ASYMPTOTICALLY SECTIONAL-HYPERBOLIC ATTRACTORS

. The notion of asymptotically sectional-hyperbolic set was recently introduced. The main feature is that any point outside of the stable manifolds of its singularities has arbitrarily large hyperbolic times. In this paper we prove the existence, on any three-dimensional Riemannian manifold, of attractors with Rovella-like singularities satisfying this kind of hyperbolicity. Further-more, we prove that asymptotically sectional-hyperbolic Lyapunov-stable sets, under certain conditions, have positive topological entropy.


1.
Introduction. During the sixties, in an attempt to predict climate behavior, Lorenz [14] simplified Saltzman's thermal convection equations obtains the following parameter-depending polynomial system:   ẋ = −σx + σẏ y = ρx − y − xż z = −βz + xy. (1) Lorenz's observations on that systems show positive orbits moving erratically to a bounded domain for parameters close to σ = 10, β = 8/3 and ρ = 28, that is, the positive orbits behavior is chaotic and presents sensitive dependence to initial conditions. In other words, equations (1) appears to show an attractor with interesting dynamic properties which contains the equilibrium point (0,0,0), making it a non-hyperbolic attractor.
In order to get a better understanding about Lorenz's systems dynamic behavior, Guckenheimer in 1976 [10] introduced the Geometric Lorenz Attractor. Its structure was widely studied by Guckenheimer and Williams [11], Afraimovich, Bykov and Shilnikov [1] and Williams [28]. Although this attractor is non-hyperbolic, it presents hyperbolic features such as transitivity and dense periodic orbits in a robust way.
In 1986, Labarca and Pacifico in [13] introduced The Singular Horseshoe, the first variation ever considered of the geometric Lorenz attractor. It was conceived as a way to disprove Palis-Smale's stability conjecture for flows on manifolds with boundary. Later, in 1993, Rovella in [26] introduced a second variation of geometric Lorenz attractor. This variation replaced the singularity by one with a central the technique given in [2] to prove that asymptotically sectional-hyperbolic attractors have positive topological entropy under certain hypothesis. More precisely, we assume the existence of a non-atomic SRB-like measure supported in Λ for the time-one map.
2. Statement of the results. From now on M will denote a differentiable manifold endowed with a Riemannian metric · . By a flow we mean the one-parameter family of maps X t induced by a C 1 vector field X of M . We will say that X is a three-dimensional vector field when dimM = 3. We denote by Sing(X) the set of singularities (i.e. zeros of X). By a periodic point we mean a point x ∈ M for which there is a minimal T > 0 such that X T (x) = x. By an orbit we mean O(x) = {X t (x) : t ∈ R} and by a periodic orbit we mean to the orbit of a periodic point. We say that Λ ⊂ M is invariant if X t (Λ) = Λ for all t ∈ R. We say that Λ is hyperbolic if there are a continuous invariant splitting x ∈ Λ and * = s, u. A singularity or periodic orbit is hyperbolic if it is hyperbolic as a compact invariant set of X. The elements of a hyperbolic periodic orbit will be called hyperbolic periodic points.
We say that a compact invariant set Λ has a dominated splitting with respect to the tangent flow if there is a continuous invariant splitting T Λ M = E ⊕ F and numbers K, λ > 0 such that for every x ∈ Λ and every t ≥ 0. In this case we say that F dominates E. A compact invariant set Λ is partially hyperbolic if it has a dominated splitting T Λ M = E s ⊕ F with respect to the tangent flow whose dominated subbundle E s is contracting (in the sense of (a) above).
According to [18] we say that a compact invariant partially hyperbolic set Λ is sectional-hyperbolic if its singularities are hyperbolic and if its central subbundle F is sectionally expanding, i.e., there are K, λ > 0 such that for every x ∈ Λ, every t ≥ 0 and every two-dimensional subspace L x of F x .
It is well known that if σ ∈ M is a hyperbolic singularity for X then it has associated its stable and unstable manifolds W s (σ), W u (σ) which are tangent at σ to stable and unstable subspaces E s σ and E u σ of T σ M respectively. Denote by W s (Sing(X)) to the union of stable manifolds W s (σ) of the hyperbolic singularities of X.
With this notation we introduce the following definition: Definition 2.1. Let Λ be a compact invariant partially hyperbolic set of a vector field X whose singularities are hyperbolic. We say that Λ is asymptotically sectionalhyperbolic if its central subbundle is eventually uniformly asymptotically expanding outside the stable manifolds of the singularities, i.e., there exists C > 0 such that for every x ∈ Λ \ W s (Sing(X)) and every two-dimensional subspace L x of F x .
Remark 1. In [4] the authors define the following 2-Riemannian metric [22]: which induces a 2-norm [9] (also called areal metric [12]) given by By using the 2-norm (3) it is possible rewrite the Definition 2.1 as follows: A compact invariant partially hyperbolic set Λ with hyperbolic singularities is asymptotically sectional-hyperbolic if there exists a positive constant C such that From [21] we can highlight the following interesting properties that satisfy the asymptotically sectional-hyperbolic sets: Remark 2. Every asymptotically sectional-hyperbolic set satisfies the Hyperbolic Lemma.
Remark 3. Every sectional-hyperbolic set is asymptotically sectional-hyperbolic but not conversely: Take for instance the union of Rovella-like singularity σ, a hyperbolic saddle-type periodic orbit O and a generic regular orbit in W s (σ) ∩ W u (O).
The example in the previous remark is far from being chaotic because it is not transitive. Therefore, it is natural to ask if there are chaotic regions which are asymptotically sectional-hyperbolic but not sectional-hyperbolic. A positive answer to that question is given by the first author and Morales in [21] exhibiting the Contracting Horseshoe, which is a slight modification of the Singular Horseshoe, replacing the Lorenz-like singularity by a Rovella-like singularity.
(such a neighborhood U is often called trapping region); • attractor if Λ is a transitive attracting set.
The attractors represent important objects in the dynamical literature. In fact, they play an important role in Smale's Spectral Decomposition Theorem [25]. Examples of (nontrivial) hyperbolic attractors for flows can be obtained by suspending the classical Plykin attractor [25]. An example which is sectional-hyperbolic but not hyperbolic is precisely the geometric Lorenz attractor. Therefore, the following question arises: Are there attractors which are asymptotically sectional-hyperbolic with Rovella-like singularities?
One of our main results gives a positive answer for the question above.
Theorem 2.3. On every three-dimensional manifold there exists a vector field X exhibiting an asymptotically sectional-hyperbolic attractor having both dense periodic orbits and a Rovella-like singularity.
In order to state our second result we recall some facts about SRB-like measures. If f : M → M is a continuous map and µ is a Borel probability measure, we say that where δ y is the Dirac measure supported at y ∈ M . We denote by pω(x) the set of measures which are accumulated for empirical probabilities associated to the orbit of x, i.e, is the set of weak * −limits of subsequences of (4). Given ε > 0 and a probability measure µ, we define the basin of ε−attraction of µ as where d * is the standard metric in the space of probability measures. An invariant measure µ is SRB-like if for all ε > 0, B ε (µ) has positive Lebesgue measure. Finally, we say that an invariant set Λ is Lyapunov stable if for every neighborhood U of Λ there is a neighborhood V of Λ such that X t (V ) ⊂ U for all t ≥ 0.
Theorem 2.4. Let Λ be an asymptotically sectional-hyperbolic Lyapunov stable set of a vector field X on a compact manifold M . If Λ supports a non-atomic SRB-like measure for the time-one map X 1 , then Λ has positive topological entropy.  The construction above implies that Λ = S 2 0 = t≥0 X t (U ), U = S 2 × (−ε, ε), is an attracting set (so Lyapunov-stable) and asymptotically sectional-hyperbolic because Λ \ W s (Sing(X)) = O and E c p = E u p ⊕ X(p) for all p ∈ O, so that (2) is satisfied. Nevertheless, all SRB-like measures for X 1 supported in Λ are given by µ = λδ P + (1 − λ)δ S , λ ∈ [0, 1]. Moreover, we have h top (X) = 0 by the construction of X.

3.
Proofs. This section will be divided in three subsections. In the section 3.1 we will construct the attractor Λ announced in Theorem 2.3. In Subsection 3.2 we prove that such attractor is asymptotically sectional-hyperbolic and in the last subsection we will give the proof of Theorem 2.4.
3.1. Construction of Λ. We begin with three 3-cells: A cube Q and two copies of a cell T like in the Geometric Lorenz Attractor (as in [11]), say T l and T r . All these sets are described in the Figure 2. As in [11] we describe the vector filed on T l and T r through the differential equations   ẋ = λ l u ẋ y = λ l ss y , λ l ss < λ l s < 0 < λ l u z = λ l s z and   ẋ = λ r u ẋ y = λ r ss y , λ r ss < λ r s < 0 < λ r u . z = λ r s z In the same way, the vector field on Q is given by the differential equation s z In that figure it is considered C l and C r as the up faces of T l and T r respectively, and B l , B r and S ± = {z = ±1} faces of Q. In each cell, (0, 0, 0) correspond to the singularities σ l , σ r and σ 0 respectively. Denoting we impose the following conditions: The first two inequalities in the above equation implies that σ 0 is a Rovella-like singularity while σ i , i ∈ {l, r}, are Lorenz-like. The third inequality is imposed in order to the first return map has an expanding condition along the central subbundle.
Next, we glue the cells T l and T r to Q by joining A l with B l and A r with B r by an appropriate transformation. This results in a 3-cell equiped with a flow as described in the Figure 3.   H(x, y)).
The map H is assumed to satisfy the following properties: (H1) 0 < |H(x, y)| < 1 2 for x = 0. (H2) ∂H ∂y (x, y) < 1 2 for all (x, y) ∈ S. To describe the map f we write and∪ denotes the disjoint union. Moreover, we set (x + , y + ) for points in S + and (x − , y − ) for points in S − . With this notation, due to the third condition in (5) (f2) The third inequality in (5) The shape of the map F is described below. . It follows that f is transitive (i.e. it has a dense positive orbit). Let Λ = {z ∈ S : F n (z) exists ∀n ∈ Z}. Using that the vertical direction is a contracting foliation (condition (H3)) and the transitivity of f we have the transitivity of F on Λ. Furthermore, the weak-l.e.o condition of f implies the density of the periodic points of F . The proof is similar to geometric Lorenz attractor's case [28]. We define Λ = t∈R X t ( Λ). Now, as usual, by using the Tubular Flow Theorem, we can construct the vector field X in Figure 4 in a such a way that it is inwardly tranverse to the boundary. It follows that X has a trapping region U , i.e., X t (U ) ⊂ U for t ≥ 0. Topologically U is a solid handlebody of genus 4 as in the Figure 6. Figure 6. The trapping region U Therefore, Λ is an attracting set (with trapping region U ). By standard topological arguments we can embed the flow X in U on any three-dimensional manifold.
Finally, using that F is transitive with dense periodic orbits we obtain that Λ is an attractor for X. Actually Λ is a homoclinic class of X (by the arguments in [3]).

3.2.
Asymptotically sectional-hyperbolicity of Λ. We start with the discussion of the partially hyperbolicity of Λ. At first glance we note the existence of a strong stable direction E s in the trapping region U which is parallel to the y axis. To obtain the central direction E c on Λ we use the canonical argument given by saturation of the central cone field defined in Λ, that is, we considerer the complement of stable π 4 −cone field at Λ (C π 4 (x, {e 1 , e 3 }), where x ∈ Λ) and, afterwards, saturates it with the flow X. This defines the partially hyperbolic splitting E s ⊕ E c on Λ.
To prove that Λ is asymptotically sectional-hyperbolic it remains to prove that E c is eventually asymptotically sectionally expanding outside the stable manifolds of the singularities, i.e., we need to find C > 0 such that Let p ∈ Λ such that p does not belong to the stable manifold of any singularity. Then there is a positive finite time t such that X t (p) ∈ S. This time is irrelevant in the asymptotic behavior of |det(DX t (p)| E c p )|, so we can suppose that p ∈ S. Denote q = F (p) = X t (p) for some positive time t. For every tangent vector u = (A, B, 0) ∈ E c p ∩ T p S, we denote v = DF (p)u and w = DX t (p)u. Since DX t (p)X(p) = X(q) we have that vol < X(q); w > vol < X(p); u > .
On the one hand, we observe that vol < X(p); u >= X(p) × u , so vol < X(p); u >= (Bλ 0 where p = (x p , y p , z p ). As X(p) × u ∈ C s π 4 (p, {e 1 , e 3 }) (as in [6]) we have (Aλ 0 where q = (x q , y q , z q ) and z q = ±1. Besides, by a straightforward computation So Next, we will prove that successive return times give exponential growth for |det(DX t (p)| E c p )|. Let V δ0 be a δ 0 open neighborhood of W s (σ) ∩ S and p = p 0 . Let t i and p i the successive return times and return points for p 0 respectively. They (a) Define T δ0 = max{t i+1 : p i ∈ A}. Note that T δ0 does not depend on n. Then (b) On the other hand, for p ∈ V δ0 the first return flight time t can be decomposed as t = τ 1 + τ 2 + τ 3 + τ 4 , where τ 1 , τ 2 , τ 3 and τ 4 are the times to go from one section to the next in the "flow order", that is: • τ 1 is the flight time to go from p to lateral side of Q.
• τ 2 is the complement flight time to go from p to the up face of T l or T r respectively. • τ 3 is the flight time to go from the up face of T l or T r to the lateral face of T l or T r respectively. ii) As p = X τ1+τ2 (p) = J r • Π loc,0 (x, y, ±1) = (x α0 , yx β0 , 1) for x > 0 then In the same way, if x < 0, then p = X τ1+τ2 (p) = J l • Π loc,0 (x, y, ±1) = (|x| α0 , y|x| β0 , −1), so that τ 3 = − α0 λ l u log |x| and From i) and ii) we have Furthermore, for δ 0 small enough we have iii) There exists a positive constant C such that max{τ 2 , τ 4 } ≤ C . We denote q = X τ1 (p), p = X τ2 (q) and q = X τ3 (p). For δ 0 is small enough, depending on the case x > 0 or x < 0, we have that log |det(DX τ2 (p)| E c ) Joining all above, and shrinking δ 0 if necessary we have that We finish the proof by putting (a) and (b) together to obtain where C = min log ρ For every x ∈ R there exists a positive integer k(x), real numbers χ 1 (x) < χ 2 (x) < · · · < χ k(x) (x) and a , depending measurably on x ∈ R, such that The numbers χ i (x) are called Lyapunov exponents of µ along F . Catsigeras et al. in [7] give a C 1 -version of the Pesin Entropy Formula for diffeomorphisms: Lemma 3.1. Let Λ be a Lyapunov stable set of a flow X. If Λ has a dominated splitting T Λ M = E ⊕F with respect to the tangent flow and µ is a SRB-like measure for the time-one map X 1 , then where χ i (x), i = 1, . . . , dimF x , are the Lyapunov exponents along F x .