SUFFICIENT CONDITIONS FOR ROBUSTLY MINIMAL FOLIATIONS

ENRIQUE R. PUJALS AND MARTÍN SAMBARINO

1. Introduction

In the theory of differentiable dynamical systems, it is always an important problem to recognize which dynamic feature of a system it is also present in all nearby systems (with respect to some topology) and which are the conditions on the initial system that guarantee this fact. In other words, we look for a dynamic property of a particular system which is robust (or stable) under perturbations. And, as a guiding principle, a robust property should be reflected into some property of the tangent map, that is, we look for a property on the tangent map that guarantee that some phenomena is robust.

There are many examples of the situation above. In particular, to start focussing on our goal, let us mention the robust transitivy: there are many examples of transitive systems that remains also transitive under C^{r} perturbations. The first one was given by Shub [Sh] on the torus 𝕋^{4}. An example by Mañé [M] followed in the torus 𝕋^{3}. Bonnatti and Diaz [BD] give a general geometric construction that leads to robustly transitive systems. All these examples are partially hyperbolic systems (although some new examples [BV] were shown to exhibit just dominated splitting): the tangent bundle T M splits into three invariant subbundles E^{ss}⊕E^{c}⊕E^{uu} where vectors in E^{ss} are forward contracted, vectors in E^{uu} are backward contracted, and vectors in E^{c} are less contracted than any vector in E^{ss} and less expanded than vectors in E^{u}. The distributions E^{ss} and E^{uu} of a partially hyperbolic system are always uniquely integrable and leads to foliations on the manifold (i.e. a partition of the manifold into C^{r} leaves) called the (strong) stable and unstable foliations.

We will show that the mentioned examples shared a strongest property: not only they are robustly transitive but also (at least) one of these foliation is robustly minimal, that is, every leaf is dense in the manifold. 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.

In this paper we are concerned with partially hyperbolic systems having one, say the stable one, foliation minimal and we will give sufficient conditions in order to this foliation remains minimal under C^{r}- perturbations.

Let us be more precise. A diffeomorphism f : M → M is said to be partially hyperbolic provided the tangent bundle split into three nontrivial sub-bundles T M = E^{ss}⊕E^{c}⊕E^{uu} which are invariant under the tangent map Df and there are numbers 0 < λ < μ < 1 such that ∀ x ∈ M

∥Df_{/E^{ss}(x)}∥ < λ, ∥Df_{/E^{uu}(x)}^{−1}∥ < λ, μ < ∥Df_{/E^{c}(x)}^{−1}∥, ∥Df_{/E^{c}(x)}∥ < μ^{−1}

It is well known (see section 2) 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. One of these foliations is said to be minimal provided every leaf of this foliation is dense in M.

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).

Definition 1.1. Let f : M → M be a partially hyperbolic diffeomorphism. We say that F^{ss}(f) is robustly minimal if there exist a C^{1} neighbourhood U(f) such that F^{ss}(g) is minimal for every diffeomorphism g ∈ U(f).

It is not difficult to see that if (for instance) F^{ss}(f) is robuslty minimal then f is robustly transitive, i.e., ever diffeomorphism C^{1} close to f is transitive.

Next, we will define the key property that guarantee the robustness of stable foliation of a partially hyperbolic diffeomorphism: Some Hyperbolicity on the central distribution E^{c} at some points. Before, 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 1.2 (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, we 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 behaviour 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).

Theorem A: 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 we have that if f satisfy the conditions of Theorem A then it is robustly transitive. The proof of Theorem A will be given in section 4.

We will use our theorem to reconstruct the examples by Shub and Mañé (see sections 5 and 6) and showing that indeed one of the strog foliations is robustly minimal. It can be proven also that the examples by Bonnati and Diaz also satisfy the property SH but to do so we have to go into the details of their geometric construction and this exceeds the purpose of this paper. Let us end this introduction by asking if a partially hyperbolic diffeomorphism robustly transitive must exhibit the property SH (at least generically). In other words, is the property SH a necessary condition for robustly transitive partially hyperbolic systems?

2. Preliminaries

In this section we recall some well-known results regarding partially hyperbolic systems (that we’ve already mentioned them in the introduction). We refer to [HPS] for a general background on the topics we will review.

As we said in the Introduction, a diffeomorphism f : M → M is partially hyperbolic provided the tangent bundle split into three nontrivial sub-bundles T M = E^{ss} ⊕ E^{c} ⊕ E^{uu} which are invariant under the tangent map Df and there are 0 < λ < μ < 1 such that ∀ x ∈ M

∥Df_{/E^{ss}(x)}∥ < λ, ∥Df_{/E^{uu}(x)}^{−1}∥ < λ
μ < ∥Df_{/E^{c}(x)}^{−1}∥, ∥Df_{/E^{c}(x)}∥ < μ^{−1}

Lemma 2.0.1. Let f ∈ Diff^{r}(M) be a partially hyperbolic diffeomorphism. Then there exist a C^{r} neighoburhood fo f, say U, 0 < λ < λ_{1} < μ_{1} < μ < 1 and continuous functions E^{ss} : U → C(M, T M), E^{c} : Diff(M) → C(M, T M) and E^{uu} : U → C(M, T M) such that for any g ∈ U and x ∈ M we get that:

(1) T M = E_{g}^{ss} ⊕ E_{g}^{c} ⊕ E_{g}^{uu}, this decomposition is invariant under Dg and no one of these subbundles is trivial.

(2) ∥Dg_{/E^{ss}(x)}∥ < λ_{1}, ∥Dg_{/E^{uu}(x)}^{−1}∥ < λ_{1}

(3) μ_{1} < ∥Dg_{/E^{c}(x)}^{−1}∥, ∥Dg_{/E^{c}(x)}∥ < μ_{1}^{−1}

The sub-bundles E_{g}^{ss} and E_{g}^{uu} are uniquely integrable and form two foliations F^{ss} and F^{uu}. Theorem 2.1. Let U be as in the lemma above. Then, for each g ∈ U there are two partitions F^{ss}(g) and F^{uu}(g) of M such that for each x ∈ M the element of the partitions that contains x, denoted by F^{ss}(x, g) and F^{uu}(x, g), are C^{1} submanifolds (called leaves) such that T_{x}F^{ss}(x, g) = E_{g}^{ss}(x) and T_{x}F^{uu}(x.g) = E_{g}^{uu}(x). These submanifolds depends continuously (on compact subsets) on x ∈ M and g ∈ U.

These submanifolds F^{ss}(x, g) and F^{uu}(x, g) inherit the riemannian metric on M. We shall denote by F_{r}^{ss}(x, g) (respectiv. F_{r}^{uu}(x, g)) the ball in F^{ss}(x, g) (respect F^{uu}(x, g)) of radius r centered at x.

The sub-bundle E^{cu} = E^{c} ⊕ E^{uu} (called center-unstable) is not (in general) integrable. However, we can choose a continuous family of locally invariant manifolds tangent to it. Let dimE^{cu} = l and denote by I_{ϵ} the ball of radius ϵ in ℝ^{l}. Lemma 2.0.2. Let U be as in Lemma 2.0.1. There exists a continuous map Φ : M × U → Emb_{1}(I_{1}, M) such that, if we set W_{ϵ}^{cu}(x, g) = Φ(x, g)I_{ϵ}, then

(1) W_{ϵ}^{cu}(x, g) ∩ W_{ϵ}^{cu}(y, g) is relatively open on both sets.

(2) T_{x}W_{ϵ}^{cu}(x, g) = E^{cu}(x, g).

(3) Given ϵ > 0 there exists r = r(ϵ) such that g^{−1}(W_{r}^{cu}(x, g)) ⊂ W_{ϵ}^{cu}(g^{−1}(x), g).

For the sake of simplicity we shall identify W_{ϵ}^{cu}(x, g) with the ball of radius ϵ in W_{1}^{cu}(x, g). Lemma 2.0.3. Let U be as in the above lemmas. Given 0 < λ < λ_{1} < 1 there exists r_{0} such that if g ∈ U and x ∈ M satisfy

∏_{j=0}^{n} ∥Dg_{/E^{cu}(g^{−j}(x))}^{−1}∥ < λ^{n}, 0 ≤ n ≤ m

then g^{−m}(W_{r_{0}}^{cu}(x, g)) ⊂ W_{λ_{1}^{m}r_{0}}^{cu}(g^{−m}(x), g).

Proof. Let c > 0 be such that cλ < λ_{1} and let ϵ be such that if dist(z, y) < ϵ then

(1)  c_{−1} < \frac{∥Dg_{/E^{cu}(z)}^{−1}∥}{∥Dg_{/E^{cu}(y)}^{−1}∥} < c.

Let ϵ be such that if z ∈ W_{ϵ_{0}}^{cu}(y, g) then dist(z, y) < ϵ. Let r_{0} = r(ϵ_{0}) from Lemma 2.0.2. It follows that g^{−1}(W_{r_{0}}^{cu})(x, g) ⊂ W_{ϵ_{0}}^{cu}(g^{−1}(x), g). On the other hand, by (1), we conclude that

g^{−1}(W_{r_{0}}^{cu})(x, g) ⊂ W_{λ_{1}r_{0}}^{cu}(g^{−1}(x), g) ⊂ W_{r_{0}}^{cu}(g^{−1}(x), g).

Arguing by induction up to m the result follows.

□

3. Robustness of property SH

In this section we prove that the property SH persist under slight perturbations.

Theorem 3.1. Let f ∈ Diff^{r}(M) be a partially hyperbolic diffeomorphisms exhibiting property SH. Then, there are U(f), C′ > 0 and 1 < σ such that for any g ∈ U follows that for any x ∈ M there exist, y^{u} ∈ F_{1}^{uu}(x, g) verifying

m{Dg_{/E^{c}(g^{m}(y^{u}))}^{n}} > C′σ^{n} for any n > 0, m > 0.

Proof. Observe that to prove the theorem is enough to find U and k_{0} such that for any x ∈ M there exist, y^{u} ∈ F_{2}^{uu}(x, g) verifying

(2)  m{Dg_{/E^{c}(g^{k_{0}m}(y^{u}))}^{k_{0}n}} > 2^{n} for any n > 0, m > 0.

Changing f by a power of itself, we can assume that there is λ_{0} > 1 such that for any x ∈ M there exists y^{u} ∈ F_{1}^{uu}(x) such that

m{Df_{/E_{f^{m}(y^{u})}^{c}}^{n}} > λ_{0}^{n}

for any n > 0, m > 0. We define the set H_{λ_{0}}^{u}(f) as

(3) H_{λ_{0}}^{u}(f) = {y ∈ M : m{Df_{/E_{f^{m}(y)}^{c}}^{n}} ≥ λ^{n} for n > 0, m > 0}

Observe that this set is closed and F_{1}^{uu}(x) ∩ H_{λ_{0}}^{u}(f) is not empty. Notice that there exist λ_{1}, 1 < λ_{1} < λ_{0}; ϵ_{0} > 0 and U_{0}(f) such that:

(4)

if x ∈ H_{λ_{0}}^{u}(f), y ∈ M and g ∈ U_{0}(f)
satisfy dist(f^{i}(x), g^{i}(y)) < ϵ_{0} for 0 ≤ i ≤ n

then m{Dg_{/E^{c}(g^{j}(y))}^{k}} > λ_{1}^{k} for 0 ≤ j ≤ n and 0 ≤ k ≤ n − j.

Next, we choose a positive integer m_{0} such that for any g ∈ U_{0} and x ∈ M follows that

(5)  F_{2}^{uu}(g^{m_{0}}(x), g) ⊂ g^{m_{0}}(F_{ϵ_{0}/2}^{uu}(x, g)).

And afterwards, we consider n_{0} such that for any g ∈ U_{0} and z ∈ M it follows that

(6)  λ_{1}^{n_{0}} sup{m{Dg_{/E_{z}^{c}}^{m_{0}}} : z ∈ M} > 2.

Now, given ϵ_{0} from (4) and n_{0} form (6) we take ϵ_{1} and U_{1} ⊂ U_{0} such that for any g ∈ U_{1} we have:

(7) if dist(x, y) < ϵ_{1} then dist(g^{i}(z), f^{i}(y)) < ϵ_{0}/2 for 1 ≤ i ≤ n_{0}.

Moreover, we can choose U_{2} ⊂ U_{1} such that

(8) if g ∈ U_{2} then dist(F_{1}^{uu}(x, g), H_{λ_{0}}(f)) < ϵ_{1} for any x ∈ M.

Set k_{0} = m_{0} + n_{0} where m_{0} and n_{0} are as in (5) and (6) respectively. We will prove that for g ∈ U = U_{2} and k_{0} we have (2) for some y^{u} ∈ F_{2}^{uu}(x, g) and any x ∈ M. In order to find such point y^{u} we will proceed by induction to construct sequences {z_{n}^{u}}_{n≥0} and {x_{n}}_{n≥0} such that

(1) z_{j+1}^{u} ∈ F_{1}^{uu}(g^{k_{0}}(z_{j}^{u}), g) for 0 ≤ j ≤ n − 1.

(2) x_{j} ∈ H_{λ_{0}}^{u}(f).

(3) dist(z_{j}^{u}, x_{j}) < ϵ_{1}.

Assume for the moment that we have already constructed such sequences and set y_{n}^{u} = g^{−k_{0}n}(z_{n}^{u}). Let us show that y_{n}^{u} ∈ F_{2}^{uu}(x, g) and for any 0 ≤ j ≤ n, 0 < i ≤ n − j we have

(9)  m{Dg_{E_{g^{k_{0}j} (y^{u}_{n})}^{c}}^{k_{0}i}} > 2^{i}.

First observe that, from (5), g^{−k_{0}}(z_{n}^{u}) ∈ F_{ϵ_{0}/2}^{uu}(z_{n−1}^{u}) ⊂ F_{2}^{uu}(g^{k_{0}}(z_{n−2}^{u})). By induction, it follows that g^{−k_{0}(j−1)}(z_{n}^{u}) ∈ F_{2}^{uu}(g^{k_{0}}(z_{n−j}^{u})) and hence y_{n}^{u} ∈ F_{2}^{uu}(x, g). Moreover, again by (5), we have that g^{−m_{0}−k_{0}(j−1)}(z_{n}^{u}) ∈ F_{ϵ_{0}/2}^{uu}(g^{n_{0}}(z_{j}^{u})). Therefore, for l = j, ..., i − 1 and d = 0, 1, ..., n_{0} we have

dist(g^{d+k_{0}l}(y_{n}^{u}), g^{d}(z_{l}^{u})) < ϵ_{0}/2.

On the other hand, since dist(z_{l}^{u}, x_{l}) < ϵ_{1}, from (7) we conclude that dist(g^{d}(z_{l}^{u}), f^{d}(x_{l})) < ϵ_{0}/2 and so

dist(g^{d+k_{0}l}(y_{n}^{u}), f^{d}(x_{l})) < ϵ_{0}.

Thus, using (4) and (6) we get m{Dg_{/E^{c}(g^{k_{0}l}(y_{n}^{u}))}^{k_{0}}} > 2 and (9) follows. The desired point y^{u} is obtained as any accumulation point of the sequence {y_{n}^{u}}.

To finish the proof of our theorem we have just to show how to construct the sequences {z_{n}^{u}} and {x_{n}}. However the construction is straightforward: since dist(F_{1}^{uu}(x, g), H_{λ_{0}}^{u}(f)) < ϵ_{1} pick z_{0}^{u} ∈ F_{1}^{uu}(x, g) and x_{0} ∈ H_{λ_{0}}^{u}(f) such that dist(z_{0}^{u}, x_{0}) < ϵ_{1}. Once we have constructed z_{j}^{u}, x_{j} pick z_{j+1}^{u} ∈ F_{1}^{uu}(g^{k_{0}}(z_{j}^{u}), g) and x_{j+1} ∈ H_{λ_{0}}^{u}(f) such that dist(z_{j+1}^{u}, x_{j+1}) < ϵ_{1}.

□

4. Proof of Theorem A

Let f be a partially hyperbolic diffeomorphism satisfying the property SH and such that the (strong) stable foliation F^{ss} is minimal, i.e., every leaf F^{ss}(x, f) is dense in M. We have to show that there is a neigbourhood U(f) such that for every g ∈ U(f) the (strong) stable foliation of g is dense, i.e., every leaf F^{ss}(x, g) is dense.

From Theorem 3.1 we know that there exist a neighbourhood U_{1}(f) and C > 0, 1 < σ such that for every g ∈ U_{1} and x ∈ M there exists a point y^{u} ∈ F_{1}^{uu}(x, g) such that

(10)  m{Dg_{/E^{c}(g^{m}(y^{u}))}^{n}} > Cσ^{n} for any n > 0, m > 0.

We may assume that C = 1 (otherwise we take a fixed power of every g ∈ U_{1}(f)). Let λ = σ^{−1} and fix 0 < λ < λ_{1} < 1 and let r_{0} be as in Lemma 2.0.2.

On the other hand, fix a number δ > 0 arbitrarily small. Since the stable foliation F^{ss}(f) is minimal there exists K > 0 such that F_{K}^{ss}x, f is δ/2-dense, meaning that F_{K}^{ss}(x, f) ∩ B(z, δ/2) ≠ ∅ for every x, z ∈ M. Moreover, if δ is small enough, we may assume without loss of generality that F_{K}^{ss}(x, f) ∩ W_{r_{0}/2}^{cu}(z, f) ≠ ∅ for every x, z ∈ M. Hence there exists a neighbourhood U(f) ⊂ U_{1}(f) such that for every g ∈ U(f) and x, z ∈ M the following holds

(11)  F_{2K}^{ss}(x, g) ∩ W_{r_{0}}^{cu}(z, g) ≠ ∅.

Let us prove that for g ∈ U(f) the stable foliation F^{ss}(g) is minimal. For this purpose, fix x ∈ M and U an open subset of M. Let z ∈ U and let β > 0 be such that F_{β}^{uu}(z, g) ⊂ U. Take n_{0} such that g^{n_{0}}(F_{β}^{uu}(z, g)) ⊃ F_{1}^{uu}(g^{n_{0}}(z), g). Consider the point y^{u} ∈ F_{1}^{uu}(g^{n_{0}}(z), g) given by Theorem 3.1 and let η > 0 be such that

(12)  g^{−n_{0}}(W_{η}^{cu}(y^{u}, g)) ⊂ U.

Next, choose a positive integer m such that λ_{1}^{m}r_{0} < η and set k = n_{0} + m. From (11) we have that

(13)  F_{2K}^{ss}(g^{k}(x), g) ∩ W_{r_{0}}^{cu}(g^{m}(y^{u}), g) ≠ ∅.

Since E^{cu} = E^{c} ⊕ E^{u} and this decomposition is dominated, there is L > 0 such that ∥Dg_{/E^{cu}}^{−n}∥ ≤ L sup{∥Dg_{/E^{u}}^{−n}∥, ∥Dg_{/E^{c}}^{−n}∥}. For the sake of simplicity, we will assume that L = 1. From (10) we know that

∏_{j=0}^{n} ∥Dg_{/E^{c}(g^{m}(y^{u}))}^{−1}∥ < λ^{n}, 0 ≤ n ≤ m

and therefore

∏_{j=0}^{n} ∥Dg_{/E^{cu}(g^{m}(y^{u}))}^{−1}∥ < λ^{n}, 0 ≤ n ≤ m.

From Lemma 2.0.2 we conclude that

g^{−m}(W_{r_{0}}^{cu}(g^{m}(y^{u}), g)) ⊂ W_{λ_{1}^{m}r_{0}}^{cu}(y^{u}, g) ⊂ W_{η}^{cu}(y^{u}, g)

and hence, using (12) we have g^{−k}(W_{r_{0}}^{cu}(g^{m}(y^{u}), g)) ⊂ U. Finally, this and (13) imply F^{ss}(x, g) ∩ U ≠ ∅ and the proof of Theorema A is completed.

5. Shub’s example on 𝕋^{4}

Here we will show that Shub’s example of a non-hyperbolic robustly transitive diffeomorphism on 𝕋^{4} can also be derived from our methods.

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}. We assume (taking a power of f if necessary) that λ < μ. Next, we

define the map on 𝕋^{4} which 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). We 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.

Finally, as we observed above for some ϵ > 0, and r > 0 we have for every (z, w) ∈ 𝕋^{2} × 𝕋^{2} that

(14)  W_{ϵ}^{uu}((z, w)) ∩ W_{r}^{s}({p} × 𝕋^{2}) ≠ ∅

which implies (since g_{p} is Anosov and DF_{/{0}×E^{c}(p,z)} is uniformly expanding in the future) that the property SH is satisfied, i.e., if y^{u} is a point in the intersection given by (14) we may find constants C > 0 and σ > 1 such that

DF_{/E^{c}(F^{m}(y^{u}))}^{n} > Cσ^{n} ∀n, m ≥ 0

(recall that here E^{c} is one dimensional). Thus, F : 𝕋^{4} → 𝕋^{4} as above is in the hypothesis of Theorem A and so the stable foliation of F is robustly minimal and therefore F is robustly transitive. Since F has a hyperbolic periodic point of index 2 and another one of index 3 we conclude that F is non-hyperbolic.

6. Mañé’s example on 𝕋^{3}

In this section we will prove that certain derived from anosov maps on 𝕋^{n} have the stable foliation robustly minimal and hence they are robustly transitive. This is an alternative approach to the one given by Mañé in [M].

Let f : 𝕋^{n} → 𝕋^{n} be an Anosov diffeomorphism such that T 𝕋^{n} = E^{ss} ⊕ E^{u} ⊕ E^{uu}, non of the above trivial (and hence n ≥ 3). Since f is Anosov (on a torus) it is transitive and so the stable foliation is minimal. Let p be a fixed point of f and let U(p) be a small open ball containg p. Let g_{t} : 𝕋^{n} → 𝕋^{n}, −1 ≤ 0 be an smooth isotopy, supported in U and such that g_{t} is Anosov for t < 0, g_{0} is not Anosov but conjugated to f, and T 𝕋^{n} = E^{ss}(g_{t}) ⊕ E^{c}(g_{t}) ⊕ E^{uu}(g_{t}). It follows that F^{ss}(g_{0}) is minimal. Let 0 < λ < 1 be such that ∥D(g_{t})_{/E^{uu}}^{−1}∥ < λ. By replacing f by a power of itself we may assume that λ < 1/2. Now, if the ball U(p) is small enough, we have that for any x ∈ 𝕋^{n} there is point z_{x} ∈ F_{1}^{uu}(x, g_{0}) such that F_{1/2}^{uu}(z_{x}) ∩ U = ∅. Therefore, we have that in any F_{1}^{uu}(x, g_{0}) there is a point y^{u} whose forward orbit never meets U. Since the maximal invariant set outside U,

⋂_{n∈ℤ}g_{0}(𝕋^{n}\U)

is hyperbolic (recall that the support of the isotopy is inside U), we conclude that g_{0} satisfy the property SH. Therefore F^{ss}(g_{0}) is robustly minimal and g_{0} is robustly transitive. Notice that there are DA maps as close as we wish to g_{0} and therefore we have DA maps on 𝕋^{n} such that their (strong) stable foliation is robustly minimal (and so they are robustly transitive).

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