CONTRIBUTIONS TO THE THEORY OF ASYMPTOTICALLY SECTIONAL HYPERBOLIC FLOWS ALEXANDER ARBIETO MIGUEL PINEDA ELIAS REGO AND KENDRY VIVAS

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CONTRIBUTIONS TO THE THEORY OF ASYMPTOTICALLY

SECTIONAL HYPERBOLIC FLOWS

ALEXANDER ARBIETO, MIGUEL PINEDA, ELIAS REGO, AND KENDRY VIVAS

Abstract. In this paper, we make several contributions to the theory of

asymptotically sectional-hyperbolic (ASH) ﬂows. First, we prove that every

star ASH attractor for a C2vector ﬁeld is, in fact, sectional-hyperbolic (SH).

Second, we establish that all ASH attractors exhibit the property of entropy

ﬂexibility. Additionally, we show that any ASH attractor for three-dimensional

vector ﬁelds is entropy-expansive and admits periodic orbits. Finally, we pro-

vide a lower bound for the growth rate of periodic orbits in an ASH attractor.

Contents

1. Introduction 2

2. Preliminaries 6

2.1. Basic setting 6

2.2. Hyperbolic, Sectional-Hyperbolic and Star Flows 7

2.3. Stable Manifolds and Homoclinic Classes 8

2.4. Ergodic Theory 8

2.5. Topological Entropy 9

3. Hyperbolic measures for ASH ﬂows 10

4. Entropy of ASH attractors and applications 13

4.1. Entropy Flexibility 13

4.2. Existence of Periodic Orbits 15

4.3. Growth of Periodic Orbits 15

4.4. Proof of Theorem A 17

5. Three-dimensional ASH attractors 21

5.1. Entropy Expansiveness 21

5.2. Positive Entropy and Periodic Orbits 33

Acknowledgments 36

STATEMENTS 36

References 37

2020 Mathematics Subject Classiﬁcation. Primary: 37C10, Secondary: 37C27.

Key words and phrases. Asymptotically sectional-hyperbolic, growth rate of periodic orbits,

entropy.

AA was partially supported by CAPES, CNPq, PRONEX-Dynamical Systems and FAPERJ

E-26/201.181/2022 “Programa Cientista do Nosso Estado from Brazil”. MP was partially sup-

ported by CAPES from Brazil. ER was supported by the European Union’s European Research

Council Marie Sklodowska-Curie grant No.101151716.

arXiv:2210.12038v3 [math.DS] 4 Nov 2024

2 ALEXANDER ARBIETO, MIGUEL PINEDA, ELIAS REGO, AND KENDRY VIVAS

1. Introduction

The study of the dynamics of ﬂows, also known as continuous-time dynamics,

from both topological and statistical viewpoints, is a signiﬁcant area of research

in mathematics. This ﬁeld traces its origins back to the work of Poincar´e, who

utilized these concepts to deepen the understanding of the topology of underlying

manifolds.

Over time, the study of continuous-time dynamics evolved into a distinct math-

ematical discipline. Following the groundbreaking contributions of Anosov and

Smale, it emerged as a particularly fruitful area of research. Speciﬁcally, the dis-

covery of the hyperbolic nature of ﬂows through Smale’s horseshoe [42] as a source

of stability, along with Anosov’s work on the hyperbolicity of the geodesic ﬂow on

negatively curved Riemannian manifolds [2], became celebrated cornerstones of the

theory. These developments introduced techniques from both diﬀerential topology

and ergodic theory, greatly advancing the understanding of the dynamics of a broad

and important class of ﬂows.

Subsequently, concepts from information theory and statistical physics, intro-

duced by Kolmogorov, Sinai, Ruelle, Bowen, and others, were incorporated to bet-

ter understand the complexity of dynamics and the relevant invariant measures

of the system, particularly those that maximize entropy and, by the variational

principle, achieve topological entropy. This integration led to a fruitful symbiosis

between topological dynamics and the ergodic theory of dynamical systems.

One particular source of chaos and positive entropy is the presence of horse-

shoes. The pursuit of identifying horseshoe-type subdynamics within more general

ﬂows became a highly active area of research ([24], [16], [17]). A more challenging

question also emerged, which can be termed the ﬂexibility of entropy: given any

value between zero and the topological entropy, can we ﬁnd a compact subset (or

an ergodic measure) whose entropy matches that value? One of the aims of this

article is to address this question for certain ﬂows that present new challenges due

to the presence of singularities, as will be discussed in the following sections.

Although hyperbolic theory is very powerful, it requires that the subspace gen-

erated by the vector ﬁeld is continuous, which implies, among other things, that

its dimension must be locally constant. Consequently, no singularity can be ap-

proached by regular orbits within the hyperbolic set. In [23], Lorenz discovered a

robust attractor (with a dense orbit) that exhibits some properties resembling those

of hyperbolic systems, yet includes singularities that are accumulated by regular

orbits within the attractor. Inspired by this model, Guckenheimer, Shilnikov and

Turaev in an independently way ([18] and [44]) introduced a geometric example

that resembles the system studied by Lorenz. Therefore, to develop a comprehen-

sive program for understanding most dynamical systems, it is essential to analyze

such open sets.

The search for such a program motivated the search for a systematic theory

to describe such dynamics, which was established by Morales and Metzger [26]

under the name sectional hyperbolicity (see Section 2). In their paper, they proved

that the geometric Lorenz attractor is sectional-hyperbolic. Moreover, in a seminal

paper by Morales, Paciﬁco, and Pujals [30], it was shown that any robust attractor

in a 3-dimensional manifold must be sectional-hyperbolic. In fact, Tucker [43]

later proved that the attractor obtained from original Lorenz equations is, in fact,

sectional-hyperbolic.

CONTRIBUTIONS TO THE THEORY OF ASH FLOWS 3

One might assume that sectional-hyperbolic systems share similar properties

with hyperbolic ones. While this is sometimes true, it is important to exercise

caution. For instance, the problem of ﬁnding a periodic orbit can yield diﬀerent

outcomes: any isolated nontrivial hyperbolic set must have a periodic orbit, but

there are isolated nontrivial sectional-hyperbolic sets without periodic orbits [27].

We will revisit this issue later. Another example is the following: Every isolated

transitive hyperbolic set is robustly transitive, but this is not true in the sectional

hyperbolic theory anymore

In his PhD thesis, Rovella constructed another ﬂow, similar to the Lorenz at-

tractor, but with a singularity exhibiting diﬀerent behavior [36]. He was able to

ﬁnd examples of attractors that, unlike the Lorenz attractor, do not exhibit robust-

ness. However, inspired by several works on the quadratic family, such as those by

Benedicks-Carleson [11] and Jakobson [19], as well as on homoclinic bifurcations

(see also the work of Palis and Yoccoz [34], he proved that such attractors persist

in a certain sense, likewise in a codimension 2 submanifold. Once again, under-

standing such examples is crucial for a comprehensive understanding of dynamical

systems.

It turns out that the Rovella attractor ﬁts within another theory with a hy-

perbolic ﬂavor: the asymptotic sectional-hyperbolic dynamics, introduced by [31].

In fact, in [37] the authors shown that the Rovella attractor is an asymptotically

sectional-hyperbolic attractor. Another example of sets satisfying this weak kind

of hyperbolicity is known as the contractive singular horseshoe [31]. It should be

noted that these examples are not sectional-hyperbolic.

In this article we continue the study of such dynamics. Let us now precise its

deﬁnition. Let Mbe a compact Riemannian manifold endowed with metric d,

induced by the Riemannian metric ∥·∥. We denote by Xto a C1vector ﬁeld on M,

and we will refer by its ﬂow on Mto the family of maps Φ = {Xt}t∈R, induced by

X. A compact subset Λ of Mis called X-invariant if Xt(Λ) = Λ, for every t∈R.

Next, recall that a compact invariant set Λ has a dominated splitting if there are

a continuous invariant splitting TΛM=E⊕Ec(respect to DXt) and constants

K, λ > 0 satisfying

∥DXt(x)|Ex∥

m(DXt(x)|Ec

x)≤Ke−λt,∀x∈Λ,∀t > 0,

where m(A) denotes the conorm of A. In this case, we say that Ecis dominated by

E. We say that Λ is partially hyperbolic if Eis a contracting subbundle , i.e.,

∥DXt(x)|Ex∥ ≤ Ke−λt,

for every t > 0 and x∈Λ. Finally, denote by Ws(Sing(X)) the union of the stable

manifolds of the singularities of X.

Deﬁnition 1.1. Let Λ be a compact invariant partially hyperbolic set of a vector

ﬁeld X. We say that Λ is asymptotically sectional-hyperbolic (ASH for short) if the

singularities of Λ are hyperbolic and its central subbundle is eventually asymptot-

ically expanding outside the stable manifolds of the singularities, i.e., there exists

C > 0 such that

(1.1) lim sup

t→+∞

log |det(DXt(x)|Lx)|

t≥C,

4 ALEXANDER ARBIETO, MIGUEL PINEDA, ELIAS REGO, AND KENDRY VIVAS

for every x∈Λ′= Λ \Ws(Sing(X)) and every two-dimensional subspace Lxof

x. We say that an ASH set is non-trivial if it is not reduced to a singularity.

Remark 1.1. ASH sets satisfy the Hyperbolic lemma i.e., any compact and invari-

ant set without singularities is hyperbolic. The proof of this result can be found in

[37].

It can be easily seen from the deﬁnition that the sectional-hyperbolic theory is

encompassed within asymptotically sectional-hyperbolic theory, which in turn con-

tains the hyperbolic theory. Moreover, the Rovella’s attractor and the contracting

singular horseshoes show that those inclusions are proper. An important consid-

eration regarding the diﬀerence between sectional hyperbolicity and asymptotic

sectional hyperbolicity is that in the case of sectional-hyperbolic dynamics, uni-

form estimates are often obtainable (which hold for nearby vector ﬁelds). However,

in the asymptotic scenario, we must exercise more caution, as in the case of the

Rovella attractor, where uniform estimates are not expected. Indeed, from ASH

property we see that for every point x∈Λ outside Ws(Sing(X)), and every plane

Lx∈G(2, F ) (the Grasmannian of two-planes contained in the subbundle F) there

is an unbounded increasing sequence of positive numbers tk=tk(x, Lx)>0, called

hyperbolic times, such that

(1.2) |det DXtk(x)|Lx| ≥ eCtk, k ≥1.

Any unbounded increasing sequence satisfying the relation (1.2) will be called a

sequence C-hyperbolic times for x.

A vector ﬁeld Xis star if there is a C1-neighborhood of Xformed by vector

ﬁelds whose all singularities and periodic orbits are hyperbolic. Star vector ﬁelds

form a key concept for dealing with global dynamics. They were introduced by

Liao and Ma˜n´e, who showed that if the dynamics cannot bifurcate through non-

hyperbolic periodic orbits, then robustly, all periodic orbits must exhibit uniform

hyperbolic strength up to their period, along with a dominated splitting arising

from the union of their hyperbolic splittings (see [22],[25]). Morales and Paciﬁco (in

dimension 3) and Shi, Gan and Wen (in dimension 4) proved that, generically, the

star property is equivalent to sectional hyperbolicity (see [28] and [40]). However,

in higher dimensions, this equivalence does not hold, as exempliﬁed by the work

of Bonatti and Da Luz [12]. Thus, one can ask when the ASH theory diverges

from the star theory. Our ﬁrst result shows that, in any dimension, when restricted

to asymptotic sectional-hyperbolic dynamics, the star property is equivalent to

sectional hyperbolicity.

Theorem A. Every asymptotically sectional-hyperbolic attractor associated to a

C2-vector ﬁeld Xon Msatisfying the star property is sectional-hyperbolic.

The crucial step in proving prove Theorem Aunder C2-regularity is the exis-

tence of a periodic orbit contained in the attractor Λ. Nevertheless, by our next

results, one can obtain periodic orbits under the assumption that Xis of class

C1with positive entropy. Moreover, as we will see later in Theorem D, this hy-

pothesis is satisﬁed in the three-dimensional scenario. Consequently, Theorem Ais

valid for any ASH attractor with positive entropy associated to a C1vector ﬁeld

and, in particular, for any ASH attractor for C1vector ﬁelds on three-dinensional

manifolds.

CONTRIBUTIONS TO THE THEORY OF ASH FLOWS 5

Next, we delve into the entropy theory of ASH ﬂows. More precisely, we address

their entropy ﬂexibility. Here, we denote by h(X) and hµ(X) the topological and

metric entropies of X, respectively (see Section 2for precise deﬁnitions). We say

that a subset Λ has entropy ﬂexibility if for every h∈[0, h(X)) there are a compact

invariant subset K⊂Λ and an invariant measure µsuch that

h=h(X|K) = hµ(X).

In our next result, we address the entropy ﬂexibility of ASH ﬂows.

Theorem B. Let Xbe a C1-vector ﬁeld on M. Suppose Mcontains an asymp-

totically sectional-hyperbolic attractor Λ for X. Then Λ has entropy fexibility.

As mentioned earlier, due to the variational principle, it is natural to inquire

whether a measure of maximal entropy exists. Bowen, in [13], introduced a property

called entropy-expansiveness to guarantee the upper semi-continuity of the entropy

map, thus ensuring the existence of measures with maximal entropy. In our next

result, we prove this property for ASH attractors in dimension three.

Theorem C. Every asymptotically sectional-hyperbolic attractor Λ associated

with a C1vector ﬁeld Xon a three-dimensional manifold Mis entropy-expansive.

Another natural question concerns the positivity of entropy. As discussed ear-

lier, this is related to the existence of horseshoe-like subdynamics, particularly the

presence of periodic orbits. As showed by [10], it is known that any attracting1

sectional-hyperbolic set has a periodic orbit. However, there are attracting ASH

sets without periodic orbits [37]. Thus, the issue of the existence of periodic orbits

remains a subtle one. Our next result addresses this problem in dimension three.

Theorem D. Any asymptotically sectional-hyperbolic attractor Λ associated to

three-dimensional vector ﬁelds Xof class C1has a periodic orbit. Actually it

contains a nontrivial homoclinic class. Thus its topological entropy is positive. If

the periodic orbits are dense on Λ, then it is a homoclinic class.

We suspect that the attractor is generally a homoclinic class. Indeed, this holds

true in the sectional-hyperbolic setting, as shown by Arroyo and Pujals [7]. How-

ever, if we consider higher regularity, this result extends to any dimension.

Theorem E. If a C2-vector ﬁeld Xon Mcontains a asymptotically sectional-

hyperbolic attractor, then Mcontains a non-trivial homoclinic class.

All the Theorems C,Dand Ehave consequences about the entropy of the at-

tractor under perturbations.

Corollary F. Let Xbe a C1vector ﬁeld on M, and let Λ be an asymptotically

sectional-hyperbolic attractor. Then, there is a neighborhood Uof Λ and a C1

neighborhood Uof Xsuch that X|Λis a point of lower semicontinuity for the

entropy function on

X1(M, U ) = 



Y|ΛY: ΛY=\

t≥0

Yt(U)



.

1A set Λ is attracting if it has a neighborhood so that every point in the neighborhood even-

tually enters and remains within the set under the dynamics, i.e., Λ = St>0Xt(U) for some

open set satisfying X1(U)⊂U.

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CONTRIBUTIONSTOTHETHEORYOFASYMPTOTICALLYSECTIONALHYPERBOLICFLOWSALEXANDERARBIETO,MIGUELPINEDA,ELIASREGO,ANDKENDRYVIVASAbstract.Inthispaper,wemakeseveralcontributionstothetheoryofasymptoticallysectional-hyperbolic(ASH)flows.First,weprovethateverystarASHattractorforaC2vectorfieldis,infact,sectional-hy...

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CONTRIBUTIONS TO THE THEORY OF ASYMPTOTICALLY SECTIONAL HYPERBOLIC FLOWS ALEXANDER ARBIETO MIGUEL PINEDA ELIAS REGO AND KENDRY VIVAS

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