Topics on homoclinic bifurcation, dominated splitting, robust transitivity and related results

Enrique R. Pujals & Martin Sambarino

December 15, 2004

Contents

1 Introduction. 2

2 A weaker form of hyperbolicity: Dominated Splitting 4 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Definition and general remarks . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Sufficient conditions for Dominated Splitting. . . . . . . . . . . . . . . . 6

3 Homoclinic tangencies 8 3.1 The dynamics near a homoclinic tangency . . . . . . . . . . . . . . . . . 8 3.2 Dominated splitting versus tangencies . . . . . . . . . . . . . . . . . . . 9

4 Surfaces diffeomorphisms 10

4.1 Dynamical consequences of the dominated splitting. Sufficient conditions for hyperbolicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.1.1 Getting rid off attracting and reppeling closed curves and periodic

attractors-repellers . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.1.2 Non existence of wandering intervals . . . . . . . . . . . . . . . . 12 4.1.3 Outline of proof of Theorem 4.1.1 in a particular case . . . . . . 16

4.2 Generic results in the C^{1} topology and consequences . . . . . . . . . . . 18 4.3 Spectral decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3.1 Sketch of proof of Theorem 4.3.3 . . . . . . . . . . . . . . . . . . 22 4.4 Dynamical determinant in the dominated splitting setting . . . . . . . . 23

5 Nonhyperbolic robust transitive systems. 26 5.1 Partial hyperbolicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2 Examples of nonhyperbolic robust transitive systems. . . . . . . . . . . . 28

5.2.1 Shub’s example. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2.2 Mañé’s example. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2.3 Bonatti-Diaz’s construction. . . . . . . . . . . . . . . . . . . . . . 31 5.2.4 Robust transitive diffeomorphisms without any hyperbolic direc-

tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.5 Sufficient conditions for robust transitivity. . . . . . . . . . . . . 33 5.3 Robust transitivity and Dominated splitting. . . . . . . . . . . . . . . . 35

5.3.1 Sketch of the proof of theorem 5.3.3. . . . . . . . . . . . . . . . . 36 5.3.2 Sketch of the proof of theorem 5.3.2. Dominated splitting versus

homotheties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.4 Similar results for conservative systems. . . . . . . . . . . . . . . . . . . 39 5.5 Robust transitivity and heterodimensional cycle. . . . . . . . . . . . . . 41

6 Flows and Singular splitting. 42 6.1 Sketch of the proof of theorem 6.0.2. . . . . . . . . . . . . . . . . . . . . 44 6.2 Dynamical consequences of singular hyperbolicity. . . . . . . . . . . . . 46

1 Introduction.

For a long time (mainly after Poincaré) it has been a goal in the theory of dynamical systems to describe the dynamics from the generic viewpoint, that is, describing the dynamics of “big sets” (residual, dense, etc.) of the space of all dynamical systems.

It was briefly thought in the sixties that this could be realized by the so-called hyperbolic ones (See Principal structures (Hasselblatt-Katok) Handbook volume 1A): systems with the assumption that the tangent bundle over the limit set L(f) (see definition in subsection 4.3) splits into two complementary subbundles T_{L(f)}M = E^{s} ⊕ E^{u} so that vectors in E^{s} (respectively E^{u}) are uniformly forward (respectively backward) contracted by the tangent map Df. Under this assumption, it is proved that the limit set decomposes into a finite number of transitive sets such that the asymptotic behavior of any orbit is described by the dynamics in the trajectories in those finite transitive sets. Moreover, this topological description allows to get the statistical behavior of the system. In other words, hyperbolic dynamics on the tangent bundle characterizes the dynamics over the manifold from a geometrical topological and statistical point of view.

Nevertheless, uniform hyperbolicity was soon realized to be a property less universal than it was initially thought: it was shown that there are open sets in the space of dynamics which are nonhyperbolic. The initial mechanisms to show this nonhyperbolic robustness (see [AS], [Sh]) were the existence of open sets of diffeomorphisms exhibiting hyperbolic periodic points of different indices (i.e. different dimension of their stable manifolds) inside a transitive set. Indeed, Shub’s construction leads to an open set of transitive diffeomorphisms on T^{4} exhibiting hyperbolic periodic points of different indices. Related to this is the notion of heterodimensional cycle where two periodic point of different indices are linked through the intersection of their stable and unstable manifolds (notice that at least one of the intersection is non-transversal).

In all the above examples the underlying manifolds must be of dimension at least three and so the case of surfaces were still unknown at the time. It was through the seminal works of Newhouse (see [N1], [N2], [N3]) that hyperbolicity were shown to be not dense in the space of C^{r} diffeomorphisms of compact surfaces (however, let us point out that in the C^{1} topology is still open!). The underlying mechanism here was the presence of a homoclinic tangency leading to the so-called nowadays “Newhouse phenomena”, i.e., residual subsets of diffeomorphisms displaying infinitely many periodic attractors.

These results naturally pushed some aspects of the theory on dynamical systems in different directions:

1. The study of the dynamical phenomena obtained from homoclinic bifurcations (i.e. the unfolding of homoclinic tangencies or heterodimensional cycles);

2. The characterization of universal mechanisms that could lead to robustly (meaning any perturbation of the initial system) nonhyperbolic behavior;

3. The study and characterization of isolated transitive sets that remain transitive for all nearby system (robust transitivity);

4. The dynamical consequences of weaker forms of hyperbolicity.

As we will show, these problems are all related to each other. In many cases, such relations provide a conceptual framework, as the hyperbolic theory did for the case of transverse homoclinic orbits.

Besides, we mentioned that two basic mechanisms were found to the obstruction of hyperbolicity, namely heterodimensional cycles and homolicinic tangencies. In the early 80’s Palis conjectured that these are very common in the complement of the hyperbolic systems:

Palis’ Conjecture:

1. Every C^{r} diffemorphism of a compact manifold M can be C^{r} approximated by one which is hyperbolic or by one exhibiting a heterodimensional cycle or by one exhibiting a homoclinic tangency.

2. When M is a two-dimensional compact manifold every C^{r} diffemorphism of M can be C^{r} approximated by one which is hyperbolic or by one exhibiting a homoclinic.

This conjecture may be thought as a start point to obtaining a generic description of C^{r}-diffeomorphisms. If it turns out to be true we may focus on the two mechanisms mentioned above in order to understand the dynamics. Nevertheless, the unfolding of these homoclinic bifurcations is still mainly a local study.

The above conjecture was proved to be true [PS1] (see section 4.2) for the case of surfaces and the C^{1} topology. When the manifold has dimension greater than two is still open despite some progress is underway, but still in the C^{1}-topology. The case of higher topologies (C^{r}, r ≥ 2) is, at this point, far from being solved.

We would like to emphasize that this notes are an attempt to give some insight on the problems mentioned above but it is not a complete survey on the subject. We apologize both to the reader and those who are not mentioned for this.

Many of the definitions used here, can be obtained in the chapter named Principal structures (B. Hasselblatt, A. Katok) of the Volume 1A.

2 A weaker form of hyperbolicity: Dominated Splitting

2.1 Introduction

In the theory of differentiable dynamics, i.e. the study of the asymptotic behavior of orbits {f^{n}(x)} when f : M → M is a diffeomorphism of a compact riemaniann manifold M, one may say that a fundamental problem is to understand how the dynamics of the tangent map Df controls or determines the underlying dynamics of f.

So far, this program has been solved for hyperbolic dynamics in the so called Spectral Decomposition Theorem (see Theorem 4.3.1 in subsection 4.3).

There were, basically, two ways to relax hyperbolicity. One, called partial hyperbolicity, allows the tangent bundle to split into Df-invariant subbundles T M = E^{s} ⊕ E^{c} ⊕ E^{u}, and the behavior of vectors in E^{s}, E^{u} is similar to the hyperbolic case, but vector in E^{c} may be neutral for the action of the tangent map. In the other, nonuniform hyperbolicity (or Pesin theory), where the tangent bundle splits for points a.e. with respect to some invariant measure, and vectors are asymptotically contracted or expanded in a rate that may depend on the base point.

Since the latter case starts on a measure-theoretical setting, one cannot expect to obtain a description from the topological dynamic point of view (see Barreira-Pesin in the present volume for an account on non-uniform hyperbolicity). In the former, there is no general theory regarding its topological dynamic consequences (although there are many important results, mostly from the ergodic point of view, see for instance the chapter Partially hyperbolic dynamical systems (Hasselblatt, Pesin) in the Handbook, [BP], [PuSh], [ABV], [CY]).

There is also another category which includes the partial hyperbolic systems: dominated splitting. Although partial hyperbolic systems arose in a natural way (time one maps of Anosov flows, frame flows, group extensions), the concept of dominated splitting was introduced independently by Mañé, Liao and Pliss, as a first step in the attempt to prove that structurally stable systems satisfy a hyperbolic condition on the tangent map. In fact, it is proved that under the assumption of C^{1}−structural stability, the closure of the periodic points exhibits dominated splitting. Nevertheless, in the last decades there has been large amount of research of this concept showing that it is not just a technical concept but appears naturally when one try to understand robust phenomena.

It is a belief that a robust dynamic phenomena must be reflected on the tangent map. This turns out to be true for instance in the case of structural stability, robust transitivity and lack of tangencies. This last two subject will be discussed in the next sections.

After that, a natural question arises: what is the feedback of a system having dominated splitting? In other words, is it possible to describe the dynamics of a system having dominated splitting?

In few words, we could say a general strategy to deal with many of the problems mentioned before is the following: first, it is shown that for some kind of robust phenomena (stability, transitivity, lack of tangencies) the limit set exhibits dominated splitting; latter, it is tried to get feedback from this kind of dynamics in the tangent bundle. However, this main issue is far from being understood except for C^{2} surface maps as we will see in 4.1 and 4.3.

2.2 Definition and general remarks

Definition 2.2.1. An f-invariant set Λ is said to have dominated splitting of index i (or just dominated splitting) if there is a decomposition over tangent bundle in two invariant subbundles T_{Λ}M = E ⊕ F, such that:

1. dimE(x) = i.

2. There exist C > 0 and 0 < λ < 1 such that

∥Df_{/E(x)}^{n}∥∥Df_{/F (f^{n}(x))}^{−n}∥ ≤ Cλ^{n}, for all x ∈ Λ, n ≥ 0.

The constant λ is called a constant of domination.

Note: It is assumed that none of the subbbundless are trivial (otherwise, the other one has a uniform hyperbolic behavior: contracting or expanding).

Let us explain briefly the meaning of the above definition. It says that, for n large, the “greatest expansion of Df^{n} on E is less than the “greatest contraction of Df^{n} on F and by a factor that becomes exponentially small with n. In other words, every direction not belonging to E must converge exponentially fast under iteration of Df to the direction F.

As in the hyperbolic case, the subbundles E and F depends continuously on the base point. The following lemma is elementary but nevertheless important.

Lemma 2.2.1. Let Λ be an f-invariant set exhibiting dominated splitting of index i. Then \overline{Λ} (the closure of Λ) exhibits a dominated splitting of index i (with the same constant of domination).

Notice that on an invariant set may coexist different dominated splittings. As a trivial example consider a periodic point p of period n having a simple Lyapunov spectrum, i.e., all the eigenvalues of Df_{/T_{p}M}^{n} have (algebraic) multiplicity one. In this case the periodic point admits dominated splitting of index i for any i = 1, ..., dimM −1.

It is straightforward to check that a hyperbolic set has a dominated splitting. On the other hand it is trivial to find examples of sets having dominated splitting without being hyperbolic. A simple one is nonhyperbolic periodic point where the spectrum is not contained in S^{1}. Another example is an attracting (or reppeling) invariant closed curve supporting an irrational rotation. More sophisticated examples are partially hyperbolic systems. See the section 5 for this kind of examples.

Another simple yet useful property is that dominated splitting can not be destroyed by small perturbations:

Lemma 2.2.2. Let f : M → M be a diffeomorphism and let Λ be a compact set having dominated splitting T_{Λ}M = E ⊕ F. Then, there exist an open set U containing Λ and a neighborhood U(f) such that for any g ∈ U(f) there is a dominated splitting over any compact g-invariant set Λ_{g} ⊂ U. Moreover the constant of domination can be chosen uniformly. The set U is called an admissible neighborhood of Λ.

2.3 Sufficient conditions for Dominated Splitting.

In this section we shall explain a technique mainly developed by Mañé to show the existence of dominated splitting.

First we recall a fundamental result in C^{1}−dynamics: the so called generic C^{1}−closing lemma proved by C. Pugh (see [Pu1] and [Pu2]).

Theorem 2.3.1. C^{1}-generically (i.e. residual) holds that Ω(f) = \overline{P er(f)}.

This result open the possibility of carrying out information on the periodic points to the whole non-wandering set. Recall that also generically follows that the periodic points are hyperbolic so for each periodic point p there is a natural hyperbolic decomposition of the tangent bundle over the periodic orbit in two complementary direction E^{s} ⊕ E^{u}. So a natural question arises: can the tangent bundle decomposition over the periodic points be extended to the non-wandering set? Here dominated splitting plays the key role. The main idea is that if the angles of the eigenspaces of hyperbolic periodic points are bounded away from zero in a robust way then it guarantees the existence of dominated splitting. Let us explain in a more precise way and we refer to [M] for details. We first introduce a simple yet powerful perturbation technique (in the C^{1} topology) due to Franks.

Lemma 2.3.1. [Fr, Lemma 1.1] Let M be a closed n-manifold and f : M → M be a C^{1} diffeomorphism, and let a neighborhood of f, U(f) be given. Then, there exist U_{0}(f) ⊂ U(f) and δ > 0 such that if g ∈ U_{0}(f), S ⊂ M is a finite set, S = {p_{1}, p_{2}, … p_{m}} and L_{i}, i = 1, …, m are linear maps L_{i} : T M_{p_{i}} → T M_{f(p_{i})} satisfying ∥L_{i} − D_{p_{i}}g∥ ⩽ δ, i = 1, …, m then there exists g̃ ∈ U(f) satisfying g̃(p_{i}) = g(p_{i}) and D_{p_{i}} g̃ = L_{i}, i = 1, ..., m Moreover, if U is any neighborhood of S then we may chose g̃ so that g̃(x) = g(x) for all x ∈ {p_{1}, p_{2} … p_{m}} ∪ (M\U).

This results says, for instance, that any small perturbation of the linear maps along a periodic orbit can be realized through a diffeomorphism nearby. Now let us introduce the definition of angle between subspaces.

Definition 2.3.1. Let E and F be two subspaces of ℝ^{d} such that E ⊕ F = ℝ^{d}. Hence dim(F) = dim(E^{⊥}) and F is the graph of the linear map L : E^{⊥} → E defined as follows: given w ∈ F there exist a unique pair of vectors v ∈ E with F. This gives a unique pair of vectors w ∈ F, u ∈ E^{⊥} and u ∈ E, such that v +u = w. Define L(v) = u obtaining that graph(L) = F. We define the angle ∠(E, F) between E and F as ∥L∥^{−1}. In particular ∠(E, E^{⊥}) = +∞.

Definition 2.3.2. Let f : M → M a diffeomorphism. The index of a hyperbolic periodic point p of f is the dimension of the stable manifold of p. We shall denote by P er_{i}(f) the set of hyperbolic periodic points of index i.

The next Theorem is essentially due to Mañe ([M2]). The techniques are rather involved so we shall give an outline of the main ideas to help the reader to go through the details.

Theorem 2.3.2. Assume that there is a neighborhood U(f) such that

∠(E^{s}(p, g), E^{u}(p, g)) > γ > 0

for every g ∈ U(f) and p ∈ P er_{i}(g). Then there is dominated splitting of index i over P er_{i}(f).

Outline of proof: The best way to understand the argument is by contraposition: absence of dominated splitting forces the presence of arbitrarily small angles.

The first step is to prove the domination at the end of the period of each periodic point, meaning the following: there exits 0 < λ < 1 such that if p is a hyperbolic periodic point of index i of f then:

∥Df_{/E^{s}(p)}^{n_{p}}∥∥Df_{/E^{u}(p)}^{−n_{p}}∥ ≤ λ^{n_{p}}

where n_{p} is the period of p. If not, it means that we do not have uniform contraction at the end of the period neither on Df_{/E^{s}(p)}^{n_{p}} nor on Df_{/E^{u}(p)}^{−n_{p}}. Thus, we can perturb, to obtain a hyperbolic periodic point p_{g} of index i of a diffeomorphism g close to f having one eigenvalue less but close to one and another eigenvalue greater but also close to one. From this it is possible to breakdown the angle between the subspaces E^{s} and E^{u}.

For the second and last step, let us assume that we do not have a dominated splitting over P er_{i}(f). Thus, it can be proved that there is a periodic point p of f whose period is arbitrarily large (say k), and for some m large we have

∥Df^{j}|E^{s}(p)∥ · ∥Df^{−j}|E^{u}(f^{j}(p)(x))∥ ⩾ 1/2 1 ≤ j ≤ m.

In some sense, this means that the action of the action of Df along the piece of orbit p, f(p), ..., f^{m}(p) is neutral, although at the end of the period we have domination, that we may assume is due to contraction at the end of the period on E^{s}.

Next define T_{i} : T_{f^{i}(p)}M → T_{f^{i}(p)}M such that T_{i}|E^{s}(f^{i}(p)) = (1 + ϵ)Id and T_{i}|E^{u}(f^{i}(p) = Id and L_{i} = T_{i} ∘ Df_{f^{i−1}(p)}. Here, the number ϵ is small enough in order to still have contraction on E^{s}(p) at the end of the period, i.e, under L_{n−1} ∘ … ∘ L_{0}. Let see the effect of this perturbation: take a space S close to E^{u}(p). Then, due to the expansion added in E^{s} along p, f(p), ..., f^{m}(p), we have that L_{m} ∘ … ∘ L_{0}(S) will “fall down to the direction of E^{s}, meaning that the angle between L_{m} ∘ … ∘ L_{0}(S) and E^{s}(f^{m}(p)) is small. On the other hand, due to the domination, at the end of the period we will have that L_{n} ∘ … ∘ L_{0}(S) is “up again, meaning that L_{n} ∘ … ∘ L_{0}(S) and E^{u}(p) are close again. Thus, adding another perturbation on L_{n} so that L_{n} ∘ … ∘ L_{0}(S) = S and using Franks lemma we find g near f so that p is hyperbolic periodic point of g and E^{s}(p, g) = E^{s}(p, f) and E^{u}(p, g) = S, and we have “destroyed the angle at g^{m}(p) = f^{m}(p) (i.e. less than γ.)

3 Homoclinic tangencies

3.1 The dynamics near a homoclinic tangency

In this section we will mention some of the dynamical phenomena related to the presences of homoclinic tangencies.

For that, first we recall that the stable and unstable sets

W^{s}(p) = {y ∈ M : dist(f^{n}(y), f^{n}(p)) → 0 as n → ∞},
W^{u}(p) = {y ∈ M : dist(f^{n}(y), f^{n}(p)) → 0 as n → −∞}

are C^{r}-injectively immersed submanifolds when p is a hyperbolic periodic point of f.

Definition 3.1.1. Let f : M → M be a diffeomorphism. We say that f exhibits a homoclinic tangency if there is a hyperbolic periodic point p of f such that the stable and unstable manifolds of p have a non-transverse intersection.

It is important to say that an homoclinic tangency is easily destroyed by small perturbation of the invariant manifolds. To get open sets of diffeomorphisms where each system exhibits an homoclinic tangency, Newhouse study system where the homoclinic tangency is associated to an invariant hyperbolic set with the property that it has large fractal dimension. In fact, he studied the intersection of the local stable and unstable manifold of a hyperbolic set (for instances, a classical horseshoes), where these kind of hyperbolic sets, roughly speaking, can be visualized as a product of two cantor set with the property that the fractal dimension of these cantor sets(more specifically, the thickness) are large. Newhouse’s construction depends on have the fractal dimension changes with perturbation of the dynamics and actually this is the maim reason that his construction works in the C^{2}−topology. In fact, Newhouses construction is based on the continuous dependence of the fractal dimension (thickness) along with C^{2} perturbations. A similar construction in the C^{1}−topology leading to same phenomena is unknown (results in the opposite direction can be found in [U]).

After the seminal works of Newhouse, any other results where obtained in the direction to understand the dynamics induced by unfolding homoclinic tangencies, specially in the case of one-parameter families . In the study of bifurcations of a generic oneparameter family of surface diffeomorphisms having generic homoclinic tangency at a parameter value, the arithmetic difference of two Cantor sets appears in a natural way (see [PT]) The Cantor sets that appear in this context are regular on the line, that is, they are defined by expansive maps and have a sort of self-similarity property, which means, roughly speaking, that a small part of them is diffeomorphic to a large part with uniformly bounded distortion. In this context, in [PT2] was obtained some results on homoclinic bifurcations associated to a basic set which ensure full density of hyperbolicity on the parameter line, provided that the Hausdorff dimension of the basic set is less than one. Palis, in [P2], conjectured that for generic pairs of regular Cantor sets of the real line, either their arithmetic difference has Lebesgue measure equal to zero or otherwise it contains an interval. The latter statement should correspond in the context of homoclinic bifurcations to open sets of tangencies. Regarding the second part of Palis’ conjecture, a partial result was obtained in [PY] and later in [MY] was proved that if the sum of the Hausdorff dimensions of two Cantor sets is greater than one in almost all cases, there exist translations of these Cantor sets whose intersection has Hausdorff dimension greater than one.

Other fundamental dynamic prototypes were found in the context of this bifurcation, namely the so called cascade of bifurcations, the Hénon-like strange attractor ([BC], [MV]) (infinitely many coexisting ones [C]), and super exponential growth of periodic points ([K]).

Nevertheless the rich dynamics that appear after the unfolding of an homoclinic tangency, Palis conjectured (see [P3]) that when a generic one parameter family of surfaces maps unfolding a homoclinic tangency, the set of parameter values corresponding to diffeomorphisms with infinitely many sinks or infinitely many Hnon-like attractors has (Lebesgue) measure zero.

In this direction in [PY2] is announced that given a surface diffeomorphisms f_{0} such that the maximal invariant set in an open set V is the union of a horseshoe and a quadratic tangency between the stable and unstable foliations of this horseshoe such that the dimension of the horseshoe is larger than but close to one, then for most diffeomorphisms f close to f_{0}, the maximal f−invariant set in V is a non-uniformly hyperbolic horseshoe, with dynamics of the same type as met in Henon attractors. In particular, most diffeomorphisms (in a measure theoretical point of view for parameters in one-parameter families) does not exhibit attracting periodic points.

In higher dimension, many of the previous result were generalized (see [PV], [R], [MPV]).

As it was mentioned of the beginning when we referred to the Newhouse’s techniques, all the previous results hold when are considered at least C^{2}−diffeomorphisms. However, in [BD2] was obtained the Newhouse’s phenomenon (coexistence of infinitely many sinks or sources) for C^{1}−diffeomorphisms of three manifolds

3.2 Dominated splitting versus tangencies

In this section we shall deal with diffeomorphism that can not be approximated by one’s exhibiting a homoclinic tangency.

Definition 3.2.1. Let f : M → M be a C^{r}-diffeomorphism. We say that f is C^{r} far from tangencies if there exists a neighborhood U(f) in the C^{r}- topology such that no g ∈ U(f) exhibits a homoclinic tangency (see definition 3.1.1)

The next theorem says that that the lack (in a robust way) of the presence of homoclinic tangency guarantees the existence of dominated splitting. It was originally proved in [PS1] for surface diffeomorphisms and extended to higher dimensions by L.Wen in [W].

Theorem 3.2.1. Let f : M → M be a diffeomorphism which is C^{1} far from tangencies. Then P er_{i}(f) (see definition 2.3.2) has dominated splitting of index i, where i = 1, ..., dimM − 1.

Idea of proof: We shall give some insight of the proof of the above theorem in the case of surface diffeomorphisms and at the end we will comment on higher dimensions. Based on Theorem 2.3.2, it is enough to show that the angles of the eigenspaces of hyperbolic periodic points of index i (index 1 for surfaces) are uniformly bounded away from zero in a C^{1} neighborhood of f. Thus, arguing by contradiction, we must show that if these angles can be arbitrarily small then we can find a diffeomorphism nearby exhibiting a homoclinic tangency. So let p be a hyperbolic periodic point such that the stable and unstable spaces have an arbitrarily small angle. For the sake of simplicity, assume that the stable and unstable manifolds of p coincides in a neighborhood of p with the stable and unstable spaces in a local chart. The idea is to push a fundamental domain in the local stable manifold to have a tangency with the unstable manifold. Nevertheless some estimates must be done and we refer to [PS1] to see the details. The idea in higher dimension is the same. However there is a main difference that makes the proof much more technically involved: in a hyperbolic periodic point of index 1 of a surface diffeorphism the size (or shape) of a fundamental domain on the local stable manifold can be estimated from the Df_{/E^{s}(p)} while in higher dimension, the shape or “eccentricity” could be out of control. To over pass this difficulty it is required that no homoclinic tangency can be created associated to any hyperbolic periodic point and not just those of index i. In other words, a bad shape or eccentricity on a fundamental domain of a hyperbolic periodic point of index i is related to a bad angle (after some perturbation) between the stable and unstable subspaces of an another periodic point of different index.

4 Surfaces diffeomorphisms

4.1 Dynamical consequences of the dominated splitting. Sufficient conditions for hyperbolicity.

The presence of homoclinic tangencies have many analogies with the presence of critical points for one-dimensional endomorphisms. On one hand, homoclinic tangecies correspond in the one dimensional setting to preperiodic critical points and it is known that its bifurcation leads to complex dynamics. On the other hand, Mañé (see [M1])showed that for regular and generic one-dimensional endomorphisms, the absence of critical points is enough to guarantee hyperbolicity. This result raises the question about the dynamical properties of surface maps exhibiting no homoclinic tangencies. As dominated splitting prevents the presence of tangencies, we could say that domination plays for surface diffeomorphisms the role that the non critical behavior does for one dimensional endomorphisms.

One may ask whether a set having dominated splitting is hyperbolic. Two necessary conditions follows trivially: all the periodic points in the set must be hyperbolic and no attracting (repelling) closed curve supporting an irrational rotation is in the set.

The next result says that these two conditions are also sufficient as long as the diffeomorphism is smooth enough. It is the analogous of a one dimensional theorem by Mañe (see [M1]).

Theorem 4.1.1. ([PS1]) Let f ∈ Diff^{2}(M^{2}) and assume that Λ ⊂ Ω(f) is a compact invariant set exhibiting a dominated splitting such that any periodic point in Λ is hyperbolic. Then, Λ = Λ_{1} ∪ Λ_{2} where Λ_{1} is a hyperbolic set and Λ_{2} consists of a finite union of periodic simple closed curves C_{1}, ...C_{n}, normally hyperbolic, and such that f^{m_{i}} : C_{i} → C_{i} is conjugated to an irrational rotation (m_{i} denotes the period of C_{i}).

The proof of the above theorem exceeds the purpose of these notes. Nevertheless, let us mention the main steps:

I: It is sufficient to prove that Λ is hyperbolic under the above conditions and assuming that Λ has neither normally hyperbolic invariant curve supporting an irrational rotation nor attracting or repelling periodic points.

II: Due to the dominated splitting, the fact that f is C^{2}, and that all the periodic points are of saddle type, we have that through each point x ∈ Λ pass two locally invariant manifolds W_{ϵ}^{cs}(x), W_{ϵ}^{cu}(x) which are of class C^{2}. One of the main problems is that, although these manifolds are locally invariant, they do not have (a priori) any dynamic meaning at all.

III: The fact that these central manifolds are one dimensional and of class C^{2} we are able to prove that they have a dynamic meaning, indeed W_{ϵ}^{cs}(x) is a stable manifold and W_{ϵ}^{cu}(x) is an unstable manifold.

IV: For every x ∈ Λ we have

∑_{n≥0} |f^{n}(W_{ϵ}^{cs}(x))| < ∞ and ∑_{n≥0} |f^{−n}(W_{ϵ}^{cu}(x))| < ∞

where | · | means length. This is enough to prove the hyperbolicity on Λ.

In the next subsection we shall give an explanation of the first one, a main technique to obtain the third one, and finally we will give a rather complete proof of the Theorem 4.1.1 in a particular case (see Theorem 4.1.4).

4.1.1 Getting rid off attracting and reppeling closed curves and periodic attractors-repellers

First it is proved that the number of closed curves that could appear in a set having dominated decomposition is finite. The proof is reduced to show that the basin of attraction of each attracting curve has a uniform size. More precisely one proves that the diameter of the curves do not go to zero (otherwise the splitting E ⊕ F must have a “singularity”). Afterwards, since f^{m_{n}} : C_{n} → C_{n} is conjugated to an irrational rotation, they support only one invariant measure with zero lyapunov exponent along the F direction and this implies (using the domination) that the fiber E is contractive, which implies that the basin of attraction of C_{n} has uniform size. So, if there are infinitely many of such curves, we will get an intersection of two different basin, which is a contradiction. Therefore, these normally hyperbolic closed curves are isolated in our set. Notice that the periodic attractors or repellers are isolated in Λ as well. Thus, if we remove all the periodic attractors or repellers and all the invariant closed curves supporting an irrational rotation we obtain a compact invariant set Λ̃ as in the next theorem:

Theorem 4.1.2. Let f ∈ Diff^{2}(M^{2}) and let Λ̃ be a compact invariant set having a dominated splitting T_{/Λ}M = E ⊕F such that the periodic points of f in Λ̃ are hyperbolic of saddle type (i.e., of index 1) and Λ̃ does not contain any periodic simple closed curve normally hyperbolic C conjugated to an irrational rotation. Then Λ̃ is a hyperbolic set.

Further explanation must be done to conclude Theorem 4.1.1 from Theorem 4.1.2: we must show that Λ had finitely many periodic attractors or repellers. This is done by showing that, if Λ had infinitely many of them, then they accumulate (in the Hausdorff sense) in Λ̃, a contradiction.

4.1.2 Non existence of wandering intervals

In this section we provide a useful tool in order to understand dynamic properties of one dimensional central manifolds under the assumption of dominated splitting. The results on this section hold not only for surface diffeomorphism but for diffeomorphisms on manifolds of any dimension, provided the dominated splitting is codimension one, that is, one of the subbundles of the splitting has dimension one, say F. Although there is no much difference, we still assume that we are working on dimension two.

Definition 4.1.1. Let f : M → M be a C^{2} diffeomorphism and let Λ be a compact invariant set having dominated splitting and let V be an admissible neighborhood (see Lemma 2.2.2). Let U be an open set containing Λ such that \overline{U} ⊂ V. We say that an C^{2}-arc I in M (i.e, a C^{2}-embedding of the interval (-1,1)) is a δ-E-arc provided the next two conditions holds:

1. f^{n}(I) ⊂ U, n ≥ 0 and |f^{n}(I)| ≤ δ for all n ≥ 0.

2. f^{n}(I) is always transverse to the E-direction.

In other words, a δ-E-arc is an arc that not growth in length in the future and always remains transversal to the E subbundle.

The next result is a fundamental tool in the understanding the behavior of the action of the differential map when expansion or contraction along a subspace is known on a large step. It is a right tool to play with set having dominated splitting. Although is purely arithmetic we shall stated in a different way for our future purposes.

Lemma 4.1.1. Pliss’s Lemma ([Pl]: Given a diffeomorphism f and 0 < γ_{1} < γ_{2} there exist N = N(γ_{1}, γ_{2}, f) and c = c(γ_{1}, γ_{2}, f) > 0 with the following property: given x ∈ M, a subspace S ⊂ T_{x}M such that for some n ≥ N we have (denoting S_{i} = Df^{i}(S))

∏_{i=0}^{n} ∥Df_{/S_{i}}∥ ≤ γ_{1}^{n}

then there exist 0 ≤ n_{1} < n_{2} < .... < n_{l} ≤ n such that

∏_{i=n_{r}}^{j} ∥Df_{/S_{i}}∥ ≤ γ_{2}^{j−n_{r}}; r = 1, ..., l; n_{r} ≤ j ≤ n.

Moreover, l ≥ cn.

The integers n_{i} from the lemma above are known in the recent literature as hyperbolic times.

The next theorem characterize the dynamic of a δ-E-arc. It says that such an arc can not be “wandering”.

Theorem 4.1.3. There exists δ_{0} such that if I is a δ-E-interval with δ ≤ δ_{0}, then one of the following properties holds:

1. ω(I) = ∪_{{x∈I}}ω(x) is a periodic simple closed curve C normally hyperbolic and f_{/C}^{m} : C → C (where m is the period of C) is conjugated to an irrational rotation,

2. ω(I) ⊂ P er(f_{/V}) where P er(f_{/V}) is the set of the periodic points of f in V. Moreover, one of the periodic points in ω(I) is not hyperbolic of saddle type.

Sketch of proof: On the sequel, λ will be the constant of domination. Step 1: First, for each n we take I_{n} the maximal δ-E-arc that contains f^{n}(I). It is proved that there are infinitely many n′_{i}s such that for I_{n_{i}} we get that E is a uniform contracting direction (which could be no true for every I_{n}). This implies that for any x ∈ I_{n_{i}} there is an stable manifold W_{ϵ}^{s}(x) of uniform size. So we can consider the box W_{ϵ}^{s}(I_{n_{i}}) = ∪_{x∈I_{n_{i}}} W_{ϵ}^{s}(x) . Furthermore, for these n′_{i}s it is possible to compare (uniformly for any box) the one-dimensional length of I_{n_{i}} with the two-dimensional volume of the box W_{ϵ}^{s}(I_{n_{i}}), i.e.: there is a constant K such, we have Kvol(W_{ϵ}^{s}(I_{n_{i}})) ≥ |I_{n_{i}}|, where K is independent of n_{i}. Step 2. It is claimed that there exist n_{i} < n_{j} such that

W_{ϵ}^{s}(I_{n_{i}}) ∩ W_{ϵ}^{s}(f^{n_{j}−n_{i}}(I_{n_{i}})) ≠ ∅.  (1)

If this is not the case, a contradiction with the maximality of the I_{n_{i}} is shown. Step 3. The fact that W_{ϵ}^{s}(I_{n_{i}}) ∩ W_{ϵ}^{s}(f^{n_{j}−n_{i}}(I_{n_{i}})) ≠ ∅ for some n_{i} < n_{j} will imply the conclusion of the Theorem.

Now we will explain these steps in more detail. Coming back to the intervals {I_{n}}, consider the sequence of positive integers n_{j} such that |I_{n_{j}}| ⩾ |I_{k}| for any k > n_{j}.

Observe that this implies that for any k > 0 there is some x_{k} ∈ I_{n_{j}} such that ∥Df_{|F_{x_{k}}}^{k} ∥ ≤ 1 and since the iterates of I_{n_{j}} remain small (less than δ) it follows that there is β small such that ∥Df_{|F_{x}}^{k}∥ < (1 + β)^{k} for any x ∈ I_{n_{j}}. Using the domination property and if β is small enough (which is obtained taking δ_{0} small), we get that

∥Df_{|E_{x}}^{k}∥ < [λ(1 + β)]^{k} = λ_{1}^{k}, for any x ∈ I_{n_{j}} and k ≥ 0  (2)

where λ_{1} < 1. Now consider all the positive integers n_{i} such that (2) holds. In particular, every point in I_{n_{i}} have a stable manifold of uniform size.

Let λ_{2} be such that λ < λ_{2} < λ_{1} < 1. Consider N = N(λ_{2}, λ_{1}) from Pliss lemma 4.1.1. It follows (assuming for simplicity that n_{i+1} − n_{i} ≥ N = 1) that the following holds:

∥Df_{|E_{x}}^{n_{i+1}−j}∥ > λ_{2}^{j} for any x ∈ f^{j}(I_{n_{i}}) and 0 ⩽ j < n_{i+1} − n_{i}.  (3)

This implies that the derivative along the F direction behave as an expanding direction for iterates between n_{i} and n_{i+1}. In fact, (3) implies that given 0 ⩽ j < n_{i+1} − n_{i} then

∥Df_{|F_{x}}^{−(n_{i+1}−j)}∥ < (\frac{λ}{λ_{2}} )^{j}

for any x ∈ I_{n_{i+1}}. In particular

|I_{−(n_{i+1}−j)}| < (\frac{λ}{λ_{2}} )^{j}|I_{n_{i+1}}|.  (4)

We are about to finish step 2. Let us assume that W_{ϵ}^{s}(I_{n_{r}}) ∩ W_{ϵ}^{s}(I_{n_{j}}) = ∅ for every r, j. Using that we can compare volume with length we have that

∑_{i>0}^{∞} |I_{n_{i}}| < ∞

and this toghether with (4) imply

∑_{n>0}^{∞} |I_{n}| < ∞

and arguing as Schwarz’s proof of the Denjoy Theorem for some n large we may find an arc J_{n} containing properly each I_{n} such that J_{n} is a δ−interval, which is a contradiction with the maximality of I_{n} for every n.

Now let us explain step 3. Let m = n_{j} − n_{i}. If |f^{km}(I_{n_{i}})| → 0 as k → ∞, then ω(I_{n_{i}}) consist of a periodic orbit. Indeed, if |f^{km}(I_{n_{i}})| → 0, then |f^{k}(I_{n_{i}})| → 0 as k → ∞. Let p be an accumulation point of f^{k}(I_{n_{i}}), that is, f^{k_{j}}(I_{n_{i}}) → p for some k_{j} → ∞, and so, f^{k_{j}+m}(I_{n_{i}}) → f^{m}(p). But by the property we are assuming, i.e., W_{ϵ}^{s}(I_{n_{i}}) ∩ W_{ϵ}^{s}(f^{n_{j}−n_{i}}(I_{n_{i}})) ≠ ∅, we have f^{k_{j}+m}(I_{n_{i}}) → p, implying that p is a periodic point. Thus, for any x ∈ I_{n_{i}} we have that ω(x) consists only of periodic orbits, and so ω(x) is single periodic orbit p. Since |f^{k}(I_{n_{i}})| → 0 we conclude that ω(I_{n_{i}}) is the orbit of the periodic point p. By the way we choose I_{n_{i}}, we have f^{n_{i}}(I) ⊂ I_{n_{i}} and so ω(I) consists of a periodic orbit, and the thesis of the Proposition requires.

On the other hand, if |f^{km}(I_{n_{i}})| does not goes to zero, we take a sequence k_{j} such that f^{k_{j}m}(I_{n_{i}}) → L for some arc L (which is at least C^{1}, and have F as its tangent direction). Now f^{(k_{j}+1)m}(I_{n_{i}}) → L′ and f^{m}(L) = L′. Moreover, by (1), L ∪ L′ is an interval (with F as its tangent direction). Let

J = ∪_{n≥0}f^{nm}(L).

We claim that there are only two possibilities: either J is an interval or a simple closed curve. To prove this, notice that f^{nm}(L) is a δ-E-interval for any n ≥ 0. In particular, for any x ∈ J there exists ϵ(x) such that W_{ϵ(x)}^{cs}(x) is stable manifold for x, and so

W (J) = ⋃_{x∈J} W_{ϵ(x)}^{cs}(x)

is a neighborhood of J.

We only have to show that, given x ∈ J, there exists a neighborhood U(x) such that U(x)∩J is an interval. This implies that J is a simple closed curve or an interval. Thus, take x ∈ J, in particular x ∈ f^{n_{1}m}(L). Take U an open interval, x ∈ U ⊂ f^{n_{1}m}(L) and let U(x) be a neighborhood of x such that U(x) ⊂ W (J) and such U(x)∩L_{1} ⊂ U where L_{1} is any interval containing f^{n_{1}m}(L), transverse to the E-direction and |L_{1}| ≤ 2δ_{0} (this is always possible if δ_{0} is small). Now let y ∈ J ∩ U(x). We have to prove that y ∈ U. There is n_{2} such that y ∈ f^{n_{2}m}(L). Since

f^{n_{1}m}(L) = lim_{j} f^{k_{j}m+n_{1}m}(I_{n_{i}})

f^{n_{2}m}(L) = lim_{j} f^{k_{j}m+n_{2}m}(I_{n_{i}})

and both have nonempty intersection with U(x), we conclude that for some j follows that f^{k_{j}m+n_{1}m}(I_{n_{i}}) and f^{k_{j}m+n_{2}m}(I_{n_{i}}) are linked by a local stable manifold. Hence f^{n_{1}m}(L) ∪ f^{n_{2}m}(L) is an interval L_{1} transverse to the E-direction with |L_{1}| ≤ 2δ_{0}. Therefore y ∈ U(x) ∩ L_{1} ⊂ U as we wish, completing the proof that J is an interval or a simple closed curve.

In case J is an interval, since f^{m}(J) ⊂ J, it follows that for any x ∈ I, ω(x) is a ω-limit point of a point in J, hence periodic orbit, completing the proof in this case. On the other hand, if J is a simple closed curve, which is of class C^{2} because is normally hyperbolic (attractive), then we have two possibilities. If f_{/J}^{m} : J → J has rational rotation number, then we can see that ω(I_{n_{i}}) consist of a union of periodic points, and the same happens to I. If f_{/J}^{m} : J → J has an irrational rotation number, then it is conjugated to an irrational rotation, and denoting C = J, we have that ω(I) is as in the first property of the thesis of the theorem.

Definition 4.1.2. We say that the point x is Lyapunov stable (in the future) if given ϵ > 0 there exists δ > 0 such that f^{n}(B_{δ}(x)) ⊂ B_{ϵ}(f^{n}(x)) for any positive integer n.

As a consequence of Theorem 4.1.3 we get:

Corolary 4.1.1. Let f : M → M be a C^{2}-diffeomorphism of a finite dimensional compact riemannian manifold M and let Λ be a set having a codimension one dominated splitting. Then there exists a neighborhood V of Λ such that if f^{n}(x) ∈ V for any positive integer n and x is Lyapunov stable, one of the following holds:

1. ω(x) is a periodic orbit,

2. ω(x) is a periodic curve normally attractive supporting and irrational rotation.

4.1.3 Outline of proof of Theorem 4.1.1 in a particular case

The proof of Theorem 4.1.1 and in particular the last “two steps” are extremely involved, so we shall explain them in a case where some of the main ideas are present. Indeed, let us give a rather complete proof of the following theorem (compare with Theorem 4.1.2):

Theorem 4.1.4. Let f ∈ Diff^{2}(M^{2}) where M^{2} is the two torus. Assume that Ω(f) = M has dominated splitting T M = E^{s} ⊕ F, E^{s} is a contractive subbundle and all the periodic points are hyperbolic. Then f is Anosov.

The proof (or outline of proof) will be done through a several lemmas. In what follows we always assume that we are under the conditions of the above theorem.

Lemma 4.1.2. For any x ∈ M passes an arc W_{ϵ}^{cu}(x) of class C^{2} and always tangent to F.

Sketch of proof: Since F is integrable, we do have an arc W_{ϵ}^{cu}(x) through x and tangent to F. In order to prove that it is of class C^{2}, by [HPS], we need to prove the so called 2-domination, that is, there exist λ < 1 and a positive constant C such that for any positive integer n follows that \frac{|Df_{|E}^{n}|}{|Df_{|F^{n}}|^{2}} < Cλ^{n}

Lemma 4.1.3. Under the above conditions the distribution F is uniquely integrable and determines an unstable foliation, i.e, if x, y belong to the same leaf we have that dist(f^{−n}(x), f^{−n}(y)) → 0 as n → ∞.

Sketch of proof: Let x ∈ M be any point and denote by W_{ϵ}^{cu}(x) any arc of length ϵ containing x and tangent to F at every point. Let us show that, given δ there exists ϵ_{1} such that |f^{−n}(W_{ϵ_{1}}^{cu}(x)| < δ for any n ≥ 0 (where |.| means length). Otherwise, there exist ϵ_{n} → 0 and m_{n} → ∞ such that

|f^{−j}(W_{ϵ_{n}}^{cu}(x))| ≤ δ, 0 ≤ j ≤ n and |f^{−m_{n}}(W_{ϵ_{n}}^{cu}(x))| = δ.

Taking an accumulation arc I of f^{−m_{n}}(W_{ϵ_{n}}^{cu}(x)) we conclude that T_{x}I = F (x) for any x ∈ I and |f^{n}(I)| ≤ δ for any n ≥ 0. That is, I is a δ-E-arc and from Theorem 4.1.3 we get a contradiction with our assumptions. In the same way it can be shown that |f^{−n}(W_{ϵ}^{cu}(x))| → 0 as n → ∞. It follows from [HPS] the conclusion of the lemma.

From the lemma above, we conclude that f is conjugated to an Anosov diffeomorphism. Indeed, a simple way to prove it is by showing that f is expansive and by [L] it follows our claim. In particular we have a Markov Partition of M, say R = {R_{1}, ..., R_{m}}. From this partition we can construct another Markov Partitions by taking preimages and intersecting with the original one, i.e, for any n > 0 we define R_{n} = {f^{−n}(R_{i}) ∩ R_{j}, 1 ≤ i, j ≤ m, R_{i} ∈ R} is a new Markov Partition (whose “unstable”size decrease with n.)

Let R_{i} be any rectangle of the Markov partition R. For x ∈ int(R_{i}) we denote by J_{i}(x) the connected component of W^{u}(x) ∩ R_{i} that contains x. This J_{i}(x) is an arc with endpoints in the “stable boundary” of R_{i}.

Lemma 4.1.4. There exists K > 0 such that if R_{i} is any rectangle of the Markov partition and x ∈ int(R_{i}) we have that

∑_{j=0}^{j=n} |f^{−j}(J_{i}(x))| < K

as long as f^{−j}(x) ∉ R_{i}, 1 ≤ j ≤ n.

Sketch of proof: Take z_{k} ∈ R_{k} for every rectangle of the Markov partition and let I_{k} = J_{k}(z_{k}). Denote by Π_{k} : R_{k} → I_{k} the projection along the stable foliation. Since the stable foliation is C^{1} (since it codimension is one) we conclude that there exists C > 0 such that if l_{u} is any arc in the unstable foliation contained in R_{k} then

C^{−1}|l_{u}| ≤ |Π(l_{u})| ≤ C|l_{u}|.

Now, let x ∈ R_{i} and let n be as in the lemma. We claim that if for some 1 ≤ j_{1} < j_{2} ≤ n we have that f^{−j_{1}}(x) and f^{−j_{2}}(x) belong to same rectangle R_{k} of the partition then Π_{k}(f^{−j_{1}}(J_{i}(x))) and Π_{k}(f^{−j_{2}}(J_{i}(x))) are disjoint arcs in I_{k}. Otherwise, from the properties of the Markov partitions we conclude that f^{−(j_{2}−j_{1})}(x) ∈ R_{i}, a contradiction. Therefore

∑_{j=0}^{j=n} |f^{−j}(J_{i}(x))| ≤ C ∑_{k=1}^{k=m} |I_{k}| = K.

Let us observe that K from the lemma above is “independent” form the Markov partition, i.e, the same K holds if in the lemma we switch the Markov partition R by the partition R_{n}.

Recall that we want to prove that f is Anosov, that is, we would like to show that the distribution F is expanding.

Lemma 4.1.5. If f is not Anosov, then there exists a set Λ_{0} which is not hyperbolic but with the property that any compact invariant proper subset of Λ_{0} is hyperbolic. Moreover Λ_{0} is a transitive set.

Sketch of proof: The existence of Λ_{0} follows from Zorns lemma. Besides, if for any x ∈ Λ_{0} we have α(x) ⊊ Λ_{0} then it follows that ∥Df_{/F (x)}^{−n}∥ → 0, i.e., Λ_{0} is hyperbolic. Thus, there is some x such that α(x) = Λ_{0}; in other words Λ_{0} is transitive.

At this point we arrive to the heart of the proof of the theorem. We shall show that a transitive set Λ_{0} such that any compact invariant proper subset is hyperbolic (and which is not an invariant circle supporting an irrational rotation) must be hyperbolic, leading to a contradiction with Lemma 4.1.5. For the sake of simplicity, we shall explain the argument assuming a supplementary condition: there is x ∈ Λ_{0} such that x ∉ ω(x).

In this case we can take a Markov partition so that if R(x) is the rectangle of the markov partition R containing x then we have that R(x) ∩ {f^{n}(x) : n ≥ 1} = ∅. On the other hand, since Λ_{0} is transitive, there exist x_{n} ∈ Λ_{0} ∩ R(x) and m_{n} → ∞ such that f^{−j}(x_{n}) ∉ R(x), 1 ≤ j < m_{n} and f^{−m_{n}}(x_{n}) ∈ R(x). Fix k = m_{n} large enough so that |f^{−n}(J_{R(x)}(y))| < r for n ≥ k, where r is small enough so that satisfy a condition to be explained later on.

Let R be the connected component of f^{−m_{n}}(R(x)) ∩ R(x) that contains x_{n}, that is R = R_{m_{n}}(x). Now, let y ∈ R ∩ Λ_{0} and assume that f^{−j}(y) ∉ R, 1 ≤ j < m and f^{−m}(y) ∈ R. Let us show that ∥Df_{/F (y)}^{−m}∥ < \frac{1}{2}. First, notice that m ≥ k. Thus,

∥Df_{/F (y)}^{−m}∥ ≤ ∥Df_{/F (f^{−(m−k)}(y))}^{−k}∥.∥Df_{/F (y)}^{−(m−k)}∥
≤ \frac{|f^{−k}(J_{R(x)}(f^{−(m−k)}(y)))|}{|J_{R(x)}(f^{−(m−k)}(y))|} exp(K_{0}K)\frac{|f^{−(m−k)}(J_{R}(y))|}{|J_{R}(y)|}exp(K_{0}K)

≤ |f^{−(m−k)}(J_{R}(y))|\frac{|f^{−k}(J_{R(x)})}{|J_{R(x)}(f^{−(m−k)}(y))|} \frac{1}{|J_{R(x)}(f^{−(m−k)}(y))|} exp(2K_{0}K)
≤ rC \frac{1}{L} exp(2K_{0}K).

Let us explain the constants above: C is the length distortion along the projection of the stable foliation, L is the minimum length of J_{R(x)}(z), z ∈ R(x) and K_{0} is a Lipschitz constant of log(Df) along the unstable foliation. Hence, if r is small enough we guarantee our claim. Now we may conclude. Let z be any point in Λ_{0}. If α(z) ⊊ Λ_{0} it follows that ∥Df_{F (z)}^{−n}∥ → 0 as n → ∞. On the other hand, if α(x) = Λ_{0} then for some n_{0} we have that y = f^{−n_{0}}(z) ∈ R and returns to it infinitely many times. Since at each return time the derivative is less than 1/2 we are done.

4.2 Generic results in the C^{1} topology and consequences

The first result we would like to state is a proof of the Palis’ conjecture for surface diffeomorphisms in the C^{1}−topology:

Theorem 4.2.1. ([PS1]) Let M^{2} be a two dimensional compact manifold and let f ∈ Diff^{1}(M^{2}). Then, f can be C^{1}-approximated either by a diffeomorphism exhibiting a homoclinic tangency or by an Axiom A diffeomorphism.

Outline of proof: Let f be a C^{1} diffeomorphism far from tanngencies (see Definition 3.2.1). We would like to show that f can be approximated by an Axiom A diffeomorphism. We may assume that f satisfies some generic (i.e. residual) conditions like satisfies Lemma 2.3.1, it is Kupka-Smale and it is a continuity point of the maps g → Γ_{i}(g) = \overline{P er_{i}(g)}, i = 0, 1, 2. From Theorem 3.2.1 we get that P er_{1}(f) has dominated splitting. From this, it may be proved that any C^{1} diffeomorphism close to f has a dominated splitting over the nonwandering set, except perhaps finitely many periodic attractors or repellers. Thus, take a g a C^{2} diffeomorphism which is C^{1} close to f, is is Kupka-Smale and has no invariant curves supporting an irrational rotation (this is C^{r}-generic). For this diffeomorphism g we apply Theorem 4.1.1 and we conclude that Ω(g) is hyperbolic. Since it is also Kupka-Smale, and g is a surface diffeomorphism we can conclude that g is an Axiom A diffeomorphism.

Similar arguments are used in [PS2] [PS4] to prove that the variation of the topological entropy leads to the unfolding of homoclinic tangencies.

An almost immediate consequence we have:

Corolary 4.2.1. Let MS be the set of Morse-Smale diffeomorphisms, denote with \overline{MS} its closure and consider U = Diff^{1}(M^{2}) − \overline{MS}. Then, there exists an open and dense set R in U such that every f ∈ R has a transverse homoclinic orbit. In particular, the closure of the interior of the set formed by the diffeomorphisms having (topological) zero entropy, is equal to \overline{MS}.

The following is a result obtained by Mañé in [M2]

Theorem 4.2.2. There exists a residual set R in Diff^{1}(M^{2}) such that for any f ∈ R one of the following hold:

1. f is Axiom A

2. f has infinitely many periodic attractors

3. f has infinitely many periodic repellers.

Sketch of proof: Let R_{1} be the residual set in Diff^{1}(M^{2}) such that any f ∈ R_{1} is a continuity point of the maps g → Γ_{i}(g) = \overline{P_{i}(g)}, i = 0, 2. If f ∈ R_{1} does not satisfies neither 2 nor 3 above then f and all nearby diffeomorphism has finitely many periodic attractors and repellers, and hence can not be approximated by one exhibiting a homoclinic tangency. So, it can be approximated by an Axiom A one.

The above theorem has a different extension to higher dimension (see [BDP])

In [PS2] it was obtained also the following:

Theorem 4.2.3. Let M^{2} be a two dimensional compact manifold and let f ∈ Diff^{∞}(M^{2}) such that its topological entropy is not locally constant. Then, f can be C^{1}-approximated by a diffeomorphism exhibiting a homoclinic tangency.

4.3 Spectral decomposition

In this section we will go further in the understanding the dynamics of dominated splitting over the limit set on surfaces.

Definition 4.3.1. The limit set of f : M → M is

L(f) = \overline{⋃_{x∈M}(ω(x) ∪ α(x))}

where ω(x) and α(x) are the ω-limit and alpha-limit sets respectively.

We will now state a classical theorem in hyperbolic dynamics (see [S] and the chapter Hyperbolic systems (Hassemblatt) in the volume 1 of Handbook.)

Theorem 4.3.1. [Spectral Decomposition Theorem] Let f : M → M be a diffeomorphism such that L(f) is hyperbolic. Then the periodic points are dense in L(f) and we can decompose the limit set into a finitely many compact invariant disjoint sets L(f) = Λ_{1} ∪ Λ_{2} ∪ ... ∪ Λ_{k}. Moreover, each Λ_{i}, i = 1, ..., k can be decomposed into finitely many compact disjoint and transitive sets Λ_{i} = Λ_{i1} ∪ ... ∪ Λ_{in_{i}} such that f(Λ_{ij}) = Λ_{ij̃} where j̃ = j + 1(modn_{i}) and f_{/Λ_{ij}}^{n_{i}} is topologically mixing (indeed Λ_{ij} is a homoclinic class).

Remark 4.3.1. The Spectral Decomposition Theorem is usually stated for Axiom A diffeomorphisms. However, we think that stating it when the limit set is hyperbolic is the right setting (notice that when f is Axiom A, then L(f) = Ω(f)).

A similar description can be obtained for surface diffeomorphisms having dominated splitting over the limit set L(f) as far as the system is smooth enough (C^{2}):

Theorem 4.3.2. ([PS3]) Let f ∈ Diff^{2}(M^{2}) and assume that L(f) has a dominated splitting. Then L(f) can be decomposed into L(f) = I ∪ L̃(f) ∪ R such that

1. I is a set of periodic points with bounded periods and contained in a disjoint union of finitely many normally hyperbolic periodic arcs or simple closed curves.

2. R is a finite union of normally hyperbolic periodic simple closed curves supporting an irrational rotation.

3. L̃(f) can be decomposed into a disjoint union of finitely many compact invariant and transitive sets. The periodic points are dense in L̃(f) and at most finitely many of them are non-hyperbolic periodic points. The (basic) sets above are the union of finitely many (nontrivial) homoclinic classes. Furthermore f/L̃(f) is expansive.

Roughly speaking, the above theorem says that the dynamics of a C^{2} diffeomorphism having dominated splitting can be decomposed into two parts: one where the dynamics consists on periodic and almost periodic motions (I, R) with the diffeomorphism acting equicontinuously; and another, where the dynamics are expansive and similar to the hyperbolic case.

Two immediate consequences follows from the previous theorem. First, any C^{2} diffeomorphism with dominated splitting over L(f) with a sequence of periodic points with unbounded periods, must exhibits a nontrivial homoclinic class and hence:

Corolary 4.3.1. The topological entropy of a C^{2} diffeomorphism of a compact surface having dominated splitting over L(f) and having a sequence of periodic points with unbounded periods is positive.

Second, using Theorem 3.2.1 and the above one, it can be proved that:

Corolary 4.3.2. Let f ∈ Diff^{2}(M^{2}) having infinitely many sinks or sources with unbounded period. Then, f can be C^{1}-approximated by a diffeomorphism exhibiting a homoclinic tangency.

Let us comment briefly the proof of Theorem 4.3.2. The starting point is Theorem 4.1.1: the breakdown of hyperbolicity of a system with dominated splitting is due either to the presence of irrational rotations or to the presence of non-hyperbolic periodic points (and they could be extremely degenerated). The following theorem provides a way to deal with the presence of non-hyperbolic periodic points.

Theorem 4.3.3. Let f : M → M be a C^{2}-diffeomorphism of a two dimensional compact riemannian manifold M and let Λ be a compact invariant set having dominated splitting. Then, there exists an integer N_{1} > 0 such that any periodic point p ∈ Λ whose period is greater than N_{1} is a hyperbolic periodic point of saddle type (i.e. index 1).

In section 4.3.1 we will give an outline of the proof of previous theorem. Now, denote by P er_{1}^{N} the set of hyperbolic periodic point of index 1 with period greater than N.

Theorem 4.3.4. Let f ∈ Diff^{2}(M^{2}) and assume that \overline{P er_{1}(f)} has dominated splitting. Then, there exists N > 0 such that \overline{P er_{1}^{N}(f)} can be decomposed into the disjoint union of finitely many homoclinic classes. Moreover, \overline{P er_{1}^{N}(f)} contains at most finitely many non-hyperbolic periodic points and f_{/\overline{P er_{1}^{N} (f)}} is expansive.

Thus, the final step in the proof Theorem 4.3.2 is to show that L̃(f) ⊂ \overline{P er_{1}^{N}(f)}. As the reader might guess, these final steps seem to be similar to the hyperbolic case. In fact they are, but let us explain why. In the hyperbolic case, the description of the dynamics follows from a fundamental tool: at each point there are transverse invariant manifolds of uniform size and these manifolds have a dynamic meaning (points in the “stable” one are asymptotic to each other in the future, and points in the “unstable” one are asymptotic to each other in the past). Under the sole assumption of dominated splitting, even if locally invariant manifolds do exist, they do not have any dynamic meaning at all. However, in the two dimensional case, using the fact that these locally invariant manifolds are one-dimensional together with smoothness, we are able to prove that these manifolds already have a dynamic meaning, perhaps not of uniform size, but enough to proceed to a description of the dynamics.

4.3.1 Sketch of proof of Theorem 4.3.3

Arguing by contradiction, assume that the conclusion of Theorem 4.3.3 is not true. Then, there exists a sequence p_{n} of periodic points whose periods are unbounded and they are not hyperbolic periodic points of saddle type. Let Λ_{0} be the set of limits points of the orbits of the points p_{n}, i.e.:

Λ_{0} = ∩_{m⩾0}\overline{∪_{n⩾m}O(p_{n})}.

This set is compact invariant and, since it is a subset of Λ, has a dominated splitting.

Assume first that either all the periodic points in Λ_{0} are hyperbolic or Λ_{0} does not contain any periodic point at all. Then, by Theorem 4.1.1, we conclude that Λ_{0} is a union of a hyperbolic set and a finite union of periodic simple closed curves normally hyperbolic. Since given a neighbourhood of Λ_{0} there exists n_{0} such that, for any n ≥ n_{0}, the orbit of p_{n} it is contained in this neighborhood, we get a contradiction. In fact, the orbits of p_{n} can not accumulate on the periodic simple closed curves since they are normally hyperbolic (attracting or repelling curves). Thus, Λ_{0} is a hyperbolic set and so the maximal invariant set in an admissible compact neighborhood of Λ_{0} is hyperbolic as well. In particular, for sufficient large n, p_{n} lies on this maximal invariant set and so it must be a hyperbolic periodic point of saddle type, a contradiction and so our assumption is false.

Therefore, Λ_{0} must contain a non-hyperbolic periodic point p, and we have that the orbits of a subsequence of {p_{n}} (with unbounded periods) accumulates on p . This contradicts the following result:

Theorem 4.3.5. Let f be a C^{2} diffeomorphism of a compact surface M and Λ ⊂ Ω(f) be a compact set having a dominated splitting. Let p ∈ Λ be a non-hyperbolic periodic point and denote by N_{p} its period. Then, there exists a neighborhood U_{p} of p such that any periodic point q ∈ Λ with period greater than 2N_{p} and whose orbit intersects U_{p} is a hyperbolic periodic point of saddle type.

Sketch of proof: Let p be a non-hyperbolic periodic point and q a periodic point whose orbit goes through a very small neighborhood of p. We would like to show that q is a hyperbolic of saddle type, that is ||Df_{|F_{q}}^{n_{q}}|| > 1 and ||Df_{|E_{q}}^{n_{q}}|| < 1 where n_{q} is the period of q.

The idea is to split the orbit of q into pieces where either is outside the neighborhood of p or inside of it.

On one hand, we show that outside any neighborhood of p, the derivative along the F -direction for any trajectory is uniform bounded away from zero, i.e.: ||Df_{/F_{x}}^{n}|| > c > 0 for f^{i}(x) ∉ U_{p}, i = 1, .., n (notice that this does not contradict that q might be a periodic attractor).

On the other hand, when a trajectory is going through a tiny neighborhood of p, not only does not loose expansion (although the derivative of p along the F −direction might be one) but it has a good expansion along the F −direction from the first time that the point goes into U_{p} until the last time that remains in it (even if the exponential rate is close to one), i.e.: for x such that f(x) ∉ U_{p}, x, ...., f^{−n}(x) ∈ U_{p} and f^{−(n+1)}(x) ∉ U_{p} then ∥Df_{|F_{x}}^{n}∥ > 2/c. Let us explain better this idea.

First we consider a small central unstable segment J containing x. Observe that since a long trajectory of x is inside a U_{p}, then J is close to the central unstable manifold of p. Let us consider a segment I in a fundamental domain of the central unstable manifold of p, obtained as the “projection of J on the central unstable manifold of p along the central stable foliation”. We show that the lengths of f^{−k}(I) and f^{−k}(J) are uniformly comparable for any 1 ⩽ k ⩽ n and we conclude that

∥Df_{/F (x)}^{−n}∥ ≤ \frac{|f^{−n}(J)|}{|J|} exp(K_{0} ∑_{j=0}^{n−1} |f^{−j}(J)|) ≈

\frac{|f^{−n}(I)|}{|I|} exp(K_{0} ∑_{j=0}^{n−1} |f^{−j}(I)|) ⩽ \frac{|f^{−n}(I)|}{|I|} exp(K_{0}|W_{ϵ}^{cu}(p)|).

Since for n large |f^{−n}(I)| is arbitrarily small we get the desired inequality.

4.4 Dynamical determinant in the dominated splitting setting

Few results have been stated in the direction to prove the existence of ergodic measure for invariant sets exhibiting dominated splitting for a surfaces map. In some cases, the scheme proposed in [HY] could be applied for systems with dominated splitting. In the mentioned paper is proved that for a system that it is Anosov in the whole torus except in a periodic point where the derivative has a neutral fixed point do not admit an SRB measure.

A u−Gibbs state is an ergodic invariant probability measure, whose induced measures along the Pesin unstable manifolds are absolutely continuous with respect to Lebesgue (see Partially hyperbolic dynamical systems, Hasselblatt-Pesin, in the present volume of the Handbook). An invariant probability measure μ is called a physical measure if there is a set of positive Lebesgue measure of points x so that \frac{1}{n}Σ^{n−1}k = 0δ_{f^{k}(x)} weak converges to μ as n → ∞. We say that a compact invariant set Λ is an attracting set if there is an open neigbourhood U such that Λ = ∩_{n>0}f^{n}(U). Let Λ_{j} be a basic set in the decomposition given in previous section, and assume that Λ_{j} is an attracting set which does not contain any periodic point with unstable eigenvalue equal to 1. In this case, the results in [BV] may be adapted to f_{|Λ_{j}} , proving that it enjoys a single uμGibbs state which is also a physical measure. Let us call SRB measure a uμGibbs state which is also a physical measure. The results in [Cas] (see also [Do]) indicate that the unique SRB measure μ furnished before has exponential rates of mixing (for Lipschitz observable).

In [BPS], it is consider a real analytic compact surface diffeomorphism f, for which the tangent space over the limit set admits a dominated splitting. It is studied its dynamical determinant d_{f}(z)

d_{f}(z) = exp − ∑_{n≥1} \frac{z^{n}}{n} ∑_{x∈Fix^{∗}f^{n}} \frac{1}{|Det(Df^{n}(x) − Id )|}

where Fix^{∗}f^{n} denotes the (finite) set of fixed points of f^{n} with no zero Lyapunov exponents. The results in the present section indicate that definition given before for d_{f}(z) is appropriate under an assumption of dominated decomposition. Moreover, it is proved that d_{f}(z) is either an entire function or an analytic function in a “slit plane. In particular, once the contribution of the hyperbolic sinks has been removed, d_{f}(z) is analytic in a disc of radius larger than one in all cases where it is known that f admits a unique exponentially mixing SRB probability measure.

The analysis of the dynamical determinant use the dichotomy proved in Theorem 4.3.2 which says that Ω = I ∪L̃(f)∪R, where R is a finite union of normally hyperbolic closed curves on which f is conjugated to an irrational rotation, I is contained in a finite union of normally hyperbolic arcs on which some iterate of f is the identity, and either f|_{Λ} is Axiom A or there is a finite spectral decomposition of Λ (Λ = ∪_{1⩽j⩽n}Λ_{j} together with local product structure) and hyperbolicity is violated on Λ (only) through the presence of finitely many periodic orbits with a single zero Lyapunov exponent.

In order to state the result more precisely in an efficient way, first notice that d_{f}(z) coincides with d_{f|Λ}(z) and then observe that Λ = Λ′ ∪ P where P is a finite union of isolated (hyperbolic) sinks and sources with multipliers (eigenvalues of Df^{P}, P ≥ 1 the period) λ_{E}, λ_{F}. It is easy to check that the dynamical determinant d_{f|P} is a finite product of:

d_{f|sink}(z) = ∏_{j=0}^{∞} ∏_{k=0}^{∞}(z^{P} − λ_{E}^{−j}λ_{F}^{−k}) , |λ_{E,F} | < 1

d_{f|source}(z) = ∏_{j=0}^{∞} ∏_{k=0}^{∞}(z^{P} − λ_{E}^{−j+1}λ_{F}^{−k+1}) , |λ_{E,F} | > 1 .

The infinite products above clearly converge, and define entire functions with an obvious zero-set (in particular, d_{f|sink}(z) is zero-free in the open unit disk and admits a single zero on the closed disk, which is simple at z = 1, while d_{f|source}(z) admits a first zero at 1/(λ_{E}λ_{F}) which is inside the open disk).

So, it may therefore concentrate on the dynamical determinant of f|_{Λ}. For that, we take the set N composed of all nonhyperbolic periodic point p of f. For each p ∈ N we take P = P (p) being the period of p, λ_{E}, λ_{F}, the eigenvalues of Df^{P} (p) and the following subset Σ(p) of ℂ which is defined as

1. {z|z^{P} ∈ [−1, 1]} if λ_{F} = −1 and |λ_{E}| < 1;

2. {z|z^{P} ∈ [min{0, Λ_{E}}, 1]} if λ_{F} = 1 and |λ_{E}| < 1;

3. {z|z^{P} ∈ λ_{F}^{−1}[−1, 1]} if λ_{E} = −1 and |λ_{F} | > 1;

4. {z|z^{P} ∈ λ_{F}^{−1}[min{0, Λ_{F}^{−1}}, 1]} if λ_{E} = 1 and |λ_{F} | > 1;

The result can now be summarized as follows:

Theorem 4.4.1. Let f : M → M be a real analytic diffeomorphism of a compact analytic riemannian surface. Assume that f admits a dominated splitting (1.1) over the nonwandering set Ω and let Λ be the (almost) hyperbolic component in the spectral decomposition of Ω. Let Λ_{j} be an element in the spectral decomposition of Λ which is not an isolated sink or source, and let N_{j} be the empty or finite set of nonhyperbolic periodic points in Λ_{j}. Then d_{f|Λ_{j}}(z) is holomorphic in the plane, slit plane, or multiply slit plane defined by

{z ∈ ℂ | 1/z ∉ ∪_{p∈N} Σ(p)} .

In other words:

1. If all periodic points in Λ_{j} are hyperbolic, then d_{f|_{Λ_{j}}} (z) is an entire function with no zeroes in the open unit disc. The point z = 1 is a zero if and only if Λ_{j} is an attractor. It is then a simple zero, and the only zero of unit modulus if Λ_{j} is mixing.

2. If there exist periodic points in Λ_{j} with multipliers λ_{E} < 1 and |λ_{F} | = ±1, then d_{f}(z) is analytic and nonzero in the disc of radius 1, with a possibly non polar singularity at z = ±1, and it admits an analytic extension to a (multiply) slit plane.

3. If there exist periodic points with multipliers |λ_{E}| = ±1 and λ_{F} > 1 in Λ_{j}, but no periodic orbits with both Lyapunov exponents non positive, letting λ_{F} be the multiplier of smallest modulus, then d_{f}(z) is analytic and nonzero in the disc of radius 1, and it may be analytically extended to the disc of radius |λ_{F} | > 1, with a possibly non polar singularity at z = λ_{F}, and a further analytic extension to a (multiply) slit plane.

Sketch of the proof: Recall the spectral decomposition stated for Ω(f) presented above. In the first case, since ℂ does not contain any periodic orbits and the periodic points in I have one Lyapunov exponent zero, the results of Rugh (see [Ru1], [Ru2]) on the dynamical determinants of hyperbolic analytic maps immediately imply that d_{f}(z) is an entire function. The key point in Rugh’s analysis, inspired by Ruelle’s [Ru] seminal study (in the case when the dynamical foliations are analytic), was to express d_{f}(z) as a quotient of the Grothendieck-Fredholm determinants of two nuclear operators, proving also that zeros in the denominator are always canceled by the numerator.

In the second case, it is used other techniques to investigate d_{f}(z). Note first that if f is real analytic with dominated decomposition over Λ, then the set I in the spectral decomposition from [PS1] is a finite union of closed curves, with I ∩ Λ = ∅. (It is easy to construct examples where I is not empty: just take a real analytic flow on the sphere with both poles as sources and the equator as limit set.) In a nutshell, d_{f}(z) is morally the determinant of a transfer operator, the building blocks of which are either “good,” i.e., of the hyperbolic type studied in [Ru1, Ru2], or approximate direct products of a one-dimensional hyperbolic operator and a one-dimensional parabolic operator describing the local jet at nonhyperbolic periodic points. For the parabolic operator, we adapt the analysis of one-dimensional analytic dynamics with neutral fixed points, also due to Rugh [Ru3], to setting. The crucial tool to do this is an approximate Fatou coordinate for parabolic points. We prove that d_{f}(z) is an analytic function in a “slit plane.”

5 Nonhyperbolic robust transitive systems.

As we said in the introduction, there are manifolds of dimension at least 3 supporting diffeomorphism which are both robustly transitive and nonhyperbolic (r ⩾ 1). Many of this examples show a weaker form of hyperbolicity: they are either partial hyperbolic or they exhibit a dominated splitting. The notion of partial hyperbolicity and different examples of it are discussed in subsection 5.1. The nonhyperbolic robustly transitive examples are outlined in subsection 5.2. In 5.3 it is proved that dominated splitting is a necessary condition for C^{1}−robust transitivity. This results are discussed in subsection 5.3. We point out that the mentioned result is only known in the C^{1}−topology.

5.1 Partial hyperbolicity.

We startr given the definition of strong partially hyperbolicity, and partially hyperbolicity.

Definition 5.1.1. Given a closed invariant set Λ we say that it is strong partially hyperbolic if T_{Λ}M = E^{s} ⊕ E^{c} ⊕ E^{u}, if there are constants 0 < σ^{−1} < γ^{−1} < 1 < γ < σ such that

∥Df_{|E_{x}^{s}}∥ < σ^{−1} < σ < ∥Df_{|E^{u}}^{−1}∥^{−1} ∀x ∈ Λ
∥Df_{|E_{x}^{s}}∥∥Df_{|E_{f^{−1}(x)}^{c}}^{−1}∥ < γ^{1} < γ < ∥Df_{|E_{x}^{c}}∥∥Df_{|E_{f^{−1}(x)}^{u}}^{−1}∥ ∀x ∈ Λ

Given a closed invariant set Λ we say that it is partially hyperbolic if T_{Λ}M = E^{s} ⊕ E^{cu}, constants 0 < σ < γ < 1 such that

∥Df_{|E_{x}^{s}}∥ < σ^{−1} < σ ∀x ∈ Λ
∥Df_{|E_{x}^{s}}∥∥Df_{|E_{f^{−1}(x)}^{c}}^{−1}∥ < γ ∀x ∈ Λ

It is well known that the sub-bundles E^{ss} and E^{uu} are uniquely integrable and hence we have two foliations in M called the (strong) stable one, denoted by F^{ss}(f) and the (strong) unstable one, denoted by F^{uu}(f) which are tangent to E^{ss} and E^{uu} respectively. We shall denote by F^{ss}(x, f) and by F^{uu}(x, f) the leaves of these foliations passing through the point x.

On the other hand, every diffeomorphism g : M → M sufficiently C^{1} close to a partially hyperbolic diffeomorphism f is also partially hyperbolic and therefore it has two invariant foliations F^{ss}(g), F^{uu}(g).

The sub-bundles E^{c} E^{cu} = E^{c}⊕E^{uu} E^{cs} = E^{ss}⊕E^{c} (called center, center stable and center-unstable respectivelly) are not (in general) integrable. However, we can choose a continuous family of locally invariant manifolds tangent to them.

For references about these results, see the chapter Partial hyperbolic systems in the present volume.

We could consider different types of construction of partial hyperbolic systems. We list some of them:

1. Product and Skew products of a hyperbolic system with a non-hyperbolic one.

2. Derived of Anosov: bifurcation of Anosov maps or maps isotopic to an Anosov system and perturbation of these maps.

3. Time one map of an Anosov flow. In this category we have to consider the following types of Anosov flows:

(a) Anosov flows which are suspensions of an Anosov map;

(b) Anosov flows which are not suspensions:

The first type of examples is built over a product manifold where acts a product diffeomorphisms. More precisely, it is taken a transitive Anosov diffeomorphisms A over M and it is taken a non-hyperbolic transitive diffeomorphisms g over a manifold N with the property that the largest and weaker expansion of Dg is dominated by the hyperbolic direction of A. In other words, given the splitting T M = E^{s} ⊕ E^{u} for A, it is assume that

\frac{||DA_{|E^{s}}||}{m\{Dg\}} < λ \frac{||Dg||}{||DA_{|E^{u}}||} < λ

for some λ < 1 and where m{L} the minimum norm of L, i.e, m{L} = ∥L^{−1}∥^{−1}. Then, it is considered the product map

A × g : M × N → M × N
(A × g)(x, y) = (A(x), g(y))

and observe that the map it is partially hyperbolic (because g is not hyperbolic) and the central subbundle is given by T N. Moreover, the central foliation has the property that each central leaf is the manifold N. This example is transitive, not necessary robust transitive, and we show in the next section that it can be perturbed to get a robust transitive system.

A similar construction is the so called skew product which consist in choosing a map

G : M → Diff^{r}(N)

with the property that for any x ∈ M follows that

\frac{||DA_{|E^{s}}||}{m\{D[G(x)]\}} < λ \frac{||D[G(x)]||}{||DA_{|E^{u}}||} < λ

Then, it is considered the map

A × G : M × N → M × N
(A × G)(x, y) = (A(x), G(x)(y))

Observe that for these type of examples, the manifold M × N is foliated by leves homeomorphic to N and each of these leaves is a normally hyperbolic embedded manifold. Then, follows that for any perturbation, the central leaves are also the manifold homeomorphic to N.

In some cases, it is possible to prove that this kind systems (as we will show later) can be either approximated by a robust transitive system or by an Axiom A system.

The second kind of examples cannot build as a product and the central leaves are not necessary compact. In fact, in some cases holds that the central foliation is uniquely integrable and each central leaf is dense; in this case, it is said that the central foliation is minimal. To obtain these examples, it is taken an Anosov map and it is performed a deformation of the system isotopic to the initial map, with the property that some direction remains hyperbolic and other fails to be hyperbolic. See the Mañé’s examples explained in next subsection.

The dynamical properties of the third type of examples depends strongly on the fact if they are or not suspensions. The suspension case, consist in to take an Anosov map A acting on M, and it is performed a suspension. Then, it is taken the flow

φ_{s}^{A}(x, t) = (A(x), s + t)

The flow exhibits an splitting E^{s} ⊕ [∂_{s}φ_{s}^{A}] ⊕ E^{u} where E^{s} and E^{u} are the invariant directions of A and [∂_{s}φ_{s}^{A}] is the flow direction.

After that, it is considered the time one map φ_{1}^{A} and observe that the time one map is not hyperbolic since the flow trajectory is neither expanded not contracted. In this case, observe that the central leaves are flow trajectories. Some of the trajectories are compact (the ones associated to periodic trajectories and induced by periodic orbits of the map A) and some other trajectories are not compact.

It is also possible to get examples Anosov flows which are not suspensions. For that, we refer to [BL] and the well known geodesic flow on manifolds of negative curvatures.

In all the previous construction it is necessary to check that the system (or at least a perturbation of it) is robust transitive.

5.2 Examples of nonhyperbolic robust transitive systems.

The first examples of robust nonhyperbolic systems (examples of robust transitive systems which are not Anosov) were given by M. Shub (see [Sh]), who considered on the 4-torus, a skew-products of an Anosov with a Derived from Anosov diffeomorphisms. Then, R. Mañé (see [M]) reduced the dimension of such examples by showing that certain Derived of Anosov diffeomorphisms on the 3-torus are robust transitive.

For a long time these examples remained the unique known ones. Later, Diaz, studying the unfold of heterodimensional cycles, showed maximal invariant set that are robust transitive and nonhyperbolic. This ideas was pushed, in [BD], where was given new classes of nonhyperbolic robustly transitive diffeomorphisms, on manifolds which are not n-dimensional torus and satisfying a weaker form of hyperbolicity. Moreover, it was also showed a general construction of maximal invariant set which are robust transitive and nonhyperbolic. In resume, the examples given in [BD] are basically:

1. Perturbations of the time-one map of any transitive Anosov flow.

2. Perturbations of (A, id_{N}) : 𝕋^{n} × N → 𝕋^{n} × N, where A is a transitive Anosov diffeomorphism on the n−dimensional torus 𝕋^{n} and N is any compact manifold. In this example the nonhyperbolic central direction is chosen tangent to the fibers {., N} providing in this way examples of robustly transitive diffeomorphism having central direction with arbitrary dimension.

3. Perturbations of (f, id_{N}) : Λ × N → Λ × N, where Λ is a hyperbolic basic piece for f and N is any compact manifold.

Later, in [BV] was introduced examples without hyperbolic subbundles

1. Examples of robustly transitive diffeomorphisms on T^{3} without any stable bundle, that is, T M = E^{cs} ⊕ E^{u}, where E^{cs} is indecomposable and nonhyperbolic.

2. Examples of robustly transitive diffeomorphisms on T^{4} without any hyperbolic bundle (stable or unstable), that is, T M = E^{cs} ⊕ E^{cu} is a dominated splitting where E^{cs}and E^{cu} are indecomposable and nonhyperbolic.

In next subsections we give a rough idea how these examples are constructed and we outline the arguments that proves that these systems are robustly transitive. We follow the construction initially done by the authors (performing some modification in some cases) and in subsection 5.2.5 it is given a condition that allows to obtain the same examples from a different point of view.

5.2.1 Shub’s example.

Let f : 𝕋^{2} → 𝕋^{2} be an Anosov diffeomorphism having two fixed points p and q. Since f is Anosov, T 𝕋^{2} = E^{ss} ⊕ E^{uu} with ∥Df_{/E^{ss}}∥ < λ < 1 and ∥Df_{/E^{uu}}^{−1}∥ < λ.

Now, consider a smooth family of torus diffeomorphisms g_{x} : 𝕋^{2} → 𝕋^{2} indexed in x ∈ 𝕋^{2} such that:

• T 𝕋^{2} = E^{s}(g_{x}) ⊕ E^{c}(g_{x}) invariant under D(g_{x}) with ∥D(g_{x})_{/E^{s}(g_{x})}∥ < μ < μ_{1} < 1 and μ < μ_{1} < ∥D(g_{x})/E^{c}(g_{x})∥ ≤ μ^{−1}.

• g_{x} ∀ x ∈ 𝕋^{2} preserves a cone field C^{s} and C^{cu}

• g_{p} is Anosov and g_{q} is a DA (derived from Anosov) map.

• g_{x}(p) = p for every x and p is an attractor for g_{q}.

It is assumed (taking a power of f if necessary) that λ < μ. Next, it is defined the map on 𝕋^{4} which is a skew product and it is the candidate to be robustly transitive:

F : 𝕋^{2} × 𝕋^{2} → 𝕋^{2} × 𝕋^{2}, F (x, y) = (f(x), g_{x}(y)).

It is not difficult to see that F is partially hyperbolic T_{(x,y)}𝕋^{4} = E^{ss}(x, y)⊕E^{s}(x, y)⊕ E^{c}(x, y) ⊕ E^{u}(x, y). Set E^{s} = E^{ss} ⊕ E^{s}. Let us show that the stable foliation (tangent to E^{ss} ⊕ E^{s}) is minimal. First observe that

W^{s}({p} × 𝕋^{2}) = ∪_{z∈𝕋^{2}}W^{ss}(p, z) = W^{ss}(p, f) × 𝕋^{2}

and hence is dense in 𝕋^{2} × 𝕋^{2}. Moreover, since g_{p} is Anosov, we have that

W^{s}(p, z) = ∪_{y∈W^{s}(z,g_{p})}W^{ss}(p, y)

is dense on 𝕋^{2} × 𝕋^{2} ∀ (p, z) ∈ {p} × 𝕋^{2}.

On the other hand, for every (z, w) ∈ 𝕋^{2} ×𝕋^{2} we have that W^{uu}((z, w))∩W^{s}({p}× 𝕋^{2}) ≠ ∅. From this it follows that the stable foliation F^{ss} (whose leaves are tangent to E^{ss}⊕E^{s}) is minimal. Later, it is proved that this property remains for the perturbation of the initial map. Observe that if the strong stable foliation is minimal then the systems is transitive. Since it is proved that the strong foliation of any perturbation of the initial system is minimal, follows that the initial system is robust transitive.

5.2.2 Mañé’s example.

Mañé showed that certain Derived of Anosov are robust transitive. To prove that, first he consider a linear Anosov map A : 𝕋^{3} → 𝕋^{3} having three real eigenvalues, 0 < λ < 1 < λ_{2} < λ_{3}. Observe that there are three invariant foliation associated to each eigenvalue which correspond to the projections over 𝕋^{3} of the eigenspaces associated to each eigenvalue. In particular, the foliation associated to the central eigenvalue (named central foliation) is minimal; i.e.: each central leaf is dense.

Later a derived of Anosov is performed in an small neighborhood D of the fixed point. More precisely, the derived of Anosov is performed in a way that the following properties hold:

1. the central foliation of the initial Anosov system is preserved,

2. the new system keeps two hyperbolic directions,

3. in the complement of D the new system is equal to the initial one,

4. in the small neighborhood D, two points of different indices appear as a consequences of a bifurcation of the fixed point.

The last item implies that the expansiveness of the central direction is lost. On one hand, observe that the new systems has the property that the central foliation remains minimal. Moreover, it is proved that it verifies a property called plaque expansive: there exists ϵ > 0 such that if {p_{n}} and {q_{n}} are two ϵ−pseudo orbits such that, f(p_{n}) and p_{n} belongs to the central leaf, f(q_{n}) and q_{n} belongs to the central leaf and dist(p_{n}, q_{n}) < ϵ, then p_{n} and q_{n} belongs to the same central leaf for any integer. As consequences of this property follows that for any perturbation there exists a unique central foliation which is also minimal. On the other hand, using that the region D is small and that the unstable leaves growth exponentially it is showed that any unstable leaf of a fixed length, has a point that its orbit remains in the complement of D and so its central direction expands. This allows to show that any central segment containing this point has its length growing to infinity by positive iterates. All this properties together implies transitivity. In fact, given an open set U, then an iterate of it will contain a point that is expansive along the central direction, and so the iterates of U will start to grow along the central direction and using the density of any central leaf, it is obtained the density of the iterates.

5.2.3 Bonatti-Diaz’s construction.

Now we will try to explain the arguments used in [D1] and [BD] to get robust transitive diffeomorphisms or sets. The authors use a geometrical construction called blender. It is said that f has a center-stable blender (center-unstable blender) Γ associated to a hyperbolic saddle p of index k if Γ is a hyperbolic invariant set contained in the homoclinic class of p and there are an open set D of embedding of the disk D^{k−1} in M and a C^{1}−neighborhood U of f such that, for every g ∈ U, follows that any disc D ∈ D intersects the closure of W_{ϵ}^{s}(Γ_{g}, g) (W_{ϵ}^{u}(Γ_{g}, g)), where Γ_{g} is the continuation of the hyperbolic set Γ for g and W_{ϵ}^{s}(Γ_{g}, g) is the local stable manifold of the invariant set Λ_{g} (W_{ϵ}^{u}(Γ_{g}, g) is the local unstable manifold of the invariant set Λ_{g}).

One way to construct a center-blender in a three dimensional manifold is to construct a hyperbolic maximal invariant set Γ associated to a periodic point p such that for the continuation of p for g close to f follows that Π^{ss}(W_{ϵ}^{s}(p_{g}) ∩ Γ) contains an interval, where Π^{ss} is the projection along the strong stable direction over some central direction contained in the local stable manifold of p.

Now, using these blenders, let us show a general strategy to guarantee robust transitivity. Let f be partially hyperbolic in the whole manifold with nontrivial splitting T M = E^{s} ⊕ E^{c} ⊕ E^{u} verifying that:

1. f has two periodic points p and q of indices one and two respectively;

2. f has a center-unstable blender Γ associated to q;

3. There are a constant L > 0 such that any strong stable segment of length larger than L, intersect the local unstable manifold Γ, and any strong unstable segment of length larger than L, intersect the local stable manifold of q.

It follows under this assumption that f is a nonhyperbolic robust transitive map. The non-hyperbolicity follows from the presence of points of different indices. Properties 2 and 3 implies transitivity: by property 3, any strong leaf of length larger than L is caught by the blender, and the dynamic is mixed inside it. In fact, given any open set we get that some future iterates will intersect the local stable manifold of the point q (the iterates growth along unstable leaves and by property 3 reach the local stable of q); for any other open set follows that some negative iterate, will intersect the blender Γ and so, the unstable manifold of q. This two facts implies that both sets will mix. On the other hand, we get that the blender is robust and property 3 also holds for a perturbation of f and this implies the robustness of the transitivity.

These previous arguments are used in [BD] to prove that time-one of a transitive Anosov flow and the product of an Anosov diffeomorphisms by the identity map defined on a compact manifold N are in the closure of the set of robustly transitive diffeomorphisms. In both cases, the dynamics is perturbed along a compact central leaf (along a closed orbit of the flow in the case of the time-one map of an Anosov flow and along a periodic central leaf for the product and skew product) to get a pair of periodic orbits of different Morse index and later it is performed with this points the construction of a blender.

5.2.4 Robust transitive diffeomorphisms without any hyperbolic direction.

We will now explain the ideas given in [BV] of the constructions of the robustly transitive diffeornorphisms without any hyperbolic directions. This construction follows closely the example of Mañé.

Consider an linear Anosov map A of the torus 𝕋^{4} having 4 real eigenvalues, 0 < λ_{1} < λ_{2} < 1 < λ_{3} < λ_{4}. Then it is fixed an A−invariant cone fields C^{cu} corresponding to the expanding eigenvalues λ_{3} and λ_{4} and a cone C^{cs} around the contracting eigenvalues. Now, take two small boxes C_{1} and C_{2}, and consider a diffeomorphism f coinciding with A outside the boxes C_{1}, and C_{2} and verifying the following:

1. f contains in C_{1} a fixed point p (of index 2) with a complex eigenvalue with eigenspace inside C^{cs} and a fixed point q, of index 1 with eigenspace also inC^{cs} . This implies that C^{cs} does not contain a hyperbolic stable direction: the existence of the contracting complex eigenvalue implies that any stable subbundle has to be of dimension 2, but the point of Morse index 1 implies that such bundle has dimension at most one.

2. f in C_{2} contains a fixed point p_{2} (of index 2) with an expanding complex eigenvalue and having a fixed point q_{2} of index 1. Arguing as before, this properties prevent the existence of hyperbolic unstable subbundle.

To assure that f is robust transitive the following properties also hold:

1. Df, (Df^{−1}) preserves the cone field C^{cu} (C^{cs}, respectively) and uniformly expands the area in this cone field.

2. The restriction of f to the complement of C_{1}, expands uniformly the vectors in C^{cu}

3. The restriction of f^{−1} to the complement of C_{2}, expands uniformly the vectors in C^{cs}

Choosing a sufficiently thin cone fields, it is verified that there exists a constant L > 0 such that every center-unstable disk D^{cu} (tangent to C^{cu}) of radius bigger than L intersects any center-stable disk D^{cs} (tangent to C^{cs}) of radius larger than L.

Arguing as in Mañé’s example, but using the uniform expansion of the area in C^{cu} instead of the uniform expansion of the vectors, it is showed that every center unstable disk D contains a point whose forward orbit remains in the complement of C_{1} and this allows to show (as in Mañé example) that, for every large n > 0, f^{n}(D) contains a center-unstable disk of radius bigger than L. The same argument shows that the large negative iterates of any center-stable disk D contains a center stable disk of radius bigger that L. And this implies the transitivity of f.

5.2.5 Sufficient conditions for robust transitivity.

The mentioned examples of robust transitive systems which are partially hyperbolic, shares a strongest property: not only they are robustly transitive but also (at least) one of the strong foliation is robustly minimal, that is, every leaf is dense in the manifold.

Definition 5.2.1. Let f : M → M be a partially hyperbolic diffeomorphism. It is said that F^{ss}(f) is robustly minimal if there exist a C^{1} neighbourhood U(f) such that Fss(g) is minimal for every diffeomorphism g ∈ U(f).

Reacall that if (for instance) F^{ss}(f) is robustly minimal then f is robustly transitive, i.e., ever diffeomorphism C^{1} close to f is transitive.

All the examples of robust transitivity considered, are based in either a property of the initial system or in a geometrical construction. But, does exist a necessary and sufficient condition among the partial hyperbolic system such that transitivity is equivalent to robust transitivity? Can this property be characterized in terms of the dynamics of the tangent map? This is clear for Anosov maps, where transitivity implies robust transitivity, but what about for the nonhyperbolic ones?

In the paper [PS5] is studied partially hyperbolic systems having at least one minimal strong foliation and it is given sufficient conditions in order to guarantee that this foliation remains minimal under C^{r}-perturbations. Before, to enunciate the sufficient condition, let us introduce a notation: if L : V → W is a linear isomorphism between normed vector spaces we denote by m{L} the minimum norm of L, i.e, m{L} = ∥L^{−1}∥^{−1}.

Definition 5.2.2 (Property SH). Let f ∈ Diff^{r}(M) be a partial hyperbolic diffeomorphism. We say that f exhibits the property SH (or has the property SH), if there exists λ_{0} > 1, C > 0 such that for any x ∈ M there exists y^{u}(x) ∈ F_{1}^{uu}(x, f) (the ball of radius 1 in F^{uu}(x, f) centered at x) verifying

m{Df_{/E^{c}(f^{m}(y^{u}(x)))}^{n}} > Cλ_{0}^{n} for any n > 0, m > 0

In other words, it is required that in any disk of radius 1 in any leaf of F^{uu}(f) there is a point y^{u} where the central bundle E^{c} has a uniform expanding behavior along the future orbit of y^{u}. A nice image of the above is the existence of a hyperbolic set (with E^{c} being part of the unstable bundle) such that the local stable manifold of this hyperbolic set is a global section to the foliation F^{uu}(f).

Using this property follows the next theorem:

Theorem 5.2.1. Let f ∈ Diff^{r}(M) be a partial hyperbolic diffemorphism satisfying the property SH and such that the (strong) stable foliation F^{ss}(f) is minimal. Then, F^{ss}(f) is robustly minimal.

A similar result for the foliation F^{uu}(f) holds provided f^{−1} satisfy the property SH. As an immediate consequence is that if f satisfies the conditions of previous theorem then it is robustly transitive.

It is also possible to use the previous theorem to reconstruct the examples by Shub, Mañé and Bonnati-Diaz.

Let us mention here that in [BDU] it is shown that C^{1}-generically, a robustly transitive diffemorphism on a three dimensional manifold M^{3} one of the (strong) foliations is minimal. On the other hand, the following natural question remains: is the property SH a necessary condition for robustly transitive partially hyperbolic systems? Sketch of the proof of theorem 5.2.1

The proof of the theorem 5.2.1 relies on the two following facts:

1. if the property SH holds then given any open set, there is a positive iterate such that it contains a central unstable disc of a fixed size;

2. the property SH is robust by perturbation;

3. fon any small perturbation of the initial map, follows that any leaf of the strong stable foliation is “almost” dense.

The last item means that given a positive constant δ then any leaf of the strong stable foliation of any small perturbation is δ−dense. On one hand, by item two the property SH holds for any perturbation so by item one follows that given any open set there is a positive iterate such that it contains a central unstable disc of a fixed size chosen for the initial system. On the other hand, by item three, any leaf of the strong stable foliation of any small perturbation is δ−dense. Therefore, if δ is small, given a close perturbation of the initial system, follows that for any strong stable leaf intersects some positive iterate of any open sets. So, the strong stable foliation of the perturbed system is also minimal.

This key property that guarantee the robustness of stable foliation of a partially hyperbolic diffeomorphism can be formulated in the following way: Some Hyperbolicity on the central distribution E^{c} at some points.

5.3 Robust transitivity and Dominated splitting.

The previous examples lead to two natural questions:

• Is there a characterization of robust transitive sets that also gives dynamical information about them?

• Can we describe the dynamics under the assumption of either partial hyperbolicity or dominated splitting?

Both question has a partial answer: it is known that C^{1}−robust transitivity implies dominated splitting, however, it is unknown in a generic setting the converse of the previous assertion.

In the following, we will say that an invariant compact set Λ for a diffeomorphisms f is a C^{1}−robust transitive set, if Λ is a transitive maximal invariant (i.e.: there is U such that Λ = ∩_{n∈ℤ}f^{n}(U) and Λ has a dense orbit) and for any g C^{1}−close to f follows that Λ_{g} = ∩_{n∈ℤ}g^{n}(U) is also a transitive set.

The next result proved in dimension two in [M2], dimension three in the paper [DPU] and in greater dimension in [BDP], shows that these two questions are closely related. In fact, some kind of dynamics on the tangent bundle is implied by the robust transitivity:

Theorem 5.3.1. Any robust transitive set of a C^{1}−diffeomorphism exhibits dominated splitting such that its extremal bundles are uniformly volume contracted or expanded.

In dimension two, it is proved that the set is hyperbolic (see [M2]); in dimension three it is proved that the sets exhibits a partial hyperbolic splitting (see [DPU]) and in higher dimension just a dominated splitting

This last theorem can also be formulated in the following way:

Theorem 5.3.2. ([BDP]) There is a residual subset of C^{1}−diffeomorphisms such that that for any diffeomorphism in the residual set, it is verified that for any homoclinic class of a periodic point (the closure of the intersection of the stable and unstable manifold of it) either has dominated splitting or it is contained in the closure of infinitely many sources or sinks.

In the particular case of dimension two, this result proved in [M2] is more general and states the following

Theorem 5.3.3. ([M2]) There is a residual subset of C^{1}−diffeomorphisms such that that for any diffeomorphism in the residual set, it is verified that the diffeomorphisms is Axiom A or it exhibits infinitely many sources or sinks.

First, we give an sketch of the proof of the theorem 5.3.3 Later we explain how the arguments work in higher dimension.

As in the proof of the dominated splitting versus tangencies, a key element in the proof is that in the C^{1}−topology any small perturbation of the derivative of a diffeomorphism f along a periodic orbit p, can be performed dynamically as a C^{1}−perturbation of f. Recall that this follows from a lemma proved in [Fr] and stated in section.

5.3.1 Sketch of the proof of theorem 5.3.3.

The proof of the theorem in dimension two follows showing that given a C^{1} diffeomorphisms f such that the number of sinks and repellers for any diffeomorphisms C^{1}−close to f is constant, then f is an Axiom A diffeomorphisms. To show that, it is necessary to prove that the limit set is hyperbolic. This is done in two steps:

• Step I: the limit set exhibits a dominated splitting;

• Step II: this dominated splitting is hyperbolic.

To get the first step, let P be a set of periodic hyperbolic saddles of a diffeomorphism f. Since we are working in the C^{1}−topology, using the C^{1}−closing lemma allow to assume that the periodic points are dense in the limit set. Over P it is considered the natural splitting T_{P}M = ∪_{{p∈P}}E^{s}(p) ⊕ E^{u}(p) where E^{s}(p) and E^{u}(p) are the eigenspaces of Df^{n_{p}} and n_{p} is the period of the periodic point p. The goal is to prove that this splitting is a dominated splitting and so can be extended to Λ. To perform that, it is showed that if the splitting over the periodic points is not uniform dominated, then with an arbitrarily small perturbations of the derivative of f along the orbit of some p ∈ P, it is obtained a matrix that has a complex eigenvalues of modulus different from one, meaning, that by a perturbation was obtained either a new sink or a new repeller, contradicting the hypothesis that no sinks or repellers can be created by perturbation. More precisely, first it is used the theorem 2.3.2 that states: if the splitting E^{s}⊕E^{u} over P is not dominated, then there is an arbitrarily small perturbation of the derivative of f along the orbit of a point p ∈ P such that the angle between the eigenspaces of the corresponding matrix is arbitrarily small.

Later, it is applied the following claim:

Asserts 5.3.1. Let A ∈ GL(2, R) having two different real eigenvalues whose eigenspaces form an angle less than ϵ. Then there is t ∈ [−ϵ, ϵ] such that the matrix R_{t} ∘ A has a pair of conjugate complex eigenvalues.

To prove the previous claim, identifying the projective space with S^{1}, observe that A acts over S^{1} as a Morse-Smale with one sink and one repeller, one close to the other. Then, composing with an small rotation it is obtained a new map with irrational rotation that correspond to the fact that R_{t} ∘ A has a complex eigenvalue.

After it is proved that the limit set has dominated splitting, using that neither sinks nor repellers can be created by perturbations, it is shown that the splitting is in fact hyperbolic. To do that, it is used the ergodic closing lemma. This lemma states that with full probability any recurrent orbit can be shadow by a periodic orbit of a system C^{1}− close to the initial. Before to state the lemma, we need some definitions.

Define B_{ε}(f, x) as the set of points y ∈ M such that d(f^{n}(x), y) ⩽ ε for some integer n and define Σ(f) to be those points x ∈ M such that for every neighborhood U ⊂ Diff^{1}(M) of f and for every ε > 0 there is a g ∈ U and a y ∈ B_{ϵ}(f, x) such that y ∈ Per(g), g = f on M − B_{ε}(f, x) and d(f^{j}(x), g^{j}(y)) ⩽ ε for all j such that 0 ⩽ j ⩽ m, where m is the g-period of y.

Theorem 5.3.4. For any f ∈ Diff^{1}(M) follows that Σ(f) has measure one for every f-invariant probability measure on the Borel sets of M.

An equivalent statement of the previous lemma is the following:

Let f ∈ Diff^{1}(M) and let μ be an invariant measure of f. Then, there exists a sequences {g_{n}} of C^{1}−diffeomorphisms converging in the C^{1}−topology to f and a sequence of measures {μ_{n}} invariant for g_{n} such that each μ_{n} has supports over a periodic orbit and the sequence of measures converges to μ in the weak topology.

Using this version, it is possible to prove that if one of the subbundle of the dominated splitting is not hyperbolic, then it is possible to create a sink or a repeller for a C^{1}−perturbation.

In fact, let us assume that F is not expanding. Then, there is an invariant measure μ such

∫ |Df_{|F (x)}| μ ⩽ 0

By the ergodic closing lemma, for any δ > 0 there is g C^{1}−close to f and an invariant measure ν supported on a periodic point such that ν is δ−close to μ in the weak topology. Using that the dominated splitting varies continuously by perturbation and that the support of ν is close to the support of mu, follows that over the support of ν there is an splitting E_{g} ⊕ F_{g} with the property that the direction F_{g} is close to F. So

∫ |Dg_{|F_{g}(x)}| ν ⩽ δ

Since the support of ν is a periodic orbit, follows that the eigenvalue along the F_{g} direction is close to 0. Then, by a C^{1}−perturbation this eigenvalue can be done smaller than zero. From the domination property follows that the other eigenvalue is also smaller than zero. And this implies that a sink has been created, which is a contradiction from the fact that it is assumed that neither sinks nor repellers can be created by perturbations.

So, it has been proved that the limit set is hyperbolic. Then, the limit set it can be decomposed in a finite union of maximal invariant transitive sets. It follows that the no cycle condition holds; if it is not the case, it is possible to create a tangency by perturbation and this lead to the appearance of either sinks or repellers. Then, if the limit set is hyperbolic and the no cycle condition holds, from the fact that we are dealing with surfaces maps follows that the limit set coincides with the non-wandering set.

5.3.2 Sketch of the proof of theorem 5.3.2. Dominated splitting versus homotheties.

In the 2-dimensional argument presented above was considered linear cocycle defined over period points an its a perturbations, without any information about the dynamics of the set of periodic points. These ideas for the two-dimensional, when they are considered in any dimension get the following lemma that has similar proof of the two dimensional case:

Lemma 5.3.1. Let P_{k} the set of periodic hyperbolic saddles of index k and let E_{k}^{s}(p) ⊕ E_{n−k}^{u}(p) the natural splitting induced by the stable and unstable directions of the points p ∈ P_{k}. If this splitting is not dominated, then by an arbitrarily small C^{1}-perturbation it is possible to change the index of some saddle .

Observe that if 2 ⩽ k ⩽ n − 2 the kind of bifurcations mentioned in the previous lemma do not break the transitivity and so they can occur in robust transitive dynamics. In fact this holds in the examples considered. Moreover, C^{1}−robust transitive diffeomorphism generically exhibits periodic points of different indexes. This follows from the fact that any diffeomorphism such that any C^{1}−perturbation has its periodic points hyperbolic, is in fact a hyperbolic diifeormorphisms.

So, in higher dimension, it is necessary to work out the dynamic generated by different periodic points acting in an homoclinic class. Using this argument it is shown that if there is not any dominated splitting then it is possible to perturb the derivative of f along a periodic orbit in order to turn the corresponding matrix into an homothety, and so, after another perturbation, we obtain a sink or a source for a diffeomorphism close to f.

To perform this argument, first, it is taken a dense subset K in H(p, f) of periodic points homoclinically related with p, and we reduce the study to this set of periodic orbits. As any dominated splitting extends to the closure, the lack of dominated splitting on H(p, f) implies the same for K. Then, given two (for simplicity fixed) hyperbolic saddles p and q, it can be supposed that they are homoclinically related(their invariant manifolds intersect transversally) and so there is a horseshoes containing p and q and some points of intersection of their invariant manifolds. Using this, follows that they are periodic points passing first arbitrarily close to p and later to q and so on. Moreover, it is possible to assume that the time spent in the complement of a neighborhood containing p and q is uniformly bounded. To be precise, given any family {n_{1}, m_{1}, ..., n_{k}, m_{k}} of arbitrarily big natural numbers there are a natural number r (independent of n_{i} and m_{i}) and a saddle z of H(p, f) of period t = kr + Σ_{i}(n_{i} + m_{i}) whose orbit spends alternately n_{i} consecutive iterates close to p and m_{i} consecutive iterates close to q. This means that the derivative of f in z is like

Df^{t}(z) = T_{2} ∘ Df^{m_{k}}(q) ∘ T_{1} ∘ Df^{n_{k}}(p) ∘ ... ∘ T 2 ∘ Df^{m_{1}}(q) ∘ T_{1} ∘ Df^{n_{1}}(p)

where T_{1} and T_{2} are called the transitions from p to q and from q to p, respectively. These two transitions correspond to a bounded number of iterations and can be adapted by small perturbations in order that their contribution to the product vanishes as n_{i} and m_{i} go to infinite, thus the noise introduced by these transitions is negligible provided n_{i} and m_{i} big enough. In other words, to multiply derivatives corresponding to different periodic points homoclinically related may make sense and the resulting linear maps (almost) correspond to the derivative at other periodic saddle of the class.

This kind of arguments also show that in any homoclinic class a there is dense subset of periodic points whose derivative at the period can be turn diagonalizable with positive eigenvalues of multiplicity one, by a small perturbation of the derivative along the orbit.

Now, if no dominated splitting is preserved, we will have the following situation: there is a linear map A ∈ GL(R, n), linear maps A_{1}, A_{2}, ..., A_{n−1} in GL(R, n) and a basis B = {v_{1}, ..., v_{n}} of R^{n} such that A is diagonal in this basis and if L_{i} is the space generated by v_{i}, then A_{j} keeps invariant the subspaces L_{1}, ..., L_{j−1}, L_{j+2}....L_{n} and permutes the spaces L_{j} and L_{j+1}. In the previous situation, the matrices A and A_{i} correspond to the derivatives of f at periodic points p, {p_{i}}. The information that L_{j} and L_{j+1} are permuted follows from the fact that we are assuming that there are not any dominated splitting: in fact, if restricted to a two dimensional subspace does not exist a dominated decomposition in two subbundles, using the assertion 5.3.1 we can get after perturbation that the derivative restricted to this direction acts as a two dimensional map with complex eigenvalue and so considering iterates of the same map we can assume that there are two invariant subspace that are interchanged. On the other hand, since that we can consider the transition, so, we can also consider all the possible products of the maps A, A_{1}, A_{2}, ...A_{n−1}. A simple result of linear algebra show that some of these product is an homothety. The idea is that, as the matrices A_{i} produce permutations of the eigenspaces of A which generate the group of all permutations of these eigenspaces, it is possible to mix the multipliers, obtaining a matrix having essentially the same rate of expansion in all directions.

The construction above shows that a robust transitive set always admits a dominated splitting (so the fines dominated splitting is defined). To get that the extremal direction are volume expansive it is applied the Ergodic Closing Lemma. Observe that, since E_{1} does not admit any dominated sub-splitting, the arguments above guarantees that (after a perturbation) we can produce a homothety in E_{1}. Now, if E_{1} is not uniformly volume contracting, by using the Ergodic Closing Lemma we get a homothety (in E_{1}) at a point which is volume expanding or (at least) whose rate of contraction of volume is close to one. So (after a new perturbation if necessary) we get a homothety which ratio greater than 1. Finally, the domination of the splitting implies that any bundle E_{j} is expanding, thus this saddle point turns to be a source.

5.4 Similar results for conservative systems.

The same kind of question and dichotomies can be formulated for conservative maps. In [BDP], was proved that for volume preserving maps, C^{1}−generically the homoclinic classes either has dominated splitting or are accumulated by periodic points such that their derivatives are the identity map.

In [AM] a so called Pasting Lemma was proved and it states that if f is a C^{1+α} volume preserving map that shows a periodic point such that Df^{n_{p}} = Id (n_{p} is the period of p and Id is the identity map) then is possible to find a C^{1} volume preserving map g C^{1}−close to f such that p also is a periodic point for g and there is a neighborhood U of p such that g^{n_{p}} is the identity map in U. As a corollary of this and the result in [BDP] it is proven in [AM]:

Theorem 5.4.1. Any robust transitive C^{1}− volume preserving diffeomorphism exhibits dominated splitting such that its extremal bundles are uniformly volume contracted or expanded.

It is worthy to mention that for volume preserving and as a consequence of a closing lemma for chain recurrent points proved in [BoC] it is concluded in the same paper that:

Theorem 5.4.2. C^{1}−generically the volume preserving diffeomorphisms are transitive.

A chain recurrent point is a point such that for every ϵ > 0 there exists a sequences of points x = x_{0}, x_{1}....., x_{n} = x_{0} with dist(f(x_{i}) − x_{i+1}) < ϵ. The closing lemma for chain recurrent points states that C^{1}−generically the chain recurrence set coincide with the closure of the periodic points

Also about volume preserving diffeomorphisms similar dichotomies as the one stated in theorem 6 can be stated. Recalling that for invariant measure we can obtain the Oseledets splitting and it is possible to calculate the Lyapunov exponents, the following theorem characterizes continuity of Lyapunov exponents for vol- ume preserving C^{1} dffeomorphisms. In fact, in [BoV] was proved that:

Theorem 5.4.3. Let μ be the normalized Lebesgue measure on a compact manifold M. If f ∈ Diff^{1}(M) is a continuity point for the map

Diff_{μ}^{1}(M) → R^{d} g → (λ_{1}(g, μ), ..., λ_{d}(g; μ))

then, for almost every point x ∈ M

1. either all Lyapunov exponents λ_{i}(f; x) = 0 for 1 ⩽ i ⩽ d

2. or the Oseledets splitting of f is dominated on the orbit of x.

Many of these results can be put togethers substituting the robust transitivity by the stable ergodicity. In [BFP], it is proven that C^{1}− stable ergodic map in the conservative category are Bernoulli maps exhibiting dominated splitting.

More precisely, Let M be a compact manifold of dimension d ≥ 2, and let μ be a volume measure in M. Take α > 0 and let Diff_{μ}^{1+α}(M) be the set of μ-preserving C^{1+α} diffeomorphisms, endowed with the C^{1} topology. Let SE ⊂ Diff_{μ}^{1+α}(M) be the set of stably ergodic diffeomorphisms (i.e., the set of diffeomorphisms such that every sufficiently C^{1}-close C^{1+α} conservative diffeomorphism is ergodic).

Theorem 5.4.4. ([BFP]) There is open and dense set R ⊂ SE such that if f ∈ R then f is non-uniformly hyperbolic, that is, all Lyapunov exponents of f are non-zero. Moreover, every f ∈ R admits a dominated splitting.

Remark 5.4.1. It is not true that every stably ergodic diffeomorphism can be approximated by a partially hyperbolic system (in the weaker sense), by the examples in [BV] and proved in [T].

The proof of theorem 5.4.4 goes trough three steps:

1. A stably ergodic (or stably transitive) diffeomorphism f must have a dominated splitting. This is true because if it doesn’t, [BDP] permits to perturb f and create a periodic point whose derivative is the identity. Then, using the Pasting Lemma proved in [AM] (for which C^{1+α} regularity is an essential hypothesis), one breaks transitivity.

2. A result in [BB] (which is a refinement of a technique developed in [SW]) gives a perturbation of f such that the sum of the Lyapunov exponents “inside” each of the bundles of the (finest) dominated splitting is non-zero.

3. Using a result proved [BoV], we find another perturbation such that the Lyapunov exponents in each of the bundles become almost equal. (If we attempted to make the exponents exactly equal, we couldn’t guarantee that the perturbation is C^{1+α}.) Since the sum of the exponents in each bundle varies continuously, we conclude there are no zero exponents.

5.5 Robust transitivity and heterodimensional cycle.

In dimension higher than two, another kind of homoclinic bifurcation breaks the hyperbolicity: the so-called heterodimensional cycles (intersection of the stable and unstable manifolds of points of different indices, see [D1] and [D2]). More precisely:

Definition 5.5.1. Let M be and let f ∈ Diff^{r}(M) (r ⩾ 1). Let us assume that f has a pair of hyperbolic saddles P and Q with different indices, that is, different dimensions of their unstable subspaces. Assume W^{s}(P) and W^{u}(Q) have non-empty intersection, and the same for W^{u}(P) and W^{s}(Q). We say that f has a heterodimensional cycle associated to P and Q.

It follows immediately from the definition that heterodimensional cycles can only exist in dimension bigger or equal to 3.

In particular, the unfold of these cycles imply the existence of striking dynamics; the appearance of nonhyperbolic robust transitive sets is the more important and explosion of homoclinic classes (see theorem 5.5.1 below). Moreover it is possible to prove the following: Let f ∈ Diff^{1}(M) and Λ_{f}(U) = ∩_{{n∈ℤ}}g^{n}(U) be a maximal invariant nonhyperbolic C^{1}−robust transitive sets. Then, there is g C^{1}−close to f such that g exhibits a heterodimensional cycles contained in Λ_{g}(U) = ∩_{{n∈ℤ}}g^{n}(U).

In the same sense, these cycles play the role for the partial hyperbolic theory as transverse intersection play for the hyperbolic theory. In the paper [BDPR] is studied the interplay done between heterodimensional cycles and robust transitive systems in any dimension.

To show the richness of the heterodimensional cycles we will expose one theorem related to them. For simplicity, we suppose that the periodic points P and Q in the cycle are actually fixed points. In addition, we always assume: a)(codimension 1) the saddles P and Q have indices p and q = p + 1, respectively, b) (quasi-transversality) the manifolds W^{s}(P) and W^{u}(Q) intersect transversely, and the intersection between W^{u}(P) and W^{s}(Q) is quasi-transverse. The next theorem asserts that the homoclinic classes of P and Q often explode, and become intermingled (non-empty intersection) when the cycle is unfolded.

A parameterized family {f_{t}}_{{t∈[1;1]}} of diffeomorphisms unfolds generically a heterodimensional cycle of f = f_{0} if there are open disks K_{t}^{u} ∈ W^{u}(P_{t}) and K_{t}^{s} ∈ W^{s}(Q_{t}), depending continuously on t such that K_{0}^{u} ∩ K_{0}^{s} contains a point of quasi- transverse intersection, and the distance between K_{t}^{s} and K_{t}^{u} increases with positive velocity when t increases. Here P_{t} and Q_{t} denote the continuations for f_{t} of the periodic points P and Q.

Theorem 5.5.1. ([D1]) There is a non-empty open set of C^{1} parameterized families of diffeomorphisms {f_{t}}_{{t∈[1;1]}} unfolding generically a heteroclicnical cycle of f = f_{0} , such that for all small positive t

1. the transverse intersection between W^{s}(P_{t}) and W^{u}(Q_{t}) is contained in the homoclinic class of Q_{t} ;

2. and the homoclinic class of P_{t} is contained in the homoclinic class of Q_{t} .

Let d be the dimension of the ambient manifold M . The key fact behind previous theorem is that, for every small positive t, the (d − p)−dimensional manifold W^{s}(P_{t}) is contained in the closure of W^{s}(Q_{t}). This makes that the stable manifold of Q_{t} , which has dimension d−p−1, behaves in fact as a manifold of dimension 1 unit bigger. The proof of the theorem, relies on a reduction of the dynamics to a family of iterated functions systems on the interval.

6 Flows and Singular splitting.

For flows, a striking example is the Lorenz attractor [Lo], given by the solutions of the polynomial vector field in R^{3}:

X(x, y, z) = \{ \begin{matrix}ẋ = −αx + αy \\ ẏ = βx − y − xz \\ ż = −γz + xy , \end{matrix}  (5)

where α, β, γ are real parameters. This equation was derived by Lorenz from the works done by Saltzman ([Sa]) concerning thermal fluid convection. Numerical experiments performed by Lorenz (for α = 10, β = 28 and γ = 8/3 ) suggested the existence, in a robust way, of a transitive strange attractor toward which tends a full neighborhood of positive trajectories of the above system. That is, the strange attractor could not be destroyed by any perturbation of the parameters. Most important, the attractor contains an equilibrium point (0, 0, 0), and hence can not be hyperbolic. The work of Lorenz raised a number of mathematical questions that are among the leitmotifs in the development of the theory of dynamical systems.

Notably, only now, three and a half decades after this remarkable work, was it proved [Tu] that the solutions of (5) satisfy such a property for values α, β, γ near the ones considered by Lorenz.

However, already in the mid-seventies, the existence of robust nonhyperbolic attractors was proved for flows introduced independently in [ABS], [Gu], (see also [GW], [W]) which we now call geometric models for Lorenz attractors. In particular, they exhibit, in a robust way, an attracting transitive set with an equilibrium (singularity).

Moreover, the properties of this geometrical models, allow one to extract very complete dynamical information.

A natural question raises: is such features present for any robust transitive set of a flow?

In [MPP] a positive answer for this question is given:

Theorem 6.0.2. C^{1} robust transitive sets with singularities on closed 3-manifolds verifies:

1. there are either proper attractors or proper repellers;

2. the eigenvalues at the singularities satisfy the same inequalities as the corresponding ones at the singularity in a Lorenz geometrical model;

3. there are partially hyperbolic with a volume expanding central direction.

The presence of a singularity prevents these attractors from being hyperbolic. But they exhibit a weaker form of hyperbolicity named singular hyperbolic splitting. This class of vector fields contains the Axiom A systems, the geometric Lorenz attractors and the singular horseshoes in ([LP]), among other systems. Currently, there is a rather satisfactory and complete description of singular hyperbolic vector fields defined on 3dimensional manifolds (but the panorama in higher dimensions remains open). A first consequence from this result is that every orbit in any robust attractor has a direction of exponential divergence from nearby orbits (positive Lyapunov exponent). Another consequence is that robust attractors always admit an invariant foliation whose leaves are forward contracted by the flow, showing that any robust attractor with singularities displays similar properties to those of the geometrical Lorenz model. In particular, in view of the result of Tucker [Tu], the Lorenz attractor generated by the Lorenz equations much resembles a geometrical one.

Moreover, related to the partial hyperbolic structure for three dimensional flows, it is proved in a sequel of works that these sets are expansive, the periodic orbits are dense (in the case that the set is an attractor), and it has a spectral decomposition (see [PPV], [BM], [AP]).

On the other hand, for the case of flows, appears a new kind of bifurcation that leads to a new dynamics distinct from the ones for diffeomorphism: the so called singular cycles (cycles involving singularities and periodic orbits, see [BLMP], [Mo], [MP] and [MPP1] for examples of dynamics in the sequel of the unfolding of it). In particular, these cycles leads to the creation of robust singular hyperbolic sets as it is shown in the papers, [MP], and in the paper [MPP1]. Systems exhibiting this cycles are dense among open set of systems exhibiting a singular hyperbolic splitting.

Moreover, recently A. Arroyo and F. Rodriguez Hertz (see [AR]) studying the dynamical consequences of the dominated splitting for the Linear Poincare flow, proved the following:

Theorem 6.0.3. any three dimensional flow can be C^{1}−approximated either by a flow exhibiting an either an homoclinic tangency, or a singular cycle, or by a hyperbolic one.

6.1 Sketch of the proof of theorem 6.0.2.

In this section we discuss the theorem 9 stated before, in other words, we will explain the dynamical structure of compact transitive sets (there are dense orbits) of flows on 3-manifolds which are robust under small C^{1} perturbations.

To state in a precise way our results, let us fix some notations and recall some definitions and results proved elsewhere.

Throughout M is a boundaryless compact manifold and X^{r}(M) denotes the space of C^{r} vector fields on M endowed with the C^{r} topology, r ⩾ 1. If X ∈ X^{r}(M), X_{t}, t ∈ R, denotes the flow induced by X. A compact invariant set Λ of X is isolated if there exists an open set U ⊃ Λ, called isolating block, such that Λ = ⋂_{t∈R} X_{t}(U). If U above can be chosen such that X_{t}(U) ⊂ U for t > 0, we say that the isolated set Λ is an attracting set.

A compact invariant set Λ of X is transitive if it coincides with the ω-limit set of an X-orbit. An attractor is a transitive attracting set. A repeller is an attractor for the reversed vector field −X. An attractor (or repeller) which is not the whole manifold is called proper. An invariant set of X is nontrivial if it is neither a periodic point nor a singularity. With this concept we define:

An isolated set Λ of a C^{1} vector field X is robust transitive if it has an isolating block U such that

Λ_{Y} (U) = ⋂_{t∈R} Y_{t}(U)

is both transitive and nontrivial for any Y C^{1} close to X.

And related to robust transitive sets it is proved the following:

Theorem 6.1.1. A robust transitive set containing singularities of a flow on a closed 3-manifold is either a proper attractor or a proper repeller.

As a matter of fact, the previous result will follow from a general result on nmanifolds, n ⩾ 3, settling sufficient conditions for an isolated set to be an attracting set: (a) all its periodic points and singularities are hyperbolic and (b) it robustly contains the unstable manifold of either a periodic point or a singularity.

The previous result is false in dimension bigger than three; a counterexample can be obtained by multiplying the geometric Lorenz attractor by a hyperbolic system in such a way that the directions supporting the Lorenz flow be normally hyperbolic. It is false as well in the context of boundary-preserving vector fields on 3-manifolds with boundary [LP]. The converse to this Theorem is also not true: proper attractors (or repellers) with singularities are not necessarily robust transitive, even if their periodic points and singularities are hyperbolic in a robust way.

To motivate the next results about singularities of robust transitive sets for flows displaying singularities, recall that the geometric Lorenz attractor L is a proper robust transitive set with a hyperbolic singularity σ such that if λ_{i}, 1 ⩽ i ⩽ 3, are the eigenvalues of L at σ, then λ_{i}, 1 ⩽ i ⩽ 3, are real and satisfy λ_{2} < λ_{3} < 0 < −λ_{3} < λ_{1} [GW]. Inspired by this property we say that a singularity σ is Lorenz-like for X if the eigenvalues λ_{i}, 1 ⩽ i ⩽ 3, of DX(σ) are real and satisfy λ_{2} < λ_{3} < 0 < −λ_{3} < λ_{1}.

If σ is a Lorenz-like singularity for X then the strong stable manifold W_{X}^{ss}(σ) exists. Moreover, dim(W_{X}^{ss}(σ)) = 1, and W_{X}^{ss}(σ) is tangent to the eigenvector direction associated to λ_{2}. Given a vector field X ∈ X^{r}(M), we let Σ(X) be the set of singularities of X. If Λ is a compact invariant set of X we let Σ_{X}(Λ) be the set of singularities of X in Λ.

The next result shows that the singularities of robust transitive sets on closed 3manifolds are Lorenz-like.

Theorem 6.1.2. Let Λ be a robust singular transitive set of X ∈ X^{1}(M). Then, either for Y = X or Y = −X, every σ ∈ Σ_{Y} (Λ) is Lorenz-like for Y and satisfies W_{Y}^{ss}(σ) ∩ Λ = {σ}.

In light of these results, a natural question arises: can one achieve a general description of the structure for robust attractors for flows exhibiting singularities? In this direction we prove: if Λ is a robust attractor for X containing singularities then it is partially hyperbolic with volume expanding central direction. To state this result in a precise way, let us introduce some definitions and notations.

Let Λ be a compact invariant transitive set of X ∈ X^{r}(M) , c > 0, and 0 < λ < 1. We say that Λ has a (c, λ)-dominated splitting if the bundle over Λ can be written as a continuous DX_{t}-invariant sum of sub-bundles T_{Λ} = E^{s} ⊕ E^{cu}, such that for every T > 0, and every x ∈ Λ, we have

(a) E^{s} is one-dimensional

(b) The bundle E^{cu} contains the direction of X, and

∥DX_{T} /E_{x}^{s}∥.∥DX_{−T} /E_{X_{T} (x)}^{cu}∥ < c λ^{T} .

E^{cu}is called the central direction of T_{Λ}.

A compact invariant transitive set Λ of X is partially hyperbolic if Λ has a (c, λ)- dominated splitting T_{Λ}M = E^{s} ⊕E^{cu} such that the bundle E^{s} is uniformly contracting, that is, for every T > 0, and every x ∈ Λ, we have ∥DX_{T} /E_{x}^{s}∥ < c λ^{T} .

For x ∈ Λ and t ∈ IR we let J_{t}^{c}(x) be the absolute value of the determinant of the linear map DX_{t}/E_{x}^{cu} : E_{x}^{cu} → E_{X_{t}(x)}^{cu}. We say that the subbundle E_{Λ}^{cu} of the partially hyperbolic set Λ is volume expanding if J_{t}^{c}(x) ⩾ c e^{λt}, for every x ∈ Λ and t ⩾ 0 (in this case we say that E_{Λ}^{cu} is (c, λ)-volume expanding to indicate the dependence on c, λ). With all this definition in mind, we introduce then following one:

Definition 6.1.1. Singular hyperbolicity: Let Λ be a compact invariant transitive set of X ∈ X^{r}(M) with singularities. We say that Λ is a singular hyperbolic set for X if all the singularities of Λ are hyperbolic, and Λ is partially hyperbolic with volume expanding central direction.

Related to this, it is proved the following result.

Theorem 6.1.3. Robust attractors of X ∈ X^{1}(M) containing singularities are singular hyperbolic sets for X.

Now we will summarize the sketch of the proof of the previous statement.

In view of the previous results one can assume that every singularity of X is Lorenzlike and has (for instance) two negative eigenvalues. Consider the splitting with three directions defined over the set of regular closed orbits of the vector field, denoted by E^{ss}⊕[X]⊕E^{uu} , where E^{ss} and E^{uu} are the stable and unstable directions and X is the flow direction. For simplicity let us assume that the set Λ is the closure of the periodic orbits of X. The objective is to extend this splitting to the closure of the regular closed orbits to get a dominated splitting. In general this is not possible, due to the presence of the equilibria; in fact for periodic orbits getting close to the singularities, the angle between the flow direction and the unstable direction is getting small. Thus it is not possible to extend this splitting in a dominated way to the whole Λ. The idea now is to consider the splitting E^{ss} ⊕ E^{c} , where E^{c} = [X] ⊕ E^{uu}, defined over the periodic orbits and extend it to its closure (that is the whole set Λ). For that it is enough to prove that such a splitting is dominated. To obtain the domination one proves that given any small neighborhood V of the finite set of singularities of Λ and any point p in a regular closed orbit that does not meet this vicinity it holds that the angle between E^{ss}(p) and E^{c}(p) is uniformly bounded from below. Otherwise, after a perturbation it can be created a repeller or a sink contradicting the robust transitivity. On the other hand, the analysis near the singularity is the following. Since all the singularities have one dimensional unstable manifolds and are Lorenz-like, there is defined a splitting E^{ss} ⊕ E^{s} ⊕ E^{uu}. Then it is proved that E^{ss}(p) → E^{ss}(σ) and E^{c}(p) → E^{s}(σ) ⊕ E^{uu}(σ) as p approaches the singularity σ. If it is not the case, it is proved after two turns nearby the singularities, by an small perturbation is created either a sink or a repeller contradicting again the robust transitivity. From the two previous facts follows that the angles between E^{ss} and E^{c} are persistently bounded away from zero and as in the case of diffeomorphisms, this allows to prove the domination of the splitting E^{ss} ⊕ E^{c}.

6.2 Dynamical consequences of singular hyperbolicity.

We shall list some of these properties that give us a nice description of the dynamics of robust transitive sets with singularities, and in particular, for robust attractors.

The first two properties do not depend either on the fact that the set is robust transitive or an attractor.

Lemma 6.2.1. Let Λ be a singular hyperbolic compact set of X ∈ X^{1}(M). Then any invariant compact set Γ ⊂ Λ without singularities is a hyperbolic set.

Recall that, given x ∈ M, and v ∈ T_{x}M, the Lyapunov exponent of x in the direction of v is

γ(x, v) = lim_{t→∞} infimum \frac{1}{t} log ∥DX_{t}(x)v∥ .

We say that x has positive Lyapunov exponent if there is v ∈ T_{x}M such γ(x, v) > 0. The next two results show that important features of hyperbolic attractors and of

the geometric Lorenz attractor are present for singular hyperbolic attractors, and so, for robust attractors with singularities:

Lemma 6.2.2. A singular hyperbolic attractor Λ of X ∈ X^{1}(M) has uniform positive Lyapunov exponent at every orbit.

Currently, there is a rather satisfactory and complete description of singular hyperbolic attractors on 3-dimensional manifolds (again, the panorama in higher dimensions remains open). In [AP] it is proved that for transitive singular hyperbolic attractors the set of periodic orbits are dense; and more, it is made of a unique homoclinic class. For that it is obtained a theorem of existence of unstable manifolds, not for the whole set but for a subset which is visited infinitely many times by a residual subset of the attractor. Such theorem has an important consequence that is a characterization of C^{1}- robustly transitive attractors with singularities.

More precisely, the Theorem 6.2.1 stated below, assert that even in the presence of equilibrium points in a singular hyperbolic attractor there are local unstable manifolds of uniform size, not for the whole attractor but for a subset of it which is visited infinitely many times by points in a residual subset of it. As a first consequence it is obtained the following theorem about the set of periodic orbits. Denote by Per(Λ) the set of periodic orbits of Φ_{t} in Λ.

Theorem 6.2.1. Let Λ be a transitive singular hyperbolic attractor for the flow Φ_{t}, then cl(Per(Λ)) = Λ.

A easy corollary is the following

Theorem 6.2.2. Let Λ be a transitive singular hyperbolic attractor for the flow Φ_{t}, then there is a periodic orbit p such that cl(H(p)) = Λ.

Although there are examples of singular hyperbolic sets which do not have any periodic orbit at all (see [MoII]). However, such sets are not attractors neither robust.

The theorem 6.2.1 is useful also in the context to obtain statistical properties (recall tha definition of SRB measure given in the volume 1A of the present Handbook). In fact, [Co] was proven the following:

Theorem 6.2.3. if Λ is a singular hyperbolic transitive attractor of a C^{1+α} vector field, α > 0, with dense periodic orbits then it has an SRB measure.

Therefore, as a consequence of this result and theorem 6.2.1 follows that:

Corolary 6.2.1. If Λ is a singular hyperbolic transitive attractor of a C^{1+α} vector field, α > 0, it has an SRB measure.

Few words about the proof of theorem 6.2.1.

Observe that for a singular hyperbolic splitting, the integrability of the strong stable bundle allow us to find stable manifolds of uniform size ε_{s} > 0 on any point of U; that is, for any x ∈ U there is a C^{1}-interval say W_{ε_{s}}^{s}(x) that any point y on it happens that d(Φ_{t}(x), Φ_{t}(y)) → 0, when t → ∞. In the case of a singularity it correspond to the stable manifold associated to the strongest contracting eigenvalue: W_{loc}^{ss}(σ).

For unstable manifolds such construction may be impossible on regular points. However, associated to any regular point x ∈ Λ there is a family of 2-dimensional sections N_{x_{t}} whose size depends on the point x_{t} and which are transversal to the flow; here x_{t} := Φ_{t}(x). According to this section we can write the family of holonomy maps between these transversal sections, for any t ∈ ℝ:

G_{x}^{t} : Dom(G_{x}^{t}) ⊂ N_{x} ⟶ N_{x_{t}}

Consider some ε > 0. The unstable manifold of size ε of a regular point x ∈ Λ is

W̃_{ε}^{u}(x) = {y ∈ M|y ∈ Dom(G_{x}^{−t}) and dist(G_{x}^{−t}(y), x_{−t}) → 0, t → ∞}

Of course, W̃_{ε}^{u}(x) ⊂ N_{x}. On the other hand, there is some ε̃ > 0 for which the central unstable manifold exists on any point x of a singular hyperbolic set Λ; according to [HPS]. Denote these central manifolds by W cu ; ε̃(x). Also denote by L(Y) the limit set of Y. The main goal of this paper is the following

Now we can state the theorem about suitable unstable manifold for a singular hyperbolic attractor.

Theorem 6.2.4. Let X ∈ X^{1}(M), Λ ⊂ M be a transitive singular hyperbolic attractor and U ⊃ Λ an open neighborhood contained in its basin of attraction. Then there is a neighborhood U(X) ⊂ X^{1}(M) such that for all Y ∈ U(X) there is a subset K(Y) ⊂ Λ_{Y} := ∩_{t⩾0}Y_{t}(U), ε_{u} > 0 and λ_{u} < 0 that

1. For any y ∈ K(Y), we have that W_{ε_{u}}^{cu}(y) ∩ N_{y} = W̃_{ε_{u}}^{u}(y).

2. For any y ∈ Ŵ_{ε_{u}}^{u}(x) there is a not bounded sequence t_{i} > 0 such that

dist(G_{x}^{−t_{i}}(y), x_{−t_{i}}) < C exp(t_{i}λ_{u}).

3. ⋃_{t>T_{0}} ⋃_{y∈K} Φ_{−t}(y) is an open and dense set in Λ_{Y} ∩ L(Y), for any T_{0} > 0.

In order to prove this theorem first is found a system of transversal sections associated to the passage through a neighborhood of each equilibrium points. Among the systems of transversal sections, some of them present an induced map with a kind of Markovian property and uniform expansion. However, to guarantee the existence of such transversal sections it is needed a deep analysis of the dynamic inside a neighborhood of singularities and also the combinatorics between them. More precisely, it is necessary to study the local holonomy maps between transversal sections in general.

References

[ABS] V. Afraimovich, V. Bykov, and L. Shil’nikov. On the appearance and structure of the Lorenz attractor. Dokl. Acad. Sci. USSR, 234 336–339, 1977.

[ABV] J. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398.

[AM] A. Arbieto and C. Matheus. A pasting lemma I: the case of vector fields. Preprint IMPA 2003.

[AP] A. Arroyo, E. Pujals; Dynamical Properties of Singular Hyperbolic Attractors. preprint, server IMPA.

[AR] A. Arroyo, F. Rodriguez Hertz, Homoclinic bifurcations and uniform hyperbolicity for threedimensional flows. Ann. Inst. H. Poincar Anal. Non Linaire 20 (2003), no. 5, 805–841.

[AS] R. Abraham, S. Smale, Nongenericity of Axiom A and Ω-stability, Global analysis-Proc. Symp.in Pure Math. Vol 14, AMS, Providence, R.I., (1970), 5-8.

[B] J. Bochi. Genericity of zero Lyapunov exponents. Ergod. Th. Dynam. Sys. 22 (2002), 1667– 1696.

[BoV] J. Bochi and M. Viana. The Lyapunov exponents of generic volume preserving and symplectic maps. Annals of Math., to appear.

[BB] C. Bonatti and A. T. Baraviera. Removing zero Lyapunov exponents. Ergod. Th. Dynam. Sys., 23 (2003), 1655-1670.

[BoC] C. Bonatti and S. Crovisier. Recurrence and genericity. C. R. Math. Acad. Sc 336 (2003), no. 10, 839–844.

[BD2] C. Bonatti, L. Daz, Connexions htroclines et gnricit d’une infinit de puits et de sources. Ann. Sci. cole Norm. Sup. (4), 32 (1999), no. 1, 135–150.

[BL] C. Bonatti, R. Langevin, Un exemple de flot d’Anosov transitif transverse un tore et non conjugu une suspension. Ergodic Theory Dynam. Systems 14 (1994), no. 4, 633–643.

[BM] S. Bautista, C. Morales. Existence of periodic orbits for singular-hyperbolic attractors, Preprint IMPA.

[BMP] S. Bautista, C. A. Morales, M. J. Pacífico. There is no spectral decomposition for singularhyperbolic flows. Preprint IMPA. (2004)

[BP] M. I.Brin, Ja. B. Pesin, . Partially hyperbolic dynamical systems. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212.

[BuDP] K. Burns, D. Dolgopyat, and Ya. Pesin. Partial hyperbolicity, Lyapunov exponents, and stable ergodicity. J. Statist. Phys. 108 (2002), 927–942.

[BPS] V. Baladi, E.R. Pujals, M. Sambarino, Dynamical zeta functions for analytic surface diffeomorphisms with dominated splitting, preprint.

[BC] M. Benedicks, L. Carleson, The dynamics of the Hénon map, Annals of Math 133 (1991), 73-169.

[BD] C. Bonatti, L. J. Diaz, Persistence of transitive diffeomorphisms, Annals of Math 143 (1995), 367-396.

[BDP] C. Bonatti, L. J. Diaz, E. R. Pujals, A C^{1}−generic dichotomy for diffeomorphisms: weak form of hyperbolicity or infinitely many sinks or sources, Annals of Math, to appear.

[BDPR] C. Bonatti, L. J. Diaz, E. R. Pujals, J. Rocha, Heterodimensional cycles and robust transitivity, Atherisques, to appear.

[BDU] C. Bonatti, L.J. Daz, R. Ures; Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms. J. Inst. Math. Jussieu 1 (2002), 4, 513–541.

[BFP] J. Bochi, B. Fayad, E. R. Pujals, Dichotomy for conservative robust ergodic maps, in preparation.

[BLMP] R. Bamon, R. Labarca, R. Mañ é, M. Pacifico, The explosion of singular cycles, Publ. Math. IHES, 78, 207-232, 1993.

[BV] C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.

[Cas] A. Castro. Backward inducing and exponential decay of correlations for partially hyperbolic attractors Preprint (2000)

[C] E. Colli, Infinitely many coexisting strange attractors, to appear in Annales de l’Inst. Henri Poincaré, Analyse Nonlinéaire.

[Co] W. Colmenares. PhD Thesis UFRJ (Universidade Federal do Rio de Janeiro), 2002.

[CY] Cowieson, B.; Young, L-S.; SRB measures as zero noise limits, Talk at Conference on Partially Hyperbolicity, Northwestern University, May 2001.

[D1] L. J. Diaz Robust nonhyperbolic dynamics at heterodimensional cycles. Ergodic Theory and Dynamical Systems, 15 (1995), 291-315.

[D2] L.J.Diaz Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations. Nonlinearity, 8 (1995), 693-715.

[Do] C. I. Doering, Persistently transitive vector fields on three-dimensional manifolds. Dynamical systems and bifurcation theory (Rio de Janeiro, 1985), 59–89,

[DPU] L. J. Diaz, E. R. Pujals, R. Ures, Partial hyperbolicity and robust transitivity, Acta Mathematica, 183 (1999), 1-43.

[DP] D. Dolgopyat and Ya. Pesin. Every compact manifold carries a completely hyperbolic diffeomorphism. Ergod. Th. Dynam. Sys. 22 (2002), 409–435.

[FP] L. Florit, E. Pujals, Partial hiperbolic system and N-central forces in celestial mechanics, in preparation.

[Fr] J. Frank, Necessary conditions for the stability of diffeomorphisms, Trans. A. M. S. 158 (1971), 295-329.

[Gu] J. Guckenheimer. A strange, strange attractor. In J.E. Marsden and M.McCracken, editors, The Hopf bifurcation theorem and its applications, pages 368–381. Springer Verlag, 1976.

[GW] J. Guckenheimer, R. Williams. Structural stability of lorenz attractors. Publ. Math. IHES, 50, 50-591979.

[HPS] M. Hirsch, C. Pugh, M. Shub, Invariant manifolds, Springer Lecture Notes in Math., 583 (1977).

[HY] H. Hu, Lai-Sang Young. Nonexistence of SBR measures for some diffeomorphisms that are almost Anosov, Ergod. Th. and Dynam. Syst., 15, 1995, 67-76.

[K] V. Kaloshin, An extension of the Artin-Mazur theorem. Ann. of Math. 2 150 , 729–741, 1999.

[KP] V. Kaloshin, E. R. Pujals, Superexponential growths of periodic orbits in heteroclinic cyles, in preparation.

[Ko] M. Komuro, Expansive properties of Lorenz attractors. The theory of dynamical systems and its applications to nonlinear problems (Kyoto, 1984), 4–26, World Sci. Publishing, Singapore, 1984.

[L] J. Lewowicz, Persistence in expansive systems. Ergodic Theory Dynam. Systems 3 (1983), 4, 567–578

[Lo] E. N. Lorenz. Deterministic nonperiodic flow. J. Atmosph. Sci., 20 130–141, 1963.

[LP] R. Labarca, M.J. Pacifico. Stability of singular horseshoes Topology, 25 337–352, 1986.

[M] R. Mañé, Contributions to the stability conjecture. Topology, 17 (1978), 386-396.

[M1] R. Mañé, Hyperbolicity, sinks and measure in one-dymensional dynamics. Comm. Math. Phys., 100, 495-524, 1985.

[M2] R. Mañé, An ergodic clossing lemma, Ann. of Math. 116 (1982), 503-540.

[Mo] C. Morales, Lorenz attractors through Saddle-node bifurcations Ann. Inst. henri Poincare (An. nonlin), 13 589-617, 1996.

[MoII] C. Morales, A note on periodic orbits for singular-hyperbolic flows. Discrete Contin. Dyn. Syst. 11 (2004), no. 2-3, 615–619.

[MoP] C. A. Morales, M. J. Pacifico, Attractors and singularities robustly accumulated by periodic orbits. International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), 64–67, World Sci. Publishing, River Edge, NJ, 2000

[MP] C. Morales, E. R. Pujals, Singular strange attractors acroos the boundary of hyperboilic systemes, Comm. Math. Phys., 211, 527-558, 1997.

[MPP] C. Morales, M. J. Pacifico, E. R. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Annals of mathematics, to appear.

[MPP1] C. Morales, M. J. Pacifico, E. R. Pujals, Strange attractors across the boundary of hyperbolic systems Comm. Math. Phys., 211, 527-558, 2000.

[MPS] R. Markarian, E. Pujals, M. Sambarino Dissipative planar billiards, in preparation.

[MV] L. Mora, M. Viana, Abundance of strange attractors, Acta Math., 171 (1993), 1-71.

[MY] C. G. Moreira, J-C. Yoccoz, Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2) 154 (2001), no. 1, 45–96.

[MPV] C. G. Moreira, J.Palis, M. Viana Homoclinic tangencies and fractal invariants in arbitrary dimension. C. R. Acad. Sci. Paris Sr. I Math. 333 (2001), no. 5, 475–480.

[N1] S. Newhouse, Non-density of Axiom A(a) on S^{2}, Proc. A.M.S. Symp. Pure Math. 14 (1970), 191-202.

[N2] S. Newhouse, Diffeomorphism with infinitely many sinks, Topology 13 (1974), 9-18.

[N3] S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. I.H.E.S. 50 (1979), 101-151.

[N4] S. Newhouse, Hyperbolic limit sets, Trans. A.M.S. 167 (1972), 125-150.

[OU] J. C. Oxtoby and S. M. Ulam. Measure-preserving homeomorphisms and metrical transitivity. Annals of Math. 42 (1941), 874–920.

[P1] J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, to appear in Astérisque.

[PT2] J. Palis, F. Takens, Hyperbolicity and the creation of homoclinic orbits. Ann. of Math. 125 (1987), no. 2, 337–374.

[P2] J. Palis, Homoclinic orbits, hyperbolic dynamics and dimension of Cantor sets. The Lefschetz centennial conference. em Contemp. Math., 58, III, (1984) 203–216,

[P3] J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors. Gomtrie complexe et systmes dynamiques (Orsay, 1995). Astrisque 261, (2000), 335–347.

[Pl] V. A. Pliss, On a conjecture due to Smale, Diff. Uravnenija, 8 (1972), 268-282.

[PPV] M. J. Pacifico, E. R. Pujals, M. Viana, Singular hyperbolic systems are sensitive to initial data,preprint

[PRS] E.R. Pujals, F. Rodriguez Hertz and M. Sambarino, Characterization of Partial hyperbolic system on the three dimensional Torus.

[PS1] E. R. Pujals, M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of math 151 (2000), 961-1023.

[PS2] E. R. Pujals, M. Sambarino, On homoclinic tangencies, hyperbolicity, creation of homoclinic orbits and variation of entropy, Nonlinearity 13 (2000) 921-926.

[PS3] E. R. Pujals, M. Sambarino, On the dynamic of dominated splitting, preprint.

[PS4] E.R. Pujals, M. Sambarino, Codimension one dominated splittings, preprint.

[PS5] E. R. Pujals, M. Sambarino, Sufficient condition for robustly mixing foliations, preprint.

[PT] J.Palis and F.Takens Hyperbolicity and sensitive-chaotic dynamics at homoclinic bifurcations Cambridge University Press, 1993.

[PV] J.Palis, M. Viana High dimension diffeomorphisms displaying infinitely many periodic attractors. Ann. of Math. (2) 140, (1994), no. 1, 207–250.

[PY] J. Palis, J-C. Yoccoz, On the arithmetic sum of regular Cantor sets. Ann. Inst. H. Poincar Anal. Non Linaire 14 (1997), 4, 439–456.

[PY2] J. Palis, J-C. Yoccoz, Fers cheval non uniformment hyperboliques engendrs par une bifurcation homocline et densit nulle des attracteurs. C. R. Acad. Sci. Paris Sr. I Math., 333, (2001), no. 9, 867–871.

[Pu1] C. Pugh, The closing lemma. Amer. J. Math. 89 1967 956–1009

[Pu2] C. Pugh, An improved closing lemma and a general density theorem. Amer. J. Math. 89 1967, 1010–1021.

[PuSh] C. Pugh, M. Shub, Stable ergodicity and julienne quasi-conformality. J. Eur. Math. Soc. (JEMS) 2 (2000), no. 1, 1–52.

[R] N, Romero, Persistence of homoclinic tangencies in higher dimensions. Ergodic Theory Dynam. Systems, 15 (1995), no. 4, 735–757.

[Ru] D. Ruelle Zeta functions for expanding maps and Anosov flows 1976 Invent. Math. 34 231– 242.

[Ru1] H.H. Rugh The correlation spectrum for hyperbolic analytic maps Nonlinearity 5 (1992) 1237– 1263

[Ru2] H.H. Rugh Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems Ergodic Theory Dynam. Systems 16 (1996) 805–819.

[Sa] B. Saltzman. Finite amplitude free convection as an initial value problem. J. Atmos. Sci 19, 329-341, 1962.

[S] S. Smale, Structurally stable systems are not dense, Amer. J. Math 88 (1966), 491-496.

[Sh] M. Shub Topologically transitive diffeomorphism of T^{4}, in Symposium on Differential Equations and Dynamical Systems (University of Warwick, 1968/69), pp. 39-40. Lecture Notes in Math.. 206. Springer-Verlag, Berlin-NewYork, 1971.

[SW] M. Shub and A. Wilkinson. Pathological foliations and removable zero exponents. Invent. Math. 139 (2000), 495–508.

[T] A. Tahzibi. Stably ergodic diffeomorphisms which are not partially hyperbolic. PhD Thesis IMPA, 2002.

[Tu] W. Tucker. The Lorenz attractor exists. C. R. Acad. Sci. Paris, 328 1197–1202, 1999.

[U] R. Ures. Abundance of hyperbolicity in the C^{1} topology. Ann. Sci. cole Norm. Sup. 4, 28, no. 6, 747–760, 1995.

[We] L. Wen. Homoclinic tangencies and dominated splittings. Nonlinearity 15 (2002), no. 5, 1445– 1469 [W] R. Williams. The structure of the Lorenz attractors. Publ math. IHES, 50, 73-99, 1979.

E. R. Pujals IMPA Rio de Janeiro, R. J. , Brazil enrique@impa.br

M. Sambarino IMERL Montevideo, Uruguay samba@fing.edu.uy