Hyperbolic sets that are not contained in a locally maximal one

In this paper we study two properties related to the structure of hyperbolic sets. First we construct new examples answering in the negative the following question posed by Katok and Hasselblatt. Let $\Lambda$ be a hyperbolic set, and let $V$ be an open neighborhood of $\Lambda$. Does there exist a locally maximal hyperbolic set $\widetilde{\Lambda}$ such that $\Lambda \subset \widetilde{\Lambda} \subset V $? We show that such examples are present in linear anosov diffeomorophisms of $\mathbb{T}^3$, and are therefore robust. Also we construct new examples of sets that are not contained in any locally maximal hyperbolic set. The examples known until now were constructed by Crovisier and by Fisher, and these were either in dimension bigger than 4 or they were not transitive. We give a transitive and robust example in $\mathbb{T}^3$. And show that such examples cannot be build in dimension 2.


Introduction
In the '60s, Anosov ([A2]) and Smale ([S]) began the study of some compact invariant sets, whose tangent space splits into invariant, uniformly contracting and uniformly expanding directions. More precisely, a hyperbolic set is defined to be a compact invariant subset of a compact manifold Λ ⊂ M of a diffeomorphism f such that the tangent space at every x ∈ Λ admits an invariant splitting that satisfies: • • there are constants C > 0 and λ ∈ (0, 1) such that for every n ∈ N one has Df n (v) ≤ Cλ n v for v ∈ E s (x) and Df −n (v) ≤ Cλ −n v for v ∈ E u (x) .
A specially interesting case is when the hyperbolic set Λ is the non-wandering set of f . Particularly when we also have that the set of periodic points of f is dense in the non-wandering set Ω(f ), we say that f is Axiom A. Given the relevance of these diffeomorphisms in the study of hyperbolic dynamics is natural to ask what kind of sets may or may not be a basic pieces of some spectral decomposition.
All basic pieces have the following property: The autor was partially supported by CSIC.
Definition 1.1. Let f : M → M be a diffeomorphism and ∆ a compact invariant hyperbolic set. We say that ∆ has locald product structure exists δ > 0 such that if x, y ∈ ∆, and d(x, y) < δ then W s ε (x) ∩ W u ε (y) ∈ ∆ where ε is as in the stable manifold theorem.
We will focus now on whether or not a set has this property. If they do not, we will be interested in studying whether or not the set is contained in an other set having this property.
Sets having this property are interesting on themselves since they can be thought of, locally, in coordinates of the stable and unstable manifold of a point. Also this property is equivalent to others that are very useful to understand the dynamics of a neighborhood of the set. Some of them are having the shadowing property or being locally maximal.
Many of the best known examples of hyperbolic compact sets have this property. Some examples could be the solenoid, the torus under an Anosov diffeomorphism or a horseshoe. Also, there are examples of simple sets that do not verify this property, for instance, the closure of the orbit of a homoclinic point. However, for a long time all the examples known of sets that did not have local product structure could be included in a set having this property. Moreover all known examples had such a set included in any neighborhood of the original one. In the 1960's Alexseyev asked the following question (that was later posed by Katok and Hasselblatt in [ [HK], p. 272]) Question 1. Let Λ be a hyperbolic set, and let V be any open neighborhood of Λ. Does there exist a locally maximal hyperbolic set Λ such that Λ ⊂ Λ ⊂ V ?
Also the following related question was unanswered: Question 2. Given a hyperbolic set Λ does there exist a hyperbolic compact invariant set with local product structure such that Λ ⊂ Λ?
Both questions remained open until 2001, when Crovisier [C] constructed an example based on an example of Shub in [HPS] that answer question 2 in the negative (and therefore question 1). This example is on the 4-torus.
Later, Fisher [Fi] constructed several other examples of this sort. He constructed robust examples in any dimension, and transitive volume preserving examples in dimension 4.
In spite of this there are still some natural questions left to answer • Does there exist an open set U (in the C 1 topology) of diffeomorphisms such that every f ∈ U possesses an invariant transitive hyperbolic set that is not contained in a locally maximal one on any manifolds? • Does there exist robust and transitive examples answering Question 1 in the negative on manifolds with dimension lower than 4? • Does there exist an example answering Question 1 in the negative but that it is contained in a bigger set having local product structure?
In section (3) we will show Theorem A. Let f A : T 3 → T 3 be an Anosov diffeomorphism. There is a connected, compact proper inavriant subset of T 3 , such that the only locally maximal set containing it is T 3 .
This answers our last question. Note that the same will be true for any g sufficiently close to f A . We also note that constructing this kind of examples is not possible for T 2 since all invariant compact proper sets are 0-dimensional and from [A1], in any neighborhood there is a locally maximal set that contains them.
In section (4) we describe a well known example by Mañe in [M] that we will use on section (5) to construct a new example of a set that is not included in any locally maximal set. This example gives a partial answer to our second question. It is robust, transitive, and it is a 3 dimensional example, which shows there are examples of this in lower dimensions. The previous examples had either tangencyes or came from skew-products so, they where not transitive or where in dim ≥ 4. In (5) we proved the following: Theorem B. There exists U ⊂ Dif f (T 3 ) such that for every g ∈ U there is a there is a compact, proper, invariant, hyperbolic subset of T 3 , such that there is no locally maximal set containing it.
In the case of 2 dimensional surfaces our first 2 questions can be combined in the following Question. If dim(M) = 2, and Λ ⊂ M is a transitive hyperbolic set and U is any given neighborhood of Λ. Does there exist compact invariant set with local product structure such that Λ ⊂ Λ ⊂ U?
We will give a positive answer to this question. In section (6) we will show: Theorem C. Let f : M → M be a diffeomorphism, M a compact surface and Λ ⊂ M a compact hyperbolic invariant set. If we also have that Ωf | Λ = Λ then for any neighborhood V of Λ, there exist Λ such that Λ is compact hyperbolic invariant and with local product structure and,

Preliminaries
Let M be a compact manifold, f a C r diffeomorphism, and Λ a hyperbolic set. For ε > 0 sufficiently small and x ∈ Λ the local stable and unstable manifolds are respectively: W u ε (x, f ) = y ∈ M| for all n ∈ N, d(f −n (x), f −n (y)) < ε . The stable and unstable manifolds are respectively: The stable and unstable manifolds are C r injectively immersed submanifolds. If two points of Λ are sufficiently close, The local stable and unstable manifolds intersect transversely at a single point.
A very useful property of hyperbolic set is the following: Theorem 2.2. (Shadowing Lemma). Let f : M → M be a diffeomorphism and Λ a compact hyperbolic set. Then, given β > 0, there exists α > 0 such that every α-pseudo orbits in Λ is β shadowed by an orbits (not necessarily in Λ).That is, if x n ∈ Λ is a α-pseudo orbit, then there exists y ∈ M such that d(f n (y), x n ) ≤ β for all n ∈ Z.
Let us recaall the following definition: Definition. 1.1 Let f : M → M be a diffeomorphism and Λ a compact invariant hyperbolic set. We say that Λ has local product structure exists δ > 0 such that if x, y ∈ Λ, and d(x, y) < δ then W s ε (x) ∩ W u ε (y) ∈ Λ. As a consequence of the shadowing theorem we have: Corollary 2.3. If in addition to the other hypothesis we have that Λ has local product structure, then every α-pseudo orbits in Λ is β shadowed by an orbits in Λ.
With this we can show a very important equivalence with having local product structure that is being locally maximal : Definition 2.4. A hyperbolic set Λ is called locally maximal (or isolated) if there exists a neighborhood V of Λ in M such that Λ = n∈Z f n (V ).
Corollary 2.5. A hyperbolic set Λ is locally maximal if and onely if Λ has local product structure.
As in [A] we name the properties we are going to be dealing with.
Definition 2.6. We say that a hyperbolic set Λ ⊂ M is premaximal, if there exists a hyperbolic set ∆ ⊂ M with local product structure (maximal invariant set) such that Λ ⊂ ∆.
Definition 2.7. We say that a hyperbolic set Λ is locally premaximal, if for every neighborhood U of Λ, there is a hyperbolic set ∆ with local product structure such that Λ ⊂ ∆ ⊂ U.

Proof of Theorem A: A set that is not locally premaximal
In this section we prove that there is a subset of the T 3 , invariant under a linear Anosov diffeomorphism f , that is not locally premaximal.
Let f be a Anosov diffeomorphism in T 3 that is induced form A ∈ GL(3, Z) which is a hyperbolic toral automorphism with only one eigenvalue grater than one, and all eigenvalues real, positive, simple, and irrational. Let π : R 3 → T 3 be such that π • A = f • π. Let us also suppose that f has two fixed points x 0 and x 1 , and π(0, 0, 0) = x 0 . As a consequence of the results in [Ha] we have: Theorem 3.1. Let f : T 3 → T 3 be a hyperbolic automorphism, we can find a path γ in T 3 , such that the set O(γ) T 3 , is compact, connected and non trivial.
This curve can also be constructed so that it's image contains a fixed point. For this γ we note Λ = O(γ).
We will prove now that in this conditions the only set with local product structure containing Λ is the whole T 3 , following mainly the ideas in [M2]. Here Mañe proves that every compact, connected, locally maximal subset of T n under a linear hyperbolic automorphism must be of the form ∆ = x + G, where x is a fixed point and G is an invariant compact subgroup. In particular in dimension 3 this implies that ∆ = T 3 or ∆ = x . We will adapt the proof to the case where ∆ is not connected but contains non trivial compact, connected, invariant set that contains a fixed point.
Definition 3.2. Let Λ ⊂ T 3 be a compact, connected and invariant, such that x 0 ∈ Λ. We say that a curve γ : We define Γ δ as the subgroup of π 1 (T 3 , e) = Z 3 generated by arcs γ : Using the continuity of A we have that, given δ there is a δ 1 , such that The idea now is to define a Γ 0 which we would naively define as the subgroup limit of Γ δ with δ going to zero. A first attempt to define it would consider δ>0 Γ δ but that set might empty and not represent what we want it to. Instead we define N δ as the subspace of R 3 generated Γ δ . We define Lemma 3.3. In the above mentioned conditions , [H] tells us that if the stable or unstable manifold are 1 dimensional then the only connected, locally connected, compact, invariant hyperbolic subsets are fixed points and the whole torus Note that since Γ 0 = Z 3 ∩ N 0 then the previous lemma implies that Γ 0 = Z 3 or Γ 0 = 0.
Lemma 3.4. If Γ 0 = Z 3 and π s : N 0 → E u is the projection associated with the Proof. To see this, note that N δ 1 ⊂ N δ 2 if δ 1 ≤ δ 2 . This implies that for some δ 0 , On the other hand dim(N 0 ) = dim(N δ ) = ran(Γ δ ), so ran(Γ δ ) = 3 and there is an If a ∈ π s (Γ δ ), and since E u + a is irrational, there is a unique a ′ ∈ Γ δ such that π s (a ′ ) = a. If there where a ′ and a ′′ such that π s (a ′ ) = π s (a ′′ ) = a, then a ′′ = E s +a ′ . This is impossible since a ′ , a ′′ ∈ Z 3 and E s + a ′ is a totally irrational plane.
Now we consider Λ to be the set described by Hancock (3.1).Then Λ is compact, connected, invariant, it contains a fixed point x 0 and is not trivial. Let us suppose there exists a set ∆ with local product structure containing Λ, and let us call it's lift ∆.
The strategy now is to see that such a ∆, must contain a dense set in the unstable manifold of x 0 (which is of dimension 1). Since ∆ is compact then ∆ = T 3 .
Definition 3.5. We say that x and y are n-ε-related in ∆ if there exists sequence of point x = x 0 , x 1 , . . . , x n = y such that: Proof. We take ε < δ with δ from the local product structure. We prove this lemma by induction. For n = 1 the property is verified by the local product structure.
Suppose now that x, y ∈ ∆ are n-ε-related. We have x = x 0 , x 1 , . . . , x n = y as in the definition. We define The following theorem implies theorem A.
Theorem 3.7. Let Λ be a compact, connected, invariant, such that x 0 ∈ Λ, and x 0 = Λ. Suppose there is ∆ such that Λ ⊂ ∆ and ∆ is compact invariant and with local product structure. Then ∆ = T 3 .
Proof. Let ∆ and Λ be the lifts of ∆ and Λ respectively.
If ∆ is compact invariant and local product structure, then by Lemma 3.6, if we have two points x, y ∈ ∆ which are n-ε-related, we have ( The goal then is to see that x 0 and any point Γ δ are n-ε-related and therefore π s (Γ δ ) ⊂ ∆. Since π s (Γ δ ) by 3.4 is dense in E u , then For this, is enough to note that Λ is in the hypothesis of the lemma 3.3. Therefore as π s (Γ δ ) is dense in E u for a δ sufficiently small, we can join x 0 with itself by a curve δ-adapted such that when lifted, it links x 0 with any point of Γ δ . For an appropriateδ , and any x ∈ Γ δ , we have that x 0 and x are n-δ-related for some n, as desired.

Mañe's robustly transitive diffeomorphisms that is not Anosov
In this section we will describe an example constructed by Mañe in [M]. This example is very well described in numerous references (see for instance [BDV], or [PS]), but we will include a description for the convenience of the reader, and because we will emphasize some properties of the example that will be useful later on. However we will not include the proofs,which can be found in any of the given references.
As in the previous section, let us starts with a linear Anosov diffeomorphism f A in T 3 that is induced form A ∈ GL(3, Z) which is a hyperbolic toral automorphism with only one eigenvalue grater than one, and all eigenvalues real, positive, simple, and irrational. Let 0 < λ s < λ c < 1 < λ u be the eigenvalues. Let F c be the foliation corresponding to the eigenvalue λ c , similarly with F s and F u . We remind you that all of these leaves are dense. We may also assume that f A has at least two fixed points, x 0 and x 1 , and that unstable eigenvalue λ u , have modulus greater than 3 (if not, replace A by some power).
Following the construction in [M] we define f by modifying f A in a sufficiently small domain C contained in B ρ 2 (x 1 ) keeping invariant the foliation F c . Where ρ > 0 is a small enough number to be determined in what follows. Let us observe We can take ρ sufficiently small so that x 0 ∈ Γ. Inside C the point x 1 undergoes a bifurcation as shown in the figure 2, in the direction of F c , which changes the unstable index of x 1 increasing it in 1. Also two other fixed points, x 2 and x 3 are created, with the same index x 1 had under f A .
As a result, we get a difeomorphism f which is strongly partially hyperbolic. That is where E s f is uniformly contracting and E u f is uniformly expanding. In fact, E s f and E u f are contained in some small cones around E s and E u respectively. Then by a well known results (see [HPS]) we get that the bundles E s f and E u f are uniquely integrable to foliations F s f and F u f called the (strong) stable and unstable foliations. Moreover, they are quasi-isometric Since we preserved the central foliation we have also uniquely integrable, by what we call the center-stable and center-unstable foliations respectively. In [M] it is shown that the leaves of F c f are dense in T 3 (see also [BDV]), and also in a robust fashion.
It is particularly relevant for us that, not onely is the central foliation minimal, but also the unstable foliation is minimal as well. This is shown for instance in the following theorem from [PS] (page 5).
Theorem 4.1. (2.0.1 in [PS]). There exists a neighborhood U of f , in the C 1 topology such that for every g ∈ U the bundles E c g , E s g and E u g , uniquely integrates to invariant foliations (F c g , F s g and F u g , respectively). Furthermore, the central and unstable foliations g ∈ U are minimal, i.e., all leaves are dense.
The following lemma is a consequence of the shadowing theorem (see [S]).
Lemma 4.2. Let A ∈ GL(3, Z) which is a hyperbolic toral automorphism and let G : R 3 → R 3 be a homeomorphism such that A(x) − G(x) ≤ r for all x ∈ R 3 . Then there exists H : R 3 → R 3 continuous and onto such that Note that H(x) = H(y) if and only if G n (x) − G n (y) ≤ 2Cr ∀n ∈ Z. This is a consequence of the uniqueness in the shadowing theorem.
Since G is isotopic to A, H induces an h : T 3 → T 3 continuous and onto such that As a consecuence of this we have: Lemma 4.3. With the above notation, H : R 3 → R 3 is uniformly continuous for every x.

Now let us see how H behaves with respect of the invariant foliations.
Lemma 4.4. For H, A and G as above we have that , Then x and y belong to the same central leaf.
These results follow mainly from the expansivity of A and the fact that H(x) − x < Cr. For a proof see [PS]. It can also be shown that h : T 3 → T 3 inherits similar properties.

Proof of Theorem B: a set that is robustly not premaximal in T 3
Let f : T 3 → T 3 be as in the previous section, the diffeomorphism form Mañe's example, and let us consider a C 1 ball around f , U. In this section we will prove that for any g ∈ U, there is a set on T 3 that cannot be included on any set with local product structure.
For this we will show that the set Λ from section 3 does not intersect some ball around x 1 So possibly taking a smaller ρ we can construct a diffeomorphism f as the one from the previous section and such that Λ ⊂ Γ = n∈Z f n (B ρ 2 (x 1 ) c ) . Note that the set Γ = n∈Z f n (B ρ 2 (x 1 ) c ) can be made to be transitive. So Λ is a compact, hyperbolic set, invariant under f since it is invariant under f A and by equation (1). For any g sufficiently close to f , there is a hyperbolic set Λ g which is the hyperbolic continuation of Λ, and that has essentially the same properties in all that concerns us. We will call both sets Λ, for simplicity. We aim to prove that if there is a set ∆ containing Λ with local product structure, then ∆ ∩ F u g (x 0 ) is dense in some small interval of F u g (x 0 ), and then ∆ is dense in T 3 in virtue of the minimality of F u g (4.1). This is a contradiction since g is not Anosov. Since in this context the unstable leaves are not parallel it would be convenient to redefine the n-ε-relation.
Let p cs x : R 3 → F u G (x) and p u x : R 3 → F cs G (x) be the projections along the center stable and unstable foliation respectively. We note as ∆ the lift of ∆.
Definition 5.1. We say that x and y are n-ε-related in ∆ if there exists sequence of point x = x 0 , x 1 , . . . , x n = y such that: The main problem which we are dealing with now, is that the lemma (3.6) relies heavily on the linearity of A. We will fix this problem by finding a tube V around (0, 0, 0) so that both the distance between the center-sable foliations of x and y and the distance between the unstable foliations in V are small when x and y are close enough. The interval of the unstable foliation in which ∆ ∩ F u G ((0, 0, 0)) will dense, will be contained in this V .
Another important difference is that 3.4 also makes a strong use of the linearity therefore we will not try to prove that the projection of all Γ δ is in ∆. It will be enough to find a point of Γ δ outside V and project the points of the δ-chain joining (0, 0, 0) with that point.
For two points x and y in the same leaf of the unstable foliation, we define l u (x, y) to be the length of the arc joining x with y. For a fixed ε, we will prove first that for any tow points x, y in R 3 , there exist a δ such that if d(x, y) < δ. Then, if we choose any z in F cs (x), then l u (z, p u y (z)) < ε.
Lemma 5.2. For any ε > 0 there exists a δ such that for every x and y ∈ F u G (x) such that l u (x, y) ≤ δ then l u (z, p u y (z)) < ε, for any z in F cs G (x). Proof. Suppose that this is not the case. Then there must exist an ε 0 such that there exist there sequences { x n } n∈N , { y n } n∈N ⊂ F u G (x) and { z n } n∈N such that l u (x n , y n ) ≤ 1/n, and l u (z n , p u y (z n )) ≥ ε 0 . Now let us recall that from (4.4) we have that H | F u G (x) is a homeomorphism, for simplicity we note H | F u G (x) = H ux . Let δ 0 be the one given by de uniform continuity . Since H ux is a homeomorphism, for δ ′ we can find a δ 0 (independent of x) such that if x and y are such that y ∈ F u G (x) and d (H ux Let us consider n 0 such that 1/n 0 < δ ′ and z ′ = H(z n 0 ). Note that z ′ ∈ F cs A (H(x n 0 )) . For perhaps a bigger n, we have that l u (x n , y n ) ≤ 1/n, and d(H(x n ), H(y n )) ≤ δ 0 , from the continuity of H. But for A, F cs A are parallel planes so since the length of the unstable segment between z ′ and p ′u H(yn) (z ′ ) is less than δ 0 (see figure 3) and therefore ε 0 > δ ′ > l u (z n , H −1 ux (p ′u H(yn) (z ′ ))) = l u (z n , p u y (z n )) ≥ ε 0 .
If two points are sufficiently close their unstable manifolds remain close in some neighborhood. This is a consequence of the continuity of the foliation.
For every ε > 0, there is β > 0 and η > 0 such that if y ∈ F cs G (x) and d(x, y) < η, then for any z ∈ F u G (x) such that l u (x, z) < β, we have that d(z, p cs y (z)) < ε. We can also take β to be uniform since the foliations are lifts of foliations in a compact set (see figure 4). Now we will put everything together. Let ε = δ p be the one from the local product structure of ∆. For this ε we find η > 0 and β from our previous observation. This will ensure us that if d(x, y) < η their unstable leaves will remain closer than δ p in a ball of radius β from x.
We can choose the ε 0 from the lemma (5.2) smaller δ p and β, So the lemma ensures us that there exists a δ 0 such that if x and y ∈ F u G (x) and l u (x, y) ≤ δ 0 then the center stable foliations of x and y will not separate more than ε 0 . Figure 4. Separation between the unstable leafs For this last δ 0 we will take a compact neighborhood of (0, 0, 0) in F u ((0, 0, 0)) that we call U u with diam(U u ) < δ 0 . In this conditions we define We have proved the following for V . Figure 5. The tube V Lemma 5.3. There exist a compact neighborhood of (0, 0, 0), U u , in F u ((0, 0, 0)), such that the set V define as V = x∈U u F cs G (x) , satisfies: • If x ∈ V and y ∈ F u G (x) ∩ V , then d(x, y) < δ p and d(x, y) < β. • For every ε there is δ such that if x ∈ V and d(x, y) < δ then d(z, p u y (z)) < ε, for any z in F cs G (x). • For every ε there is δ such that if x ∈ V and d(x, y) < δ then d(z, p cs y (z)) < ε, for any z in F u G (x) ∩ V . The following implies Theorem B: Theorem 5.4. Let f : T 3 → T 3 be as in Mañe's example and let g ∈ U(f ), be a difeomorphism sufficiently close to f . There exists Λ a compact, connected,non trivial, g-invariant, hyperbolic set such that x 0 ∈ Λ and Λ is not included in any set with local product structure.
Proof. We first recall that for every g ∈ U(f ) the existence of a set Λ which is a compact, connected, non trivial, g-invariant, hyperbolic set, has already been stated.
As before we start by supposing that there is a set ∆ with local product structure containing Λ. Recall that the strategy to see that such a ∆ can not exist, is to find an interval of the unstable foliation in which ∆ ∩ F u G (x 0 ) will dense. Let V ⊂ R 3 be an open tube as defined in 5.3, Figure 6. A n-ε-relation between (0, 0, 0) and x j .
Let us take any point q of Γ δ which is not in V . (0, 0, 0) is n-δ/2-related to q. We call x j+1 the firs element of the sequence relating (0, 0, 0) to q that is not in V . As in the proof of the assertion, we can construct a new sequence from the sequence that j-ε-relates (0, 0, 0) to x j as follows.
Recall that since x i ∈ V for i = 1, . . . , j and d(x i , x i+1 ) ≤ δ, so by 5.3 we have: • For any z in F cs This proves that the new sequence we have defined is in the hypothesis of our assertion so, p cs x 0 (x i ) ∈ ∆ for i = 0, . . . , j − 1. But then p cs x 0 (x i ) ∈ ∆ for i = 0, . . . , j. Let U u+ and U u− , be the positive and negative sub intervals of U u . Arguing by contradiction we suppose that ∆ ∩ U u is not dense in any of these sub intervals. If this is so, there must be some gaps of size at least γ in each sub interval, for which there are no points of ∆ ∩ U u in these gaps. But choosing ε ≤ γ/2, δ for this ε and q a point in the corresponding Γ δ , (0, 0, 0) is n-δ-related to q.
Therefore for j as before, p cs x 0 (x i ) ∈ ∆ for i = 0, . . . , j. Since x j+1 is out of V , one of the subintervals, U u+ or U u− has at least one point of ∆ in every gap of size 2ε leading to a contradiction with our previous assumption. Iterating U u and since ∆ is invariant, we have then that

in dimension 2 transitive sets are locally premaximal
In this section we prove Theorem C, allowing us to complete the answer to the question of whether or not it is possible to construct transitive sets which are not locally premaximal in dimensions smaller than 4. For this, we rely on a result by Anosov [A1] (for a proof in an other context see also [BG] proposition 4.3) and one by Fisher in [Fi], which we state below. Remark 6.3. Since M is a Hausdorff locally compact space, (we will have a compact surface actually), then zero dimensional subsets are exactly the totally disconnected subsets.
We will prove some lemmas that will imply Theorem C. In what follows f : M → M will be a diffeomorphism, of M a compact surface and Λ ⊂ M will be a compact hyperbolic invariant set such that Ω(f | Λ ) = Λ Definition 6.4. Let Λ ⊂ M be a compact hyperbolic invariant set. We note Λ 0 is the union of all points p in Λ such that the connected component of p in Λ is p itself. We define Λ 1 as Λ 1 = Λ \ Λ 0 .
Note that Λ 0 is totally disconnected and therefore 0 dimensional.
As before, for a sufficiently small neighborhood of x ∈ Λ 1 we define p s x : U(x) ∩ Λ 1 → W u loc (x) to be the local projection along the stable manifolds. We define analogously p u x : U(x) ∩ Λ 1 → W s loc (x). Recall that both stable and unstable manifolds are one dimensional.
Lemma 6.5. Periodic points are dense in Λ 1 . Moreover for any x ∈ Λ 1 we have that W u loc (x) ⊂ Λ 1 or W s loc (x) ⊂ Λ 1 (or both). Proof. We note the local connected component of x in Λ 1 as lcc(x) and from the definition of Λ 1 we have that lcc(x) is not trivial. Then for any x ∈ Λ 1 we have that either p s x (lcc(x)) contains a nontrivial connected set (an arc) or p u x (lcc(x)) contains a nontrivial connected set. This will also be true for any smaller U(x).
Since Ω(f | Λ ) = Λ , using the shadowing theorem we have that if x ∈ Λ then it is approximated by periodic points (which a priori would not be in Λ ).
Let us suppose that p s x (lcc(x)) contains an arc. Then we define and we have thatV is open and not empty. We take z ∈V ∩ lcc(x) and p s (z) is in the interior of p s x (lcc(x)) in the relative topology of W u (x). If { p n } n∈N is such that p n → z (where p n are periodic points), then p n ∈V for all n greater than some n 0 . Since Λ is invariant, and f a diffeomorphism it follows that Λ 1 is invariant too ( otherwise it would take a nontrivial connected component into a point). This implies that ω(p n ) ∈ Λ 1 but then p n ∈ Λ 1 for all n > n 0 . Now we take p n sufficiently close to z such that p s x (p n ) is in the interior of p s x (lcc(x)) in the relative topology of W u (x). Then W s loc (p n ) ∩ lcc(x) = ∅. Iterating for the future f n (lcc(x)) accumulates on the unstable manifold containing p n in its interior and since diam(lcc(x)) > 0, f n (lcc(x)) accumulates on an arc of W u loc (p n ). Therefore this arc of W u loc (p n ) is contained in Λ, since Λ is compact. Such an arc must be contained in Λ 1 because it clearly does not belong to Λ 0 . The invariance of Λ 1 implies that W u (p n ) ⊂ Λ 1 and as p n → z implies that W u (z) ⊂ Λ 1 . We can take now a sequence of z n as before that are contained in neighborhoods U n (x) which are each time smaller, and z n → x so W u (x) ⊂ Λ 1 .
The situation is analogous if p u x (lcc(x)) is a stable arc.
1 and Λ s 1 are compact sets with local product structure. Proof. We will prove first that Λ u 1 is closed and therefore compact. Let { x n } n∈N ⊂ Λ u 1 and x n → y then the unstable manifold of y is in Λ since Λ is compact. Hence y ∈ Λ u 1 . Let x, y ∈ Λ u 1 such that d(x, y) < δ for some δ appropriate such that W u loc (x) ∩ W s loc (y) = ∅. Since W u (x) ⊂ Λ 1 , and W u (y) ⊂ Λ 1 , W u loc (x) ∩ W s loc (y) ∈ W u (x) ⊂ Λ 1 . The situation is analogous for Λ s 1 .
Corollary 6.7. The set Λ 1 is compact, invariant and has local product structure.
Proof. The sets Λ u 1 and Λ s 1 defined in lemma 6.6 are such that Λ 1 = Λ u 1 ∪ Λ s 1 , as a consequence of lemma 6.5. Therefore Λ 1 is compact and has local product structure.
Corollary 6.8. Either Λ 1 is the disjoint union of an attractor Λ s 1 and a repeller Λ u 1 , or Λ = M = T 2 and f is Anosov.
Proof. Suppose that there is x ∈ Λ u 1 ∩ Λ s 1 . Then W u (x) and W s (x) ∈ Λ 1 . Since Λ 1 has local product structure this implies that y∈W s loc (x) W u loc (y) ⊂ Λ 1 , and so Λ 1 has non empty interior. It follows from Theorem 6.2 that Λ = Λ 1 = M = T 2 and f is Anosov.
Lemma 6.9. The set Λ 0 is compact and disjoint from Λ 1 . Moreover for any neighborhood V of Λ 0 , we can find a set Λ ′ 0 with local product structure such that Λ ′ 0 ∩ Λ 1 = ∅ and Λ 0 ⊂ Λ ′ 0 ⊂ V . Proof. From its definition, Λ 0 is disjoint from Λ 1 . Suppose that Λ 0 is not empty (the lemma holds trivially if it is empty). Suppose that there is a sequence { x n } n∈N ⊂ Λ 0 such that x n → y ∈ Λ 1 . By corollary 6.8 Λ 1 is the disjoint union of attractors and a repellers so suppose that y ∈ Λ s 1 . For a sufficiently big n, x n must be in the basin of attraction of Λ s 1 and therefore f m (x n ) does not return to a neighborhood of x n , which is impossible since Ω(f | Λ ) = Λ. Now we are in the right conditions to prove: Theorem 6.10. Let f : M → M be a diffeomorphism, M a compact surface and Λ ⊂ M a compact hyperbolic invariant set. If we also have that Ω(f | Λ ) = Λ then for any neighborhood V of Λ, there exist Λ a compact hyperbolic invariant set with local product structure such that, Proof. Let V be any open set containing Λ and V ′ = V \ Λ 1 . Note that V ′ is open and contains Λ 0 .
We conclude that Λ = Λ ′ 0 ∪ Λ 1 has local product structure, is contained in V and contains Λ.

Acknowledgment
This work is part of master thesis at the Universidad de la Repbulica Uruguay under the guidance of Martín Sambarino, I would not have been able to write this paper without his help support and infinite patience. I can not find a way to describe just how much he contributed to this work or how much he contributed to my education in general, so probably I should just say thanks for all. He also helped me greatly in the writing of the paper, dedicating a lot of work to correcting and re-correcting it. I would like to thank Rafael Potrie for repeatedly listening to me, and helping me understand. Also for suggesting ideas and encouraging me to work harder. To Javier Correa who helped with the writing of the paper. And finally to Todd Fisher and Keith Burns who kindly listened to the results and showed interest in it, also they suggested me ideas that I am still thinking about in order to possibly understand these examples further.