^⊥)and F is the graph of the linear mapL:E^⊥→Edefined as follows: givenw∈F there exists a unique pair of vectors v∈E,u∈E^⊥, such thatv+u=w. DefineL(v)=uobtaining that graph(L)=F. We define, as it is done in [9], the angle (E, F )betweenEandF asL⁻¹In particular (E, E^⊥)= +∞.

Remark 3.1.IfEandF are subspaces of a vector spaceWwith an inner product and such thatE∩F = {0}then we can define the angle between them as before just settingV =E⊕F with the inner product inherited fromW. Proposition 3.2.LetΛbe as in Theorem B. There exist a neighborhoodV(f )off, a neighborhoodU ofΛ,γ >0 andη >0such that for anyg∈V(f )and anyq∈Per^SD_n₋₁(g, U )∩SDg(1+η)it holds that

E_q^s, E_q^u

> γ .

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With the above two propositions we prove Theorem B and this is the content of our last proposition in this section.

Proposition 3.3.Under the assumptions of Theorem B, Propositions3.1and3.2imply that the setΛ\P₀(f_/Λ)has contractive codimension one dominated splitting.

The proofs of these propositions are done in the next subsections. Before given the proof, we state now a classical C¹perturbation technique known as Franks’ lemma.

Lemma 3.0.1.(See [4, Lemma 1.1].) LetMbe a closedn-manifold andf :M→Mbe aC¹diffeomorphism, and letU(f )a neighborhood off. Then, there existU1(f )⊂U(f )and >0such that ifg∈U1(f ),S⊂Mis a finite set,S= {p₁, p₂, . . . , p_m}andL_i,i=1, . . . , m, are linear mapsL_i:T_p_iM→T_{f (p}_i₎MsatisfyingL_i−D_p_ig, i=1, . . . , m, then there existsg˜∈U(f )satisfyingg(p˜ _i)=g(p_i)andD_p_ig˜=L_i,i=1, . . . , m. Moreover, ifUis any neighborhood ofSthen we may choseg˜so thatg(x)˜ =g(x)for allx∈ {p₁, p₂, . . . , p_m} ∪(M\U ).

3.1. Proof of Proposition 3.1

We first recall a version of the closing lemma (see [9, Lemma I.2]).

Lemma 3.1.1.Givenf ∈Diff¹(M),x∈M, >0and a neighborhoodU(f )there existr >0,ρ >1such that ifw∈ B_r(x)with0< rrandf^m(w)∈B_r(x)for somem >0then there exist0m₁< m₂mandg∈U(f )such that f^mⁱ(w)∈B_ρr,i=1,2,g^m²⁻^m¹(f^m²(w))=f^m²(w)andd(g^j(f^m²(w)), f^j(f^m¹(w)))for0jm₂−m₁. Corollary 3.1.LetΛbe as in Theorem B and letx∈Λ. Then, there exist sequencesg_n→f,q_n→x andη_n→0 such thatq_n∈Per(gn, U )∩SDgn(1+η_n).

Proof. Letη_nbe a sequence of positive real numbers decreasing to 0. Then there exist_n0 andUn(f )such that ifg∈Un(f )andzsatisfies thatd(z, Λ)2thenz∈SDg(1+η_n). This follows by a standard continuity argument sinceΛ⊂SDf(1). We may assume thatUn+1⊂Unand _nUn=f.

Now, letx∈Λ. Ifx is periodic then there is nothing to prove. Assume thatx is not periodic. SinceΛis isolated withinL(f ), we may assume that for all_nit holds that L(f )∩ {z: d(z, Λ)_n} =Λ. It follows that ifω(y)(or α(y)) intersects{z: d(z, Λ)_n}then it is contained inΛand henced(f^j(y), Λ)_n for allj j₀(orjj₀).

Now fixn. Sincex∈L(f )there existy_kandz_k∈ω(y_k)∪α(y_k)such thatz_k→x. Then, the result follows by direct application of the preceding lemma by settingwas an appropriate iterate ofy_kforklarge enough. 2

In order to finish the proof of the proposition we have to prove that ifx∈Λis not a periodic attractor, then the sequences given by the above corollary can be chosen so thatq_nis also sectionally dissipative periodic point ofgof indexn−1.

We may assume without loss of generality that the periodic pointsq_nofg_nare hyperbolic with simple spectrum and denote bym_nthe period ofq_n.

Assume first thatq_n are saddles for infinitely manyn’s (and we may assume without loss of generality that this holds for any n), and let λ_u=max{|λ|: λeigenvalue ofDg^m_nⁿ(q_n)}. Since q_n∈SDgn(1+η_n) it follows that the product of any two eigenvalues is less than(1+ηn)^mⁿ. We now choose a numberρ:

• Ifλu> (1+ηn)^mⁿthenρ=(1+ηn)^mⁿ.

• Otherwise choose 1< ρ < λuthatρ²is bigger than the product of any two distinct eigenvalues.

Letμ=ρ^mn¹ . Apply Franks’ lemma toL_i =μ⁻¹Dg_n(g_nⁱ(q_n)). Thus we obtaing˜_nsuch thatq_nis a periodic point ofg˜_nandO(q_n, g_n)=O(q_n,g˜_n). Notice thatg˜_n→f. The largest eigenvalue ofq_nwill now have modulus equals to λ_u/ρand henceq_nis a saddle. On the other hand, eitherq_n∈SDg_˜_n(1)or the product of any two distinct eigenvalues is equal to the product of any two distinct eigenvalues ofDg_n^mⁿ divided by ρ². In any case the periodic point is sectionally dissipative. Moreover it is always true thatq_n∈SDg_˜n(1+η_n).

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It remains to prove the proposition in case the periodic points given by Corollary 3.1 are periodic attractors for every largen.

The periods of the periodic pointsq_n must be unbounded. Otherwise, the pointx∈Λis also periodic and cannot be an attractor by our hypothesis. It cannot be a saddle because otherwise the pointsq_nare also saddle forg_n. Thus, x is nonhyperbolic periodic point contradicting the assumption that any periodic point off inΛis hyperbolic.

It is left to prove the proposition in the case the period of the periodic attractorsq_nare unbounded. For this we need a result which essentially due to Pliss [16].

Theorem 3.1.Letg_nbe a sequence of diffeomorphisms converging tof∈Diff¹(M). Assume that there is a sequence q_nsuch thatq_nis a periodic attractor ofg_nand whose periodsk_nare unbounded. Then, for every sequence_m0 there exist a subsequencen_mand a sequenceg˜_msuch that:

1. g^j_n_m(qn_m)= ˜g_m^j(qn_m)for0jkn_m.

2. Dg_n_m(g_n^j_m(q_n_m))−Dg˜_m(g^j_m(q_n_m))< _mfor0jk_n_m. 3. q_n_mis a saddle hyperbolic periodic point ofg_m.

4. g_m→f.

Before giving the (outline of) the proof of this theorem let us remark that with it, the proof of our Proposition 3.1 can be finished since we fall again in the case where the sequence of pointsq_n in Corollary 3.1 can be chosen as hyperbolic saddles and then we finish the proof as it was done before.

Proof. As we said this is essentially due to Pliss. We will give an outline of the proof so that the reader could complete it by itself. Fix the sequence _m. For everm we have to findq_n_m and g˜_m. Fix m and set=_m. It is enough, by a direct application of Franks’ lemma, to show that for somenmit holds that there are linear maps L_i:T_gi

n(q_n)M→T_gi+1

n (q_n)M,i=0, . . . , kn−1, such thatL_i −Dg_n(gⁱ_n(q_n))< andk_n−1

i=0 L_i has an eigenvalue of modulus equal to one. Arguing by contradiction, assume that this does not hold. This means (following [9]) that the family of sequence of periodic matrices induced by{Dg_n(g_nⁱ(q_n)): i∈Z, nm}is uniformly attracting. It follows by Lemma II.5 of [9] that there exitK₀, 0< λ <1 andm₀such that

k−1

j=0

Dg^m_n⁰

g_n^m⁰^j(q_n)K₀λ^k wherek= [k_n/m₀].

To continue we need a lemma known as Pliss’ lemma [16] (see also [8]):

Lemma 3.1.2.LetH >0and0< λ₂< λ₁<1be given. Then there exist a positive integerNand0< c <1such that given positive real numbersa_j,j =0, . . . , k−1, withk−1N such thata_jH forj=0, . . . , kand satisfying

k−1

j=0

a_jλ^k₂

then there exist0j1< j2<· · ·< jlk−1such that p

j=0

aj+j_iλ^p₁ for any1pk−1−ji andi=1,2, . . . , l.

Moreover,lck.

Continuing with the proof of the theorem we observe that givenm0there is a constantHsuch thatDg_n^m⁰(x)H for anyx∈Msinceg_n→f. Since the periodsk_nof the periodic pointsq_nare unbounded (and we may assume that k_n → ∞) we may choose 0< λ₂< λ₁<1 such that K₀λ^k < λ^k₂ where k= [k_n/m₀] for everyn large enough.

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Applying Pliss’ lemma, we have that for every largenthat there exist 0j₁(n) < j₂(n) <· · ·< j_l_n(n)[k_n/m₀] −1 such that

p

j=0

Dg_n^m⁰

g^(j_n⁺^jⁱ^)m⁰(q_n)λ^p₁ ∀0p[k_n/m₀] −1−j_i.

By this uniform contraction of rateλ1 we have that there existγ >0 and λ1< ρ <1 such that for anyx, y∈ Bγ(g^j_nⁱ(qn))it holds that

d

gn^pm⁰(x), gn^pm⁰(y)

ρ^pd(x, y) ∀0p[kn/m0] −1−ji.

Letp₀be such thatρ^p< γ /4 for allpp₀. Now, since the number of “times” 0j₁(n) < j₂(n) <· · ·< j_l_n(n) [k_n/m₀] −1 goes to infinity asngrows, we may findnlarge enough and 0i < t < l_nsuch that

j_t(n)−j_i(n)p₀ and d

g_n^jⁱ^m⁰(q_n), g^j_n^t^m⁰(q_n)

< γ /4.

Therefore, settingp=j_t−j_i we have that g^pm_n ⁰ :Bγ

g_n^jⁱ^m⁰(qn)

→Bγ /4

g^(p_n ⁺^jⁱ^)m⁰(qn)

⊂Bγ

g_n^jⁱ^m⁰(qn)

is a contraction and hence every point inB_γ(g^j_nⁱ^m⁰(q_n))under iteration ofg^pm_n ⁰converge to the unique fixed point of this contraction. This is not possible because the pointg^j_nⁱ^m⁰(q_n)is periodic ofg_n of periodk_nand cannot be fixed byg^pm_n ⁰. This is a contradiction and the proof is completed. 2

3.2. Proof of Proposition 3.2

Recall thatf/Λis far from sectionally dissipative tangencies and so there are neighborhoodsU(f )andU (Λ)such that there are no homoclinic tangencies associated to points inPer^SD_n₋₁(g, U )for anyg∈U(f ).

Proposition 3.2 asserts that there existV(f ), γ >0 andη >0 such that for anyg∈V(f )andq∈Per^SD_n₋₁(g, U )∩ SDg(1+η)then

E^s_q, E^u_q

> γ .

We state first the key lemma of this section which establishes the relationship between small angle of stable and unstable subspaces and homoclinic tangencies. It is a straightforward adaptation of Lemma 2.2.1 of [17]. Compare also with Lemma 4.2 of [24] where an explicit proof can be found.

Lemma 3.2.1.Let >0and letg∈Diff¹(M). Assume thatp∈Mis a hyperbolic periodic point ofgwith periodm.

Assume that there areE₁^u⊂E_p^uandE_p^s invariant underDg^m(p)and such thatDg_/E^ms

1< λ <1andDg_/E⁻^mu 1⁻¹>

σ >1. Assume thatλσ <1and letγ= (E^s₁, E₁^u). If γ <σ−1

σ+1 2

then there isg -C˜ ¹close togsuch thatpas a hyperbolic periodic point ofg,˜ gⁱ(p)= ˜gⁱ(p)fori=0,1, . . . , mand

˜

gexhibits a homoclinic tangency associated top. FurthermoreDg^m(p)=Dg˜^m(p).

Although the last part is not included in the original bidimensional statement it follows from the proof since the support of the perturbation is disjoint of the orbit ofp. Hence the orbit remains the same and if for instance pis sectionally dissipative still it is after the perturbation.

Now letU0(f )be a neighborhood off and₁such that ifg∈U0(f )andg˜is₁−C¹close togtheng˜∈U(f ).

LetU1(f )andbe from Lemma 3.0.1 applied toU0(f ).

Lemma 3.2.2.There existsm₀ such that ifg∈U1(f )andp is a periodic point ofgwhose orbit is inU of period m_pm₀such that:

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1. All eigenvalues ofDg^m^p:T_pM→T_pMhas modulus1.

2. There is a bidimensional subspaceP ⊂T_pMsuch thatDg_/P^m^p=Id.

Then, there existsg˜∈U(f )such thatp∈Per^SD_n₋₁(g, U )˜ and has a homoclinic tangency associated top.

Proof. LetK >1 and chooseγ >0 such thatγ < ^K_K⁻₊¹₁₂¹. Consider alsoC=sup{Dg: g∈U0(f )}and set= /C. Finally, letm0be such that

m₀

4(K+1)>1.

We will show that m₀ as above satisfies the lemma. So, letg andp satisfying the conditions of the lemma. By performing a very small perturbation we may assume thatDg^m^p(p)is diagonalizable and thatKer(Dg^m^p(p)−Id)= P and alsog∈U1(f ). LetEandF be two one-dimensional subspaces ofP such that (E, F ) < γ. After performing a very small perturbation, we obtaing1still inU1(f )such that:

• gⁱ(p)=gⁱ₁(p)fori=0, . . . , mp−1.

• p∈Per^SD_n₋₁(g₁, U ).

• EandF are invariant byDg₁^m^p(p).

• Dg₁^m^p_/E =λ <1; Dg^m₁^p_/F =σ >1 andλσ <1.

Notice thatE^u_p=F andE⊂E_p^s. Letγ₁= (E, F )and we may assume without loss of generality that (E, F )= min{ (Dgⁱ₁(E), Dgⁱ₁(F )), i=0, . . . , mp−1}. Moreover we may assume thatγ₁ (E^s(gⁱ₁(p)), E^u(g₁¹(p))),i= 0, . . . , mp−1.

If

γ₁<(σ−1)1

(σ+1)2

then, by Lemma 3.2.1 we are done. Otherwise we setδ=γ₁/2 and for 0im_p−1 considerT_i :T_gi

1(p)M→

T_gi+1

1 (p)M such that T_i/E^s

gi1(p)=(1−δ)Id and T_i/E^u

gi1(p)=(1+δ)Id.

It follows thatTi−Id< . For 0imp−2 letLi=Ti+1◦Dg1(g₁ⁱ(p))andLm_p−1=T0◦Dg1(g₁^m^p⁻¹(p)). It holds thatL_i−Dg₁(gⁱ₁(p))< and by applying Lemma 3.0.1 we obtaing˜∈U0(f )such thatg₁ⁱ(p)= ˜gⁱ(p)and Dg(˜ g˜ⁱ(p))=Li. It is straightforward to check that:

• p∈Per^SD_n₋₁(g, U ).˜

• E⊂E_p^s,F ⊂E_p^u.

• ˜λ= Dg˜^m_/E^p =λ(1−δ)^m^pandσ˜Dg˜_/F^m^p =σ (1+δ)^m^p.

• (E, F )=γ₁.

Ifσ˜Kthen it follows thatγ1⁽₍^σ^˜_σ_˜⁻₊¹⁾₁₎₂¹ and by Lemma 3.2.1 we are done. On the other hand, ifσ˜ Kby the way we choosem0we have that:

(σ˜ −1)1

(σ˜+1)2 =((1+δ)^m^pσ−1)1

((1+δ)^m^pσ+1)2 ((1+m_pδ)σ−1)1

(K+1)2 (m_pδσ )1

(K+1)2 m0

₁

(K+1)4γ1> γ1

and again by Lemma 3.2.1 the proof is now complete. 2

Corollary 3.2.Under the assumptions of Theorem B, there existδ >0and a neighborhoodU2(f )such that for any g∈U2(f )andp∈Per^SD_n₋₁(g, U )then there exists only one eigenvalue ofDg^m(p)(wheremis period ofp)of modulus

> (1−δ)^mand it is real and simple.