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                    UNIVERSIDADE FEDERAL DE ALAGOAS
INSTITUTO DE MATEMÁTICA
PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA UFAL-UFBA

JOSÉ EDUARDO MILTON DE SANTANA

EQUILIBRIUM STATES AND INDUCED SCHEMES WITH SPECIAL HOLES FOR
NON-UNIFORMLY EXPANDING MAPS

Maceió/AL
October 2018

JOSÉ EDUARDO MILTON DE SANTANA

EQUILIBRIUM STATES AND INDUCED SCHEMES WITH SPECIAL HOLES FOR NONUNIFORMLY EXPANDING MAPS

Thesis presented to IM-UFAL, Institute of
Mathematics of the Federal University of
Alagoas, as partial fulfillment of requirements for
the degree of Doctor in Mathematics and
approved on October 05, 2018.
Advisor: Krerley Irraciel Martins Oliveira

Maceió/AL
October 2018

Catalogação na fonte
Universidade Federal de Alagoas
Biblioteca Central
Divisão de Tratamento Técnico
Bibliotecária Responsável: Helena Cristina Pimentel do Vale CRB4 - 661
S232e

Santana, José Eduardo Milton de.
Equilibrium states and induced schemes with special holes for non- uniformly
expandanding maps / José Eduardo Milton de Santana. – 2019.
46 f.
Orientador: Krerley Irraciel Martins Oliveira.
Tese (doutorado em Matemática) – Universidade Federal de Alagoas. Instituto
de Matemática. Programa de Pós-Graduação de Doutorado em Matemática
Interinstitucional UFBA/UFAL. Maceió, 2018.
Bibliografia: f. 43-46.
1. Sistemas dinâmicos. 2. Estados de equilíbrio. 3. Formalismo termodinâmico.
4. Esquemas induzidos (Matemática). I. Título.
CDU: 517.9

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ACKNOWLEDGMENTS

This is not such an easy part of the thesis because I am grateful to so
many people and I do not want to forget anyone.
Firstly, according to my Catholic faith, I thank God for blessing me since
my birth, by giving me a marvelous family, friends and teachers. They
supported me in times of troubles and encouraged me all the time. Also, I
thank God for giving me health and opportunities to study and to do in life
what I really love to do: Mathematics!
I wish to specially thank some people:
Family: My mother Maria, for always being present in my life and light
on my path. My father João, for always being an example of courage and
strength. My brother Júnior and sister Ionara for their friendship and for
taking care of our parents when I am not with them. My aunt Milta, for being
an example of fight and dedication in studies. They all are examples of fight
in life. I thank all the support and tender all the time!
Friends: I thank Antônio Larangeiras, Isnaldo Barbosa, Carllos Eduardo
Holanda, Davi Lima, Micael Dantas, Rodrigo Lima, Wagner Xavier, Jamerson
Douglas, Diogo Santos, Ailton Campos, André Timótheo, Sergio Ibarra, Ítalo
Melo, Abraão Mendes, Rafael Alvarez, Anderson Lima, Rafael Lucena, Pedro
Carvalho, Matheus Martins, Giovane Ferreira, Marlon Oliveira, Vitor Alves,
Alexander Arbieto, Carlos Matheus Santos, Mauricio Polletti, Sandoel Vieira,
Alex Zamudio, Wanderson Costa, Hugo Araújo, Ermerson Araújo, Fernando
Lenarduzzi, Anuar Montalvo, Gabriela Estevez, Yaya Tall, Cássia Monalisa,
Isania Soares, Catarina Silva, Juliana Theodoro de Lima, Cícero Calheiros,
Alyson Pereira, Denilson Tenório, Abinaldo Aluizio. Thank you all for the
support and moments of fun!
Professors: Firstly, I thank my advisor, Krerley Oliveira, for his friendship
and constant support and incentive (it was a great pleasure having him as an
advisor). I thank the thesis jury members (Stefano Luzzatto, José F. Alves,

Vilton Pinheiro and Wagner Ranter) for their valious comments and
suggestions. I thank Enrique Pujals for letting me participate of his seminars
at IMPA and for his advices; Mikhail Belolipetsky, for his support and
encouragement; Carlos Gustavo Moreira (Gugu) for the financial support
during a staying at IMPA and Paulo Ribenboim for his friendship, support and
advices.
I also thank Valdenberg Silva, Ali Golmakani, Fernando Micena, André
Contiero, Yuri Lima, Feliciano Vitório, Márcio Batista, Marcos Petrúcio, Amauri
Barros and Luis Guillermo. In general, I thank all the people who taught me
every single thing in my life to make me who I am.
I thank Institute of Mathematics (IM / UFAL) for letting me to be away to
work on my thesis at IMPA and people of IMPA for well receive me in all my
staying there. Finally, I thank all the people that contributed in any way to
help me in this journey. Thank you so much, you all are amazing!
Finally, I thank CAPES for the financial support.

"Everything happens in the
right time!"

ABSTRACT

We prove existence of equilibrium states and induced schemes with special holes for
continuous non-uniformly expanding maps. The potential we consider is the hyperbolic
one, which means that the pressure emanates from the expanding set. For zooming
maps, we consider the open dynamics with hole obtained from small balls by erasing
intersections with regular pre-images and construct induced schemes with the property
of respecting the hole.

Keywords: Equilibrium state; Induced scheme; Open Dynamics; Zooming Map; Hole;
Hyperbolic Potential.

RESUMO

Provamos a existência de estados de equilíbrio e esquemas induzidos com buracos
especiais para mapas não-uniformemente expansores contínuos. O potencial
considerado é o hiperbólico, o que significa que a pressão emana do conjunto
expansor. Para mapas zooming, nós consideramos a dinâmica aberta com buraco
obtido a partir de bolas suficientemente pequenas, ao retirar-se interseções com
pré-imagens regulares e construímos esquemas induzidos com a propriedade de
respeitar o buraco.

Palavras-chave: Estado de equilíbrio; Esquema induzido; Dinâmica Aberta; Mapa
Zooming; Buraco; Potencial Hiperbólico.

Contents
1 Introduction

3

2 Equilibrium States for Hyperbolic Potentials

5

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Definitions and Statement of the Result . . . . . . . . . . . . . . . . . .

6

2.2.1

Non-Uniformly Expanding Maps

. . . . . . . . . . . . . . . . .

6

2.2.2

Induced Schemes . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2.3

Markov Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2.4

Topological Pressure . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2.5

Hyperbolic Potentials . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2.6

Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2.7

Statement of Main Result . . . . . . . . . . . . . . . . . . . . .

10

Equilibrium States for the Lifted Dynamics . . . . . . . . . . . . . . . .

10

2.3.1

Hyperbolic Potentials and Expanding Measures . . . . . . . . .

10

2.3.2

Equilibrium States for Markov Shifts . . . . . . . . . . . . . . .

11

Finiteness of Ergodic Equilibrium States . . . . . . . . . . . . . . . . .

14

2.4.1

Measures with Low Free Energy . . . . . . . . . . . . . . . . . .

14

2.4.2

The Inducing Time is Integrable . . . . . . . . . . . . . . . . . .

17

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.5.1

Viana maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.5.2

Benedicks-Carleson Maps

. . . . . . . . . . . . . . . . . . . . .

22

2.5.3

Rovella Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.3

2.4

2.5

3 Induced Schemes with Special Holes

25

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.2

Definitions and Statement of the Result . . . . . . . . . . . . . . . . . .

26

1

3.2.1

Markov Maps and Induced Schemes . . . . . . . . . . . . . . . .

26

3.2.2

Open Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.2.3

Statement of Main Result . . . . . . . . . . . . . . . . . . . . .

27

Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.3.1

Nested Collections . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.3.2

Constructing Nested Sets and Collections . . . . . . . . . . . . .

29

3.3.3

Zooming Sets and Measures . . . . . . . . . . . . . . . . . . . .

30

Induced Schemes Respecting Special Holes . . . . . . . . . . . . . . . .

32

3.4.1

Existence of Nested Collections . . . . . . . . . . . . . . . . . .

32

3.4.2

Existence of Zooming Nested Collections . . . . . . . . . . . . .

33

3.4.3

Finding Zooming Returns . . . . . . . . . . . . . . . . . . . . .

34

3.4.4

Constructing an Induced Scheme . . . . . . . . . . . . . . . . .

34

3.4.5

A Dense Partition . . . . . . . . . . . . . . . . . . . . . . . . . .

36

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.5.1

Hyperbolic Times . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.5.2

Viana maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.5.3

Benedicks-Carleson Maps

. . . . . . . . . . . . . . . . . . . . .

40

3.5.4

Rovella Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.3

3.4

3.5

2

Chapter 1
Introduction
In this thesis, we prove two results. In chapter 2 we prove the existence of finitely
many ergodic equilibrium states that are expanding measures, in the context of continuous non-uniformly expanding maps and hyperbolic potentials. This class of maps
extends the notion of differentiable non-uniformly expansion, where some expansion is
obtained in the presence of hyperbolic times. The hyperbolic potentials are important
because the pressure emanates from the expanding set, which is the set of points with
positive frequence of hyperbolic times. This means that is where we obtain most of
expansion. So, in the study of the thermodynamic formalism, we pay attention to the
expanding measures, which give total mass for the expanding set.
The strategy is to use induced schemes constructed by Pinheiro in [25] to code the
dynamics. Pinheiro proves that there are finetely many induced schemes for which we
can lift an ergodic expanding measure. Once we code the dynamics, we use the results
by Sarig in [28], [29], [30], [31] to obtain equilibrium states for the associated countable
shift. In this context, the equilibrium state is unique and also it is a Gibbs measure.
Then we can get at most finitely many ergodic equilibrium states that are expanding
measures. It is obtained by showing that the inducing time is integrable, by following
ideas of Iommi and Todd in [18]. Finally, we get equilibrium states for the hyperbolic
potentials and they are, at most, finitely many.
In chapter 3 we consider open zooming maps and construct induced schemes that
respects a certain type of holes. This idea appears in [13] for Collet-Eckmann maps.
A zooming map generalizes the notion of non-uniformly expansion by generalizing
the notion of hyperbolic times, obtaining the zooming times. In the same way, we get

3

4
expansion of certain type, in the context of continuous maps. It is the key ingredient
to construct the induced schemes. We follow ideas of Pinheiro in [25]. The idea
is considering disjoint balls sufficiently small located in certain regions of the phase
space and erase intersections with regular pre-images, getting a special open set. The
presence of points with zooming returns allows us to take regular pre-images inside this
open set, constructing elements of the partition in one of the connected components.
At the end, we have and induced scheme and a special hole, where the induced scheme
respects the hole.
The class of maps introduced by Viana in [34], and called Viana maps, is an important example for our settings.

Chapter 2
Equilibrium States for Hyperbolic
Potentials
2.1

Introduction

The theory of equilibrium states was firstly developed by Sinai, Ruelle and Bowen
in the sixties and seventies. It was based on applications of techniques of Statistical
Mechanics to smooth dynamics.
Given a continuous map f : M → M on a compact metric space M and a continuous
potential φ : M → R, an equilibrium state is an invariant measure that satisfies an
Variational Principle, that is, a measure µ such that
Z
hµ (f ) +

φdµ =

sup



Z
hη (f ) + φdη ,

η∈Mf (M )

where Mf (M ) is the set of f -invariant probabilities on M and hη (f ) is the so-called
metric entropy of η.
In the context of uniform hyperbolicity, which includes uniformly expanding maps,
equilibrium states do exist and are unique if the potential is Hölder continuous and the
map is transitive. In addition, the theory for finite shifts was developed and used to
achieve the results for smooth dynamics.
Beyond uniform hyperbolicity, the theory is still far from complete. It was studied
by several authors, including Bruin, Keller, Demers, Li, Rivera-Letelier, Iommi and
Todd [11],[10],[13]),[17],[18],[19] for interval maps; Denker and Urbanski [14] for rational
maps; Leplaideur, Oliveira and Rios [20] for partially hyperbolic horseshoes; Buzzi,
5

2.2 Definitions and Statement of the Result

6

Sarig and Yuri [12],[36], for countable Markov shifts and for piecewise expanding maps
in one and higher dimensions.
For local diffeomorphisms with some kind of non-uniform expansion, there are results due to Oliveira [21]; Arbieto, Matheus and Oliveira [7]; Varandas and Viana [33].
All of whom proved the existence and uniqueness of equilibrium states for potentials
with low oscillation. Also, for this type of maps, Ramos and Viana [27] proved it for
potentials so-called hyperbolic, which includes the previous ones. The hyperbolicity of the potential is characterized by the fact that the pressure emanates from the
hyperbolic region.
Our result is similar to the Ramos and Viana, but for maps presenting some kind
of non-uniform expansion and critical points. However, our strategy is completely
different, since we do not use the analytical approach of the Transfer Operator in order
to obtain conformal measures. We use results on countable Markov shifts by Sarig
for the “coded” dynamics in induced schemes constructed by Pinheiro in [25], where
a Markov structure is constructed. We prove that there exist finitely many ergodic
equilibrium states that are expanding measures.

2.2

Definitions and Statement of the Result

We begin by defining non-uniformly expanding maps in a non-differentiable context.

2.2.1

Non-Uniformly Expanding Maps

Let M be a connected compact metric space, f : M → M a continuous map and µ
a reference Borel measure on M . Firstly, we will define hyperbolic times.
Fix σ ∈ (0, 1), δ > 0 and x ∈ M .
Definition 2.2.1 We say that n ∈ N is a (σ, δ)-hyperbolic time for x if
n
• There exists a neighbourhood Vn (x) of x such that f|V
: Vn (x) → Bδ (f n (x)) is
n (x)

a homeomorphism;
• d(f i (y), f i (z)) ≤ σ n−i d(f n (y), f n (z)), ∀ y, z ∈ Vn (x), ∀ 0 ≤ i ≤ n − 1.

2.2 Definitions and Statement of the Result

7

The sets Vn (x) are called hyperbolic pre-balls and their images f n (Vn (x)) =
Bδ (f n (x)), hyperbolic balls.
We say that x ∈ M has positive frequency of hyperbolic times if
1
lim sup #{0 ≤ j ≤ n − 1 | j is a hyperbolic time for x} > 0
n→∞ n
We denote by
H = {x ∈ M | the frequency of hyperbolic times of x is positive}.
We call H the expanding set and we say that the reference measure µ is expanding if µ(H) = 1.
Definition 2.2.2 Given a measure µ on M , its jacobian is a function Jµ f : M →
R
[0, +∞) such that µ(f (A)) = A Jµ f dµ for every A domain of injectivity, that is,
a measurable set such that f (A) is measurable and fA : A → f (A) is a bijection.
Definition 2.2.3 Given a measure µ with a jacobian Jµ f , we say that the measure
has bounded distortion if there exists ρ > 0 such that
Jµ f n (y)
≤ ρd(f n (y), f n (z))
log
n
Jµ f (z)
for every y, z ∈ Vn (x), µ-almost everywhere x ∈ M , for every hyperbolic time n of x.
A map with an expanding measure with bounded distortion associated is called
non-uniformly expanding.
Remark 2.2.1 We observe that a hyperbolic time in our context is a type of zooming
time in the context of [25].

2.2.2

Induced Schemes

We recall the definition of an induced scheme. It is useful to code the dynamics in
order to use results concerning symbolic dynamics.
Definition 2.2.4 Given an open set U ⊂ M and P = {P1 , . . . , Pn , . . . } a partition of
open subsets in U, if there exists a map F : U → U such that F|Pi : Pi → U is an
homeomorphism for all i ∈ N and if for every element Pi there exists τi ∈ N such that
τi
F|Pi = f|P
, F is said to be an induced map and we call the pair (F, P) an induced
i

2.2 Definitions and Statement of the Result

8

scheme on U. The function τ : U → N such that τ|Pi := τi is called the inducing
time.
Definition 2.2.5 Given an induced scheme (F, P) and an invariant probability µ, we
say that µ is liftable to (F, P) if there exists a measure µ such that for every measurable
set A,
µ(A) =

∞ τX
k −1
X

µ(f −j (A) ∩ Pk )

k=1 j=0

The next result, due to Pinheiro, guarantee that every ergodic expanding measure
can be lifted to some induced scheme.
Theorem 2.2.1 (Pinheiro [25]) There exist finitely many induced schemes
(F1 , P1 ), . . . , (Fs , Ps ),
such that every ergodic probability µ for which µ(H) = 1 is liftable to some of these
induced schemes, with uniformly bounded integral of inducing time.

2.2.3

Markov Shifts

Now we recall the basic definitions of symbolic dynamics. Given a countable set
S, we define the space of symbols Σ := {(x1 , x2 , . . . , xn , . . . ) | xi ∈ S, ∀i ∈ N}. The
shift map σ : Σ → Σ is defined by σ((x1 , x2 , . . . , xn , . . . )) = (x2 , x3 , . . . , xn , . . . ). A
cylinder is a set of the form Cn := {x ∈ Σ : x1 = a1 , . . . , xn = an }.
When an induced scheme (F, P) is given, we can define a space of symbols by the
following rule. Let x ∈ U be a point such that F k (x) is well defined for all k ∈ N. To
obtain a sequence (x1 , x2 , . . . , xn , . . . ), we put xi = j if F i (x) ∈ Pj . So, we can see that
the map F is conjugate to the shift map. The advantage here is that we can use the
theory of symbolic dynamics to obtain results for our original map.

2.2.4

Topological Pressure

We recall the definition of relative pressure for non-compact sets by dynamical balls,
as it is given in [4].
Let M be a compact metric space. Consider f : M → M and φ : M → R. Given
δ > 0, n ∈ N and x ∈ M , we define the dynamical ball Bδ (x, n) as the set
Bδ (x, n) := {y ∈ M | d(f i (x), f i (y)) < δ, for 0 ≤ i ≤ n}.

2.2 Definitions and Statement of the Result

9

Consider for each N ∈ N, the set
FN = {Bδ (x, n) | x ∈ M, n ≥ N }.
Given Λ ⊂ M , denote by FN (Λ) the finite or countable families of elementes in FN
that cover Λ. Define for n ∈ N
Sn φ(x) = φ(x) + φ(f (x)) + · · · + φ(f n−1 (x)).
and
Rn,δ φ(x) =

sup

Sn φ(y).

y∈Bδ (x,n)

Given a f -invariant set Λ ⊂ M , not necessarily compact, define for each γ > 0
 X

−γn+Rn,δ φ(y)
mf (φ, Λ, δ, N, γ) = inf
e
.
U ∈FN (Λ)

Bδ (y,n)∈U

Define
mf (φ, Λ, δ, γ) = lim mf (φ, Λ, δ, N, γ).
N →+∞

and
Pf (φ, Λ, δ) = inf{γ > 0 | mf (φ, Λ, δ, γ) = 0}.
Finally, define the relative pressure of φ on Λ as
Pf (φ, Λ) = lim Pf (φ, Λ, δ).
δ→0

The topological pressure of φ is, by definition, Pf (φ) = Pf (φ, M ) and satisfies
Pf (φ) = sup{Pf (φ, Λ), Pf (φ, Λc )}

(2.1)

where Λc denotes the complement of Λ on M . We refer the reader to [24] for the
proof of 2.1 and for additional properties of the pressure. See also [35] for a proof of
the fact that

Pf (φ) =

sup

Z
hµ (f ) +


φdµ .

µ∈Mf (M )

2.2.5

Hyperbolic Potentials

Definition 2.2.6 We say that a real continuous function φ : M → R is a hyperbolic
potential if the topological pressure Pf (φ) is located on H,i.e.,
Pf (φ, H c ) < Pf (φ).

2.3 Equilibrium States for the Lifted Dynamics

10

In [19] H. Li and J. Rivera-Letelier consider other type of hyperbolic potentials for
one-dimensinal dynamics that is weaker than ours. In their context φ is a hyperbolic
potential if
Z
sup

φdµ < Pf (φ).

µ∈Mf (M )

2.2.6

Equilibrium States

Given a continuous map f : M → M and a Hölder potential φ : M → R, an
equilibrium state is an invariant measure that satisfies an Variational Principle,
that is, a measure µ such that
Z
hµ (f ) + φdµ =

sup



Z
hη (f ) + φdη ,

η∈Mf (M )

where Mf (M ) is the set of invariant probabilities on M and hη (f ) is the so-called
metric entropy of η.

2.2.7

Statement of Main Result

Now, we state our main result on the existence of equilibrium states.
Theorem A Given a non-uniformly expanding map f : M → M and a Hölder hyperbolic potential φ : M → R with finite topological pressure Pf (φ), there exist finitely
many ergodic equilibrium states and they are expanding measures.

2.3

Equilibrium States for the Lifted Dynamics

In this section we begin the proof of Theorem A. The strategy is lifting the dynamics, finding equilibrium states and then projecting them.

2.3.1

Hyperbolic Potentials and Expanding Measures

The next proposition and the Theorem 2.2.1 guarantee that every ergodic expanding
measure with “high free energy” and, in particular, those which are candidates to be
equilibrium states can be lifted to some induced scheme.
Proposition 2.3.1 If φ is a hyperbolic potential, given an ergodic probability µ such
R
that hµ (f ) + φdµ > Pf (φ, H c ), then µ(H) = 1.

2.3 Equilibrium States for the Lifted Dynamics

11

Proof. Since H is an invariant set and µ is an ergodic probability, we have µ(H) = 0
or µ(H) = 1. But,
Z



c

φdµ > Pf (φ, H ) ≥ sup

hµ (f ) +



Z
hν (f ) +

ν(H c )=1

φdν
Hc

(For the second inequality, see Pesin [24], Theorem A2.1)
So, we can not have µ(H c ) = 1 and we obtain µ(H) = 1,i.e., µ is an expanding
measure.


2.3.2

Equilibrium States for Markov Shifts

Definition 2.3.1 Given a potential φ : M → R and an induced scheme (F, P), we
τ (x)−1
X
define the induced potential as φ(x) =
φ(f j (x)).
j=0

Definition 2.3.2 Given a potential φ : Σ → R, we say that φ is locally Hölder if
there exist A > 0 and θ ∈ (0, 1) such that for all n ∈ N the following holds
Vn (φ) := sup{| φ(x) − φ(y) |, ∀x, y ∈ Cn } ≤ Aθn .
Proposition 2.3.2 If φ : M → R is a Hölder potential, then φ : Σ → R is a locally
Hölder potential.
Proof. As φ is Hölder, there are constants ρ, α > 0 such that | φ(x) − φ(y) |≤
ρd(x, y)α . We must show that there are constants A > 0 and θ ∈ (0, 1) such that
| φ(x) − φ(y) |≤ Aθn , ∀x, y ∈ Cn . In fact, given x, y ∈ Cn , there are Pi0 , Pi1 , . . . , Pin
such that F k (x), F k (y) ∈ Pik . Then, we have
τi0 −1

| φ(x) − φ(y) |=

X

τi0 −1
j

j

φ(f (x)) − φ(f (y)) ≤

j=0

≤ρ

| φ(f j (x)) − φ(f j (y)) |≤

j=0

τi0 −1

X

X

τi0 −1
j

j

α

d(f (x), f (y)) ≤ ρ

j=0

X

(σ τi0 −j d(F (x), F (y))α ≤

j=0

≤ρ

∞
X

(σ α )j d(F (x), F (y))α = Aθn .

j=0

where A := ρ(

P∞

j=0 (σ

α j

) /σ α )δ, θ := σ α .



As a consequence of Theorem 2.2.1, there exist an induced scheme (F, P) and a
R
sequence µn of liftable ergodic probabilities such that hµn (f ) + φdµn → Pf (φ).

2.3 Equilibrium States for the Lifted Dynamics

12

Proposition 2.3.3 (Abramov’s Formulas [37]) Given a measure µ liftable to µ, we
have the formulas
hµ (F )
,
hµ (f ) = R
τ dµ

Z

R
φdµ
φdµ = R
.
τ dµ

As a consequence, we have
Z


Z
Z
hµ (f ) + φdµ =
τ dµ hµ (f ) + φdµ .
Definition 2.3.3 Given a Markov shift (Σ, σ), we define the Gurevich Pressure as
1
PG (φ) := lim log
n→∞ n



X

e

φn (x)


n−1
X
, where φn (x) =
φ(f j (x)).

σ n (x)=x,x0 =a

j=0

Theorem 2.3.1 (Sarig [28]) If (Σ, σ) is topologically mixing and φ is locally Hölder,
then the Gurevich Pressure is well defined and independent of a.
Theorem 2.3.2 (Approximation Property [28])(Sarig) If (Σ, σ) is topologically mixing
and φ is locally Hölder, we have
PG (φ) = sup{Ptop (φ|Y ), Y ⊂ Σ is a topologically mixing finite Markov shift}.
Theorem 2.3.3 (Variational Principle)(Iommi-Jordan-Todd [16]) Let (Σ, σ) be a topologically mixing countable Markov shift and φ : Σ → R a potential with summable
P
variations (i.e., ∞
n=1 Vn (φ) < ∞). Then we have


Z
Z
PG (φ) = sup hν (F ) + φdν : − φdν < ∞ .
Given φ : M → R a hyperbolic potential, set ϕ := φ − Pf (φ). So, ϕ is a hyperbolic
potential such that Pf (ϕ) = 0.
Proposition 2.3.4 Given a Hölder hyperbolic potential ϕ such that Pf (ϕ) = 0, we
have that PG (ϕ) = 0.
Proof. PG (ϕ) ≥ 0: By Pinheiro’s Theorem, there exist a sequence µn of liftable
ergodic probabilities. By the Abramov’s Formulas, we have that
Z


Z
Z
hµn (f ) + φdµn =
τ dµn
hµn (f ) + φdµn .
R
ϕdµn → Pf (ϕ) = 0 and the sequence
τ dµn n is uniformly
R
bounded, we obtain hµn (F ) + ϕdµn → 0, which means 0 ≤ P (ϕ).
Since hµn (f ) +

R

2.3 Equilibrium States for the Lifted Dynamics

13

PG (ϕ) ≤ 0: By taking the finite Markov subshift with symbols P1 , P2 , ..., PN , deR
noted by (ΣN , σN ), we obtain an equilibrium state ν N such that hνN (F ) + ϕdνN =
Ptop (ϕ|ΣN ) → PG (ϕ) (By the approximation property). By contradiction, if PG (ϕ) > 0,
for some N big enough we obtain
Z


Z
Z
0 < hν N (f ) + ϕdν N =
τ dν N
hνN (f ) + ϕdνN .
that implies
Z
0 < hνN (f ) +

ϕdνN < Ptop (ϕ) = 0.

a contradiction. So, PG (ϕ) ≤ 0.
Finally, by the Variational Principle, we obtain 0 ≤ Ptop (ϕ) = PG (ϕ) ≤ 0 and
PG (ϕ) = 0.



Definition 2.3.4 If we denote the matrix transition of (Σ, σ) by T = (tij ), the Big
Images and Preimages (BIP) Property is
∃ b1 , . . . , bN ∈ S such that ∀a ∈ S ∃ i, j ∈ {1, . . . , N } such that tbi a tabj = 1.
Clearly, if (Σ, σ) is a full shift, it has the BIP property.
We are able to use the Sarig’s results on the existence and uniqueness of conformal
measures, Gibbs measures and equilibrium states for countable Markov shifts. The
original works are [28], [29], [30], but the results can all be found in [31]. They can be
summarized as follows.
Theorem 2.3.4 (Sarig [31]) Let (Σ, σ) be topologically mixing and ϕ of summable
variations. Then ϕ has an invariant Gibbs measure µϕ if, and only if, it has the BIP
property and PG (ϕ) < ∞. Moreover, the Gibbs measure has the following properties:
R
• If hµϕ (σ) < ∞ or − ϕdµϕ < ∞, then µϕ is the unique equilibrium state (in
R
particular, PG (ϕ) = hµϕ (σ) + ϕdµϕ ).
• µϕ is finite and dµϕ = hϕ dmϕ , where L(hϕ ) = λhϕ and L∗ (mϕ ) = λmϕ , λ =
R
ePG (ϕ) . It means that m(σ(A)) = A eϕ−log λ dmϕ .
• This hϕ is unique and mϕ is the unique (ϕ−log λ)-conformal probability measure.
From this Theorem, we obtain a unique (ϕ − log λ)-conformal probability measure
mϕ and a unique equilibrium state µϕ , which is also a Gibbs measure.
We now need to show that the inducing time is integrable in order to project the
Gibbs measure.

2.4 Finiteness of Ergodic Equilibrium States

2.4

14

Finiteness of Ergodic Equilibrium States

We will prove, by following ideas of Iommi and Todd, that the inducing time is
integrable with respect to the Gibbs measure µϕ . As a consequence, we can project it
to a measure µϕ , which will be shown to be an equilibrium state for the original system
(f, ϕ).

2.4.1

Measures with Low Free Energy

We state the results we adapted from ([18]) and whose proofs are exactly the same.
We reproduce the main steps here.
Proposition 2.4.1 Given a hyperbolic potential ϕ with Pf (ϕ) = 0 and an induced
scheme (F̃ , P), there exists k ∈ N such that replacing (F̃ , P) by (F, P), where F = F̃ k ,
the following holds. There exists γ0 ∈ (0, 1) and for any cylinder Cn ∈ PnF , any n ∈ N,
a constant δn < 0 such that any measure µF ∈ MF with
µF (Cn ) ≤ (1 − γ0 )mϕ (Cn ) or mϕ (Cn ) ≤ (1 − γ0 )µF (Cn ),
where mϕ is the conformal measure for the system (F, ϕ), must have hµF (F ) +
R

ϕdµF < δn .
The following lemma will allow us to choose k in the above proposition.

Lemma 2.4.1 Suppose that we have an induced scheme (F, P) and a locally Hölder

P∞
potential ϕ with distortion constant K := exp
j=1 Vj (ϕ) and PG (ϕ) = 0. We let
mφ denote the conformal measure for the system (F, ϕ). Then, for any Cn ∈ PnF and
any n ∈ N,
mϕ (Cn ) ≤ e−λn
where λ := − log(K supC1 ∈P1F mϕ (C1 )).
Suppose that the distortion of the potential ϕ̃ for the scheme (F̃ , P) is bounded by
K ≥ 1. We first prove that measures giving cylinders very small mass compared to mϕ
must have low free energy. Note that for any k ∈ N, the potential ϕ for the scheme
(F, P), where F = F̃ k also has distortion bounded by K. We will choose k later so
that λ for (F, P), as defined in Lemma 2.4.1, is large enough to satisfy the conditions
associated to 2.3,2.4 and 2.6. Note that as in [[32], Lemma 3] we also have PG (ϕ) = 0.

2.4 Finiteness of Ergodic Equilibrium States

15

In Lemma 2.4.3 below, we will use the Variational Principle to bound the free energy
of measures for the scheme which, for some γ, have µ(Cni ) ≤ Kmϕ (Cni )(1 − γ)/(1 −
mϕ (Cni ))n in terms of the Gurevich pressure. However, instead of using ϕ, which, in the
computation of Gurevich pressure weights points x ∈ Cni by eϕ(x) , we use a potential
which weighs points in Cni by (1 − γ)eϕ(x) .
We define the potential ϕ[ as

ϕ[ (x) =


 ϕ(x) + log(1 − γ),

if x ∈ Cni

 ϕ(x)

if x ∈ Cnj , for j 6= i

Lemma 2.4.2 We have that PG (ϕ[ ) = log(1 − γmϕ (Cn )).
R
We define MF (ϕ) as the set of F -invariant measures such that − ϕdµ < ∞.
Lemma 2.4.3 We have that MF (ϕ) = MF (ϕ[ ) and for any cylinder Cni ∈ PnF the
following holds.


R
K(1−γ)
sup hF (µ) + ϕdµ : µ ∈ M(ϕ), µ(Cni ) < 1−mϕ (C i )n µϕ (Cni )
n


R [
K(1−γ)
[
i
i
≤ sup hF (µ) + ϕ dµ : µ ∈ M(ϕ ), µ(Cn ) < 1−mϕ (C i )n µϕ (Cn )
n


log(1−γ)
− K(1−γ)
µϕ (Cni )
i )n
1−mϕ (Cn


K(1−γ) log(1−γ)
[
i
≤ PG (ϕ ) −
µϕ (Cn ) .
1−mϕ (C i )n
n

Note that we can prove that the final inequality is actually an equality, but we
don’t require this here.
Lemmas 2.4.2 and 2.4.3 imply that any measure µF with µF (Cni ) < K(1−γ)mϕ (Cni /(1−
mϕ (Cni )n must have

Z
hF (µ) +


K(1 − γ) log(1 − γ)
i
ϕ dµ ≤ PG (ϕ ) −
µϕ (Cn )
1 − mϕ (Cni )n
[



[


K(1 − γ) log(1 − γ)
i
= log(1 − γmϕ (Cn )) −
µϕ (Cn ) .
1 − mϕ (Cni )n

(2.2)



(2.3)

If mϕ (Cni ) is very small then log(1 − γmϕ (Cn )) ≈ −γmϕ (Cn ) and so choosing
γ ∈ (0, 1) close to 1 the above is strictly negative. By Lemma 2.4.1, µϕ (Cni ) < e−λn so

2.4 Finiteness of Ergodic Equilibrium States

16

Cni is small if λ large. Hence, if λ is sufficiently large then we can set γ = γ̃ [ ∈ (0, 1)
so that
[ −λn

log(1 − γ̃ e

K(1 − γ̃ [ ) log(1 − γ̃ [ ) −λn
)−
e
(1 − e−λn )n




is strictly negative for all n ∈ N. This implies that 2.3 with γ = γ̃ [ is strictly
negative for any Cni ∈ PnF and any n, so we set 2.3 to be the value δni,[ .
For the upper bound on the free energy of measures giving Cni relatively large mass,
we follow a similar proof, but with the potential ϕ] defined as

 ϕ(x) − log(1 − γ), if x ∈ C i
n
ϕ] (x) =
j
 ϕ(x)
if x ∈ C , for j 6= i
n

Similarly to above, one can show that





γ
γ
≤ mϕ (Cn ) 1−γ
.
Lemma 2.4.4 We have that PG (ϕ] ) = log 1 + mϕ (Cn ) 1−γ
Also, one can show that
Lemma 2.4.5 We have that MF (ϕ) = MF (ϕ] ) and for any cylinder Cni ∈ PnF the
following holds.

(
sup hF (µ) +

)
R

1

ϕdµ : µ ∈ M(ϕ), µ(Cni ) >





K(1−γ) 1+mϕ (Cn )

γ
1−γ

n µϕ (Cni )

(
≤ sup hF (µ) +

)
R

1

ϕ] dµ : µ ∈ M(ϕ] ), µ(Cni ) >





K(1−γ) 1+mϕ (Cn )

"

#
i)
log(1−γ)µϕ (Cn

+





K(1−γ) 1+mϕ (Cn )

γ
1−γ

n
#

"
≤ PG (ϕ] ) +

i)
log(1−γ)µϕ (Cn





K(1−γ) 1+mϕ (Cn )

γ
1−γ

n .

Therefore, if
µ(Cni ) >
we have

m (C )
 ϕ n

n
γ
K(1 − γ) 1 + mϕ (Cn ) 1−γ

γ
1−γ

n µϕ (Cni )

2.4 Finiteness of Ergodic Equilibrium States

Z
hF (µ) +

ϕdµ ≤ mϕ (Cni )



γ
1−γ



17

log(1 − γ)µϕ (Cni )

n .

+
γ
K(1 − γ) 1 + mϕ (Cn ) 1−γ

(2.4)

If λ is sufficiently large then we can choose γ = γ̃ ] ∈ (0, 1) so that this is strictly
negative and can be fixed to be our value δni,] . This can be seen as follows: let γ =
p/(p + 1) for some p to be chosen later. Then the right hand side of 2.4 becomes

mϕ (Cni )(p + 1)




p
log(p + 1)
−
.
p + 1 K(1 + pe−λn )n

(2.5)

If λ is sufficiently large, then there exists some large λ0 ∈ (0, λ) such that (1 +
0

pe−λn )n ≤ 1 + pe−λ n for all n ∈ N. Hence with this suitable choice of λ we can choose
p so that the quantity in the square brackets in 2.5 is negative for all n. So we can
choose δni,] < 0 to be 2.4 with γ = γ̃ ] .
We let

γ ] = 1 − (1 − γ̃ ] ) 1 + e−λn

γ̃ ]
1 − γ̃ ]

!!n
.

(2.6)

For appropriately chosen λ this is in (0, 1).
We set γ00 := max{γ [ , γ ] } and for each Cni ∈ PnF we let δni := max{δni,[ , δni,] . The
proof of the proposition is completed by setting γ0 := 1 − K(1 − γ00 ), which we may
assume is in (0, 1).

2.4.2

The Inducing Time is Integrable

To finish the proof of Theorem A we will use the following proposition.
Proposition 2.4.2 Given a hyperbolic potential ϕ with Pf (ϕ) = 0, there exist finitely
many ergodic equilibrium states for the system (f, ϕ) and they are expanding.
In order to prove the above proposition, we take a sequence of f -invariant measures
R
{µn }n such that hµn (f ) + ϕdµn → 0 and liftable with respect to the same induced
scheme. We will show that the set of lifted measure is tight (see definition below and [22]
for details) and has the Gibbs measure as its unique accumulation point with respect
to which the inducing time is integrable. Finally, the Gibbs measure is projected to an
equilibrium state for (f, ϕ).

2.4 Finiteness of Ergodic Equilibrium States

18

Definition 2.4.1 We say that a set of measures K on X is tight if for every  > 0
there exists a compact set K ⊂ X such that η(K c ) <  for every measure η ∈ K.
Lemma 2.4.6 Let us consider the lifted sequence {µn }n . The set {µn } is tight.
Proof.
By Pinheiro’s Theorem, if τ̃ is the inducing time of F̃ , there exists θ > 0 such that
R

τ̃ dµn < θ, ∀n ∈ N. We claim that this implies that the set {µn } is tight. It is enough

to show that, given j ∈ N, we can find a compact set Kj such that µn (Kjc ) < θj , ∀n ∈ N.
If fact,
Z

θ
τ̃ dµn < θ ⇒ µn ({τ̃ > j}) < , ∀n ∈ N.
j
{τ̃ >j}

jµn ({τ̃ > j}) <

It remains to show that Kj = {τ̃ ≤ j} is compact. In fact, Kj is the union of
finitely many cylinders, which are compact. So, the set {µn } is tight, as we claimed.

Proof. (of Proposition 2.4.2): Proposition 2.4.1 implies that there exists K 0 > 0 such
that, given a cylinder Cn ∈ PnF , there exists kn ∈ N such that for k ≥ kn we have
1
µ (Cn )
≤ Sk ϕ(x) ≤ K 0 , ∀x ∈ Cn .
0
K
e k
From Lemma 2.4.6 the set {µn } is tight and we obtain a convergent subsequence,
which we keep writing {µn }n . We can see that the limit is a Gibbs measure and, by
uniqueness, is the measure µϕ . Now, we can see that the inducing time τ̃ is integrable
with respect to µϕ .
First of all, we remind that F = F̃ k and denote µn as µF,n if we look at the map
R
R
F and µF̃ ,n if we look at the map F̃ . Then, note that τ dµF,n = τ̃ k dµF̃ ,n ≤ θk.
For the purpose of this proof, we let τN := min{τ, N }. By the Monotone Convergence
Theorem, we obtain.
Z

Z
τ dµϕ ≤ lim

N →∞

Z
τN dµϕ ≤ lim lim sup
N →∞

n→∞

τN dµF,n ≤ θk.

Finally, since τ is integrable, we can project the Gibbs measure µϕ and obtain
an invariant measure µ for f . By the Abramov’s formulas we can see that µ is an
equilibrium state for the system (f, ϕ).

2.5 Applications

19

As there exists an equilibrium state, we also can find an ergodic one. Also, if ν is
an ergodic equilibrium state, we can lift it to an equilibrium state for the shift, which
is the Gibbs measure µϕ . So, the projection of it is ν. It shows that there exists at
most one ergodic equilibrium state for each induced scheme. Then, they are, at most,
finitely many.

2.5



Applications

In order to give examples of maps that satisfies our hypothesis, we begin with some
definitions as they are given in [1].
Definition 2.5.1 Let M be a compact Riemannian manifold of dimension d ≥ 1 and
f : M → M a continuous map defined on M . The map f is called non-flat if it is a
local C 1+α , (α > 0) diffeomorphism in the whole manifold except in a non-degenerate
set C ⊂ M . We say that C ⊂ M is a non-degenarate set if there exist β, B > 0 such
that the following two conditions hold.
(x)vk
≤ Bd(x, C)−β for all v ∈ Tx M .
• B1 d(x, C)β ≤ kDfkvk

For every x, y ∈ M \C with d(x, y) < d(x, C)/2 we have
B
• | log k Df (x)−1 k − log k Df (y)−1 k|≤ d(x,C)
β d(x, y).

In what follows, we give an example of a non-flat map.

2.5.1

Viana maps

Example 2.5.1 (Viana maps) We recall the definition of the open class of maps with
critical sets in dimension 2, introduced by Viana in [34]. We skip the technical points.
It can be generalized for any dimension (See [1]).
Let a0 ∈ (1, 2) be such that the critical point x = 0 is pre-periodic for the quadratic
map Q(x) = a0 − x2 . Let S 1 = R/Z and b : S 1 → R a Morse function, for instance
b(θ) = sin(2πθ). For fixed small α > 0, consider the map
f0 : S 1 × R −→ S 1 × R
(θ, x) 7−→ (g(θ), q(θ, x))

2.5 Applications

20

where g is the uniformly expanding map of the circle defined by g(θ) = dθ(modZ)
for some d ≥ 16, and q(θ, x) = a(θ) − x2 with a(θ) = a0 + αb(θ). It is easy to check
that for α > 0 small enough there exists an interval I ⊂ (−2, 2) for which f0 (S 1 × I)
is contained in the interior of S 1 × I. Thus, any map f sufficiently close to f0 in the
C 0 topology has S 1 × I as a forward invariant region. We consider from here on these
maps f close to f0 restricted to S 1 × I. Taking into account the expression of f0 it is
not difficult to check that for f0 (and any map f close to f0 in the C 2 topology) the
critical set is non-degenerate.
The main properties of f in a C 3 neighbourhood of f are summarized below (See
[1], [6], [25]):
(1) f is differentiable non-uniformly expanding , that is, there exist λ > 0 and
a Lebesgue full measure set H ⊂ S 1 × I such that for all point p = (θ, x) ∈ H,
the following holds
n−1

1X
log k Df (f i (p))−1 k−1 < −λ.
lim sup
n→∞ n
i=0
(2) Its orbits have slow approximation to the critical set, that is, for every
 > 0 the exists δ > 0 such that for every point p = (θ, x) ∈ H ⊂ S 1 × I, the
following holds
n−1

lim sup
n→∞

where

1X
− log distδ (p, C) < .
n i=0


 dist(p, C), if dist(p, C) < δ
distδ (p, C) =

1
if dist(p, C) ≥ δ

(3) f is topologically mixing : for every open set U ⊂ M , there exists n(U ) ∈ N
n
such that f n(U ) (U ) = ∩∞
n=0 f (M ).

(4) f is strongly topologically transitive: for all open set U ⊂ M , we have
n
M = ∪∞
n=0 f (U ).

(5) it has a unique ergodic absolutely continuous invariant (thus SRB) measure;
(6) the density of the SRB measure varies continuously in the L1 norm with f .

2.5 Applications

21

The idea of hyperbolic times for differentiable maps is a key notion on the study of
non-uniformly hyperbolic dynamics and it was introduced by J. Alves et al. This is a
powerful tool in order to get expansion in the context of non-uniform expansion.
In the following, we recall the definition of a hyperbolic time for differentiable maps
see([3], [25]).
Definition 2.5.2 (Hyperbolic times). Let us fix 0 < b = 13 min{1, 1/β} < 12 min{1, 1/
β}. Given 0 < σ < 1 and  > 0, we will say that n is a (σ, )-hyperbolic time
for a point x ∈ M (with respect to the non-flat map f with a β-non-degenerate critical/singular set C) if for all 1 ≤ k ≤ n we have
n−1
Y

k(Df ◦ f j (x)−1 k ≤ σ k and dist (f n−k (x), C) ≥ σ bk .

j=n−k

We denote de set of points of M such that n ∈ N is a (σ, )-hyperbolic time by
Hn (σ, , f ).
Proposition 2.5.1 (Positive frequency). Given λ > 0 there exist θ > 0 and 0 > 0
such that, for every x ∈ U and  ∈ (0, 0 ],
#{1 ≤ j ≤ n; x ∈ Hj (e−λ/4 , , f )} ≥ θn,
whenever n1

Pn−1

i
−1 −1
≥ λ and n1
i=0 log k(Df (f (x))) k

Pn−1

λ
i=0 − log dist (x, C) ≤ 16β .

If f is non-uniformly expanding with slow approximation to the critical set, it
follows from the Proposition 2.5.1 that the points of U have infinitely many moments
with positive frequency of hyperbolic times. In particular, they have infinitely many
hyperbolic times.
Proposition 2.5.2 Given σ ∈ (0, 1) and  > 0, there is δ, ρ > 0, depending only on σ
and  and on the map f , such that if x ∈ Hn (σ, , f ) then there exists a neighbourhood
Vn (x) of x with the following properties:
(1) f n maps Vn (x) diffeomorphically onto the ball Bδ (f n (x));
(2) dist(f n−j (y), f n−j (z)) ≤ σ j/2 dist(f n (y), f n (z)), ∀y, z ∈ Vn (x) and 1 ≤ j < n.
n

Df (y)|
(3) log |det
≤ ρd(f n (y), f n (z)).
|det Df n (z)|

2.5 Applications

22

for all y, z ∈ Vn (x).
The sets Vn (x) are called hyperbolic pre-balls and their images f n (Vn (x)) =
Bδ (f n (x)), hyperbolic balls.
From the above facts we can see that the Viana maps are included in our setting.
Here the Lebesgue measure is expanding.

2.5.2

Benedicks-Carleson Maps

We study a class of non-hyperbolic maps of the interval with the condition of
exponential growth of the derivative at critical values, called Collet-Eckmann Condition. We also ask the map to be C 2 and topologically mixing and the critical points
to have critical order 2 ≤ α < ∞.
Given a critical point c ∈ I, the critical order of c is a number αc > 0 such that
f (x) = f (c) ± |gc (x)|αc , ∀x ∈ Uc where gc is a diffeomorphism gc : Uc → g(Uc ) and Uc
is a neighbourhood of c.
Let δ > 0 and denote C the set of critical points and Bδ = ∪c∈C (c − δ, c + δ). Given
x ∈ I, we suppose that
• (Expansion outside Bδ ). There exists κ > 1 and β > 0 such that, if xk =
f k (x) 6∈ Bδ , 0 ≤ k ≤ n − 1 then |Df n (x)| ≥ κδ (αmax −1) eβn , where αmax =
max{αc , c ∈ C}. Moreover, if x0 ∈ f (Bδ ) or xn ∈ Bδ then |Df n (x)| ≥ κeβn .
• (Collet-Eckmann Condition). There exists λ > 0 such that
|Df n (f (c))| ≥ eλn .
• (Slow Recurrence to C). There exists σ ∈ (0, λ/5) such that
dist(f k (x), C) ≥ e−σk .

2.5.3

Rovella Maps

There is a class of non-uniformly expanding maps known as Rovella Maps. They
are derived from the so-called Rovella Attractor, a variation of the Lorenz Attractor.
We proceed with a brief presentation. See [5] for details.

2.5 Applications

23

Contracting Lorenz Attractor
The geometric Lorenz attractor is the first example of a robust attractor for a flow
containing a hyperbolic singularity. The attractor is a transitive maximal invariant
set for a flow in three-dimensional space induced by a vector field having a singularity
at the origin for which the derivative of the vector field at the singularity has real
eigenvalues λ2 < λ3 < 0 < λ1 with λ1 + λ3 > 0. The singularity is accumulated by
regular orbits which prevent the attractor from being hyperbolic.
The geometric construction of the contracting Lorenz attractor (Rovella attractor)
is the same as the geometric Lorenz attractor. The only difference is the condition
(A1)(i) below that gives in particular λ1 + λ3 < 0. The initial smooth vector field X0
in R3 has the following properties:
(A1) X0 has a singularity at 0 for which the eigenvalues λ1 , λ2 , λ3 ∈ R of DX0 (0)
satisfy:
(i) 0 < λ1 < −λ3 < 0 < −λ2 ,
(ii) r > s + 3, where r = −λ2 /λ1 , s = −λ3 /λ1 ;
(A2) there is an open set U ⊂ R3 , which is positively invariant under the flow, containing the cube {(x, y, z) :| x |≤ 1, | y |≤ 1, | x |≤ 1} and supporting the Rovella
attractor
Λ0 =

\

X0t (U ).

t≥0

The top of the cube is a Poincaré section foliated by stable lines {x = const} ∩ Σ
which are invariant under Poincaré first return map P0 . The invariance of this
foliation uniquely defines a one-dimensional map f0 : I\{0} → I for which

f 0 ◦ π = π ◦ P0 ,
where I is the interval [−1, 1] and π is the canonical projection (x, y, z) 7→ x;
(A3) there is a small number ρ > 0 such that the contraction along the invariant
foliation of lines x =const in U is stronger than ρ.
See [5] for properties of the map f0 .

2.5 Applications

24

Rovella Parameters
The Rovella attractor is not robust. However, the chaotic attractor persists in
a measure theoretical sense: there exists a one-parameter family of positive Lebesgue
measure of C 3 close vector fields to X0 which have a transitive non-hyperbolic attractor.
In the proof of that result, Rovella showed that there is a set of parameters E ⊂ (0, a0 )
(that we call Rovella parameters) with a0 close to 0 and 0 a full density point of E, i.e.
| E ∩ (0, a) |
= 1,
a→0
a
lim

such that:
(C1) there is K1 , K2 > 0 such that for all a ∈ E and x ∈ I
K2 | x |s−1 ≤ fa0 (x) ≤ K1 | x |s−1 ,
where s = s(a). To simplify, we shall assume s fixed.
(C2) there is λc > 1 such that for all a ∈ E, the points 1 and −1 have Lyapunov
exponents greater than λc :
(fan )0 (±1) > λnc , ∀n ≥ 0;
(C3) there is α > 0 such that for all a ∈ E the basic assumption holds:
| fan−1 (±1) |> e−alphan , ∀n ≥ 1;
(C4) the forward orbits of the points ±1 under fa are dense in [−1, 1] for all a ∈ E.
Definition 2.5.3 We say that a map fa with a ∈ E is a Rovella Map.
Theorem 2.5.1 (Alves-Soufi [5]) Every Rovella map is non-uniformly expanding.

Chapter 3
Induced Schemes with Special
Holes
3.1

Introduction

In this chapter we deal with open zooming maps. A zooming map is a map
which extends the notion of non-uniform expansion, where we have a type of expansion,
obtained in the presence of hyperbolic times. The zooming times extend this notion
beyond exponential contractions. In our context, a map is said to be open when the
phase space is not invariant. In other words, we begin with a closed map f : M → M
and consider a Borel set H ⊂ M , in order to study the orbits with respect to H (called
the hole of the system): if a point x ∈ M is such that its orbit pass through H, we
consider the point x escapes from the system. In particular, we study the set of points
that never pass through H. Also, once a reference measure m is fixed, we study the
escape rate defined by:


1
n−1 −j
E(f, m, H) = − lim log m ∩j=0 f (M \H) .
n→∞ n
The study of open dynamics began with Yorke and Pianigiani in the late 1970s. The
abstract concept of an open system leads immediately to the notion of a conditionally
invariant measure and escape rate along with a host of detailed questions about how
mass escapes or fails to escape from the system under time evolution. For interval
maps, induced schemes are studied in [15] and changes are made in [9] to study induced
schemes for interval maps with holes.
25

3.2 Definitions and Statement of the Result

26

In order to study the escape rate, Young towers are quite useful and the induced
schemes as well. In the context of open dynamics, the induced schemes need to have
the property of respecting the hole, as considered in [13], for example . It means
that an element of the partition can only escape entirely. It is an important property
to construct the Young tower with hole.
In this work, we prove the existence of induced schemes respecting holes of a special
type. The context is the zooming maps and the hole is obtained from small balls, by
erasing intersections with regular pre-images. Our construction mimics the work of
Pinheiro in [25], where the main ingredient is the zooming times.

3.2

Definitions and Statement of the Result

In this section we give some definitions and state our main result. We define induced
schemes and what we mean by an induced scheme respecting a hole.

3.2.1

Markov Maps and Induced Schemes

We recall the definitions of Markov partition, Markov map and induced Markov
map (See [25] for details).
Consider a measurable map f : M → M defined on the metric space M endowed
with the Borel σ-algebra.
Definition 3.2.1 (Markov partition). Let f : U → U be a measurable map defined
on a Borel set U of a compact, separable metric space M . A countable collection
P = {P1 , P2 , P3 , . . . } of Borel subsets of U is called a Markov partition if
(1) int(Pi ) ∩ int(Pj ) = ∅, if i 6= j;
(2) if f (Pi ) ∩ int(Pj ) 6= ∅ then f (Pi ) ⊃ int(Pj );
(3) #{f (Pi ); i ∈ N} < ∞;
(4) f|Pi is a homeomorphism and it can be extended to a homeomorphism sending Pi
onto f (Pi );
(5) lim diam(Pn (x)) = 0, ∀ x ∈ ∩n≥0 f −n (∪i Pi ),
n→∞

3.2 Definitions and Statement of the Result

27

where Pn (x) = {y; P(f j (y)) = P(f j (x)), ∀ 0 ≤ j ≤ n} and P(x) denotes the
element of P that contains x.
Definition 3.2.2 (Markov map). The pair (F, P), where P is a Markov partition of
F : U → U , is called a Markov map defined on U . If F (P ) = U, ∀ P ∈ P, (F, P) is
called a full Markov map.
Note that if (F, P) is a full Markov map defined on an open set U then the elements
of P are open sets (because F (P ) = U and F |P is a homeomorphism, ∀ P ∈ P).
Definition 3.2.3 (Induced Markov map). A Markov map (F, P) defined on U is called
an induced Markov map for f on U if there exists a function R : U → N =
{0, 1, 2, 3, . . . }(called inducing time) such that {R ≥ 1} = ∪P ∈P P , R |P is constant
∀ P ∈ P and F (x) = f R(x) (x), ∀ x ∈ U .
If an induced Markov map (F, P) is a full Markov map, we call (F, P) an induced
full Markov map. We will also call the pair (F, P) an induced scheme.

3.2.2

Open Dynamics

For the classical dynamical systems the phase spaces are invariant and called closed.
When we consider systems where the phase space is not invariant, they are called open.
It is done by considering holes in a closed dynamical system.
Definition 3.2.4 (Hole). Given a dynamical system f : M → M , we say that an open
set with finitely many connected components is a hole for the system.
Definition 3.2.5 (Dynamics that respects the hole). Given a hole H ⊂ M , we say
that an induced scheme (F, P) respects the hole H if we have f k (P ) ∩ H 6= ∅ ⇒
f k (P ) ⊂ H, ∀ P ∈ P, ∀ 0 ≤ k < R(P ).

3.2.3

Statement of Main Result

Now, we state our main result on the existence of induced schemes respecting holes
of a special type.
Theorem B Let f : M → M be a zooming map. There exists a ball Br (p0 ) with radius
r > 0 sufficiently small such that for any pairwise disjoint finite collection of balls

3.3 Preliminary Results

28

A = {Br (pi ), i = 1, 2, . . . , k} contained in Br (p0 )c , there are an open set U ⊂ Br (p0 ),
an induced scheme (F, P) constructed in U and a hole H, such that
∪ki=1 B ri (pi ) ⊂ H ⊂ ∪ki=1 Bri (pi )
2

and the induced scheme respects the hole H.

3.3

Preliminary Results

In order to construct an induced scheme that respect holes of a certain type, we
need the notions of linked sets, nested collections and zooming times, introduced by V.
Pinheiro in [25]. Zooming times generalize the notion of hyperbolic times, which are
fundamental for our construction. The elements of the partition will be regular preimages of a certain open set and the hole will be obtained from small balls by deleting
linked regular pre-images of the considered balls.

3.3.1

Nested Collections

We recall some definitions and results from [25], that will help us to show that the
induced map we will build respects a certain type of hole.
Definition 3.3.1 (Linked sets). We say that two open sets A and B are linked if
both A − B and B − A are not empty.
We introduce the following notation.
Notation: We write A ↔ B to mean that A and B are linked and A 6↔ B to mean
that A and B are not linked.
Definition 3.3.2 (Regular pre-images). Given V an open set, we say that P is a
regular pre-image of order n of V if f n sends P homeomorphically onto V . Denote
by ord(P ) the order of P (with respect to V ).
Let us fix a collection E0 of open sets. For each n we consider En (V ) as the collection
of pre-images of order n of V . Set En = (En (V ))V ∈E0 . We call the sequence E = (En )n a
dynamically closed family of regular pre-images if f k (E) ∈ En−k , ∀ E ∈ En , ∀ 0 ≤
k ≤ n.

3.3 Preliminary Results

29

Let E = (En )n be a dynamically closed family of regular pre-images. A set P is
called an E-pre-image of a set W if there is n ∈ N and Q ∈ En such that W ⊂ f n (Q)
−n
and P = f|Q
(W ).

Definition 3.3.3 (Nested sets). An open set V is called E-nested if it is not linked
with any E-pre-image of it.
Definition 3.3.4 (Nested collections). A collection A of open sets is called an Enested collection of sets if every A ∈ A is not linked with any E-pre-image of an
element of A with order bigger than zero. Precisely, if A1 ∈ A and P is an E-pre-image
of some A2 ∈ A, then either A1 6↔ P or P = A2 .
It follows from the definition of an E-nested collection of sets that every subcollection of an E-nested collection is also an E-nested collection. In particular, each
element of an E-nested set collection is an E-nested set.
Lemma 3.3.1 (Main property of a nested collection). If A is an E-nested collection of
open sets and P1 and P2 are E-pre-images of two elements of A with ord(P1 ) 6= ord(P2 )
then P1 6↔ P2 .
Corollary 3.3.1 (Main property of a nested set). If V is an E-nested set and P1 and
P2 are E-pre-images of V then P1 6↔ P2 . Furthermore,
(1) if P1 ∩ P2 6= ∅ then ord(P1 ) 6= ord(P2 );
(2) if P1 ⊂6= P2 with ord(P1 ) < ord(P2 ) then V is contained in an E-pre-image of
itself with order bigger than zero, f ord(P2 )−ord(P1 ) (V ) ⊂ V .

3.3.2

Constructing Nested Sets and Collections

Let A be a collection of connected open subsets such that the elements of A are
not contained in any E-pre-image of order bigger than zero of an element of A.
Definition 3.3.5 (Chains of pre-images). A finite sequence K = {P0 , P1 , . . . , Pn } of
E-pre-images of A is called a chain of E-pre-images beginning in A ∈ A if
• 0 < ord(P0 ) ≤ ord(P1 ) ≤ · · · ≤ ord(Pn );

3.3 Preliminary Results

30

• A ↔ P0 ;
• Pj−1 ↔ Pj , 1 ≤ j ≤ n;
• Pi 6= Pj if i 6= j.
For each A ∈ A define the open set
A? = A\

[

[

Pj

(Pj )j ∈chE (A) j

where chE (A) is the set of chains.
Proposition 3.3.1 (An abstract construction of a nested collection). For each A ∈ A
such that A? 6= ∅ choose a connected component A0 of A? . If A0 = {A0 ; A ∈ A and A? 6=
∅} is not an empty collection then A0 is an E-nested collection of sets.

3.3.3

Zooming Sets and Measures

For differentiable dynamical systems, hyperbolic times are a powerful tool to obtain
a type of expansion in the context of non-uniform expansion. As it was given in the previous chapter, it may be generalized for systems considered in a metric space, also with
exponential contractions. The zooming times generalizes it beyond the exponential
context. Details can be seen in [25].
Let f : M → M be a measurable map defined on a connected, compact, separable
metric space M .
Definition 3.3.6 (Zooming contractions). A zooming contraction is a sequence of
functions αn : [0, +∞) → [0, +∞) such that
• αn (r) < r, ∀ n ∈ N, ∀ r > 0.
• αn (r) < αn (s), if 0 < r < s, ∀ n ∈ N.
• αm ◦ αn (r) ≤ αm+n (r), ∀ r > 0, ∀ m, n ∈ N.
• sup

∞
X

αn (r) < ∞.

r∈(0,1) n=1

Let (αn )n be a zooming contraction and δ > 0.

3.3 Preliminary Results

31

Definition 3.3.7 (Zooming times). We say that n ∈ N is an (α, δ)-zooming time
for p ∈ X if there exists a neighbourhood Vn (p) of p such that
• f n sends Vn (p) homeomorphically onto Bδ (f n (p));
• d(f j (x), f j (y)) ≤ αn−j (d(f n (x), f n (y))) for every x, y ∈ Vn (p) and every 0 ≤ j <
n.
We call Bδ (f n (p)) a zooming ball and Vn (p) a zooming pre-ball.
We denote by Zn (α, δ, f ) the set of points in X for which n is an (α, δ)- zooming
time.
Definition 3.3.8 (Zooming measure) A f -non-singular finite measure µ defined on
the Borel sets of M is called a weak zooming measure if µ almost every point has
infinitely many (α, δ)-zooming times. A weak zooming measure is called a zooming
measure if
1
lim sup {1 ≤ j ≤ n | x ∈ Zn (α, δ, f )} > 0,
n→∞ n
µ almost every x ∈ M .
Definition 3.3.9 (Zooming set) A positively invariant set Λ ⊂ M is called a zooming
set if the above inequality holds for every x ∈ Λ.
Definition 3.3.10 (Bounded distortion) Given a measure µ with a Jacobian Jµ f , we
say that the measure has bounded distortion if there exists ρ > 0 such that
log

Jµ f (y)
≤ ρd(f n (y), f n (z))
Jµ f (z)

for every y, z ∈ Vn (x), µ-almost everywhere x ∈ M , for every hyperbolic time n of x.
The map f with an associated zooming measure is called a zooming map.
Let us introduce the following notation.
Notation: We denote by EZ = (EZ ,n )n the collection of all (α, δ)-zooming pre-balls,
where (EZ ,n )n = {Vn (x)|x ∈ Zn (α, δ, f )}. Observe that this collection is a dynamically
closed family of pre-images.
With the notation of Corollary 3.4.1, let Ai be contained in a zooming ball for all
i = 0, 1, 2, . . . , k. This corollary implies that we have A0 an EZ -nested collection, if the
chains are small enough.

3.4 Induced Schemes Respecting Special Holes

32

Definition 3.3.11 (Zooming nested collection). We call A0 an (α, δ)-zooming nested
collection.
Definition 3.3.12 (Backward separated map) We say that a map f : M → M is
backward separated if for every finite set F ⊂ M we have

d F, ∪nj=1 f −j (F )\F > 0, ∀ n ≥ 1.
Observe that if f is continuous and sup{#f −1 (x), x ∈ M } < ∞, then f is backward
separated.

3.4

Induced Schemes Respecting Special Holes

In this section, we proceed with the proof of Theorem B.
We will use the notation and results of the previous sections to prove some preliminaries results. We will follow ideas from [25].

3.4.1

Existence of Nested Collections

The following Lemma is proved in [25] for the case of a collection with one ball as
corollary of Proposition 3.3.1. We prove it here for a collection with finitely many open
sets, pairwise disjoint.
Lemma 3.4.1 (Existence of nested collections). Let  ∈ (0, 1) and A = {A0 , A1 , A2 , . . . , Ak }
be a finite collection of pairwise disjoint open sets. Given pi ∈ Ai , i = 0, 1, 2, . . . , k, set
ri = d(pi , ∂Ai ), i = 0, 1, 2, . . . , k. If we have
• f n (Ai ) 6⊂ Aj , ∀ n ≥ 1, i, j = 0, 1, 2, . . . , k;
• Every chain of E-pre-images of A has diameter less than m0 = min{ri , i =
0, 1, 2, . . . , k};
Then the set A?i contains the ball Bri (1−) (pi ), i = 0, 1, 2, . . . , k. Moreover, setting
A0i as the connected component of A?i that contains pi , we have that A0 = {A0i , i =
0, 1, 2, . . . , k} is an E-nested collection.

3.4 Induced Schemes Respecting Special Holes

33

Proof. Since f n (Ai ) 6⊂ Aj , ∀ n ≥ 1, i, j = 0, 1, 2, . . . , k, we have that Ai is not contained in any E-pre-image of A (with order bigger than zero) i = 0, 1, 2, . . . , k. Let
[
ΓAi be the collection of all chains intersecting ∂Ai . If (Pj )j ∈ ΓAi , then
Pj is a
j

connected open set intersecting ∂Ai with diameter less than m ≤ ri . So,

[

Pj ⊂

j

Bri (∂Ai ), ∀(Pj )j ∈ ΓAi .

As a consequence, we obtain A?i

= Ai \

[

[

Pj ⊃

(Pj )j ∈chE (Ai ) j
?
0
Ai \Bri (∂Ai ) ⊃ Bri (1−) (pi ). Then, Ai ⊃ Bri (1−) (pi ). Setting Ai the connected compo-

nent of A?i that contain pi , by the previous proposition we have that A0 is an E-nested
collection.

3.4.2



Existence of Zooming Nested Collections

The following lemma gives a sufficient condition to guarantee that the chains are
small enough, in order to show that A0 in the previous section is an (α, δ)-zooming
nested collection. It is proved in [25] for the case of a nested set.
Lemma 3.4.2 (Existence of zooming nested collection). Let  ∈ (0, 1) and set M0 =
max{diam(Ai ), i = 0, 1, 2, . . . , k} and m0 = min{ri , i = 0, 1, 2, . . . , k}. Suppose that
diam(Ai ) > αn (diam(Aj )), i, j = 0, 1, 2, . . . , k, ∀n ∈ N
• If we have

P

?
?
n≥1 αn (M0 ) ≤ m0 , then Ai is well defined and Ai ⊃ B(1−)ri (pi ), i =

0, 1, 2, . . . , k.
• If f is backward separated, supr>0

P

n≥1 αn (r)/r

< ∞ and M0 is sufficiently

small, M0 /2 ≤ m0 , then A?i is well defined and A?i ⊃ B(1−)ri (pi ), i = 0, 1, 2, . . . , k.
Proof. Firstly, observe that f n (Ai ) 6⊂ Aj , ∀ n ≥ 1, i, j = 0, 1, 2, . . . , k, since diam(Ai ) >
αn (diam(Aj )), i, j = 0, 1, 2, . . . , k, ∀n ∈ N. Moreover, every chain has diameter less
P
than n≥1 αn (an ) for some (am )m ∈ {diam(A0 ), diam(A1 ), diam(A2 ), . . . , diam(Ak )}N
P
P
and also, n≥1 αn (an ) < n≥1 αn (M0 ) ≤ m0 . By Lemma 3.4.1, the first part is done.
P
For the second part supr>0 n≥1 αn (r)/r < ∞, then there exists n0 ∈ N such
P
that
n≥n0 αn (M0 )/M0 < /2. If f is backward separated, let γ > 0 such that
0
d(F0 , ∪nj=1
f −j (F0 )\F0 ) > γ, where F0 = {p0 , p1 , . . . , pk }. Set rγ = 31 min{, γ}. Sup-

pose that M0 < 2rγ . Note that if j < n0 then Ai ∩ P = ∅, ∀P ∈ EZ ,j (because

0
P ∩ ∪nj=1
f −j (F0 )\F0 6= ∅ and diam(P ) < diam(Ai ) < 2rγ < γ, , ∀ i). Thus, every

3.4 Induced Schemes Respecting Special Holes

34

chain of EZ -pre-images of A begins with a pre-image of order bigger than n0 . Observe
P
that the diameter of any chain is smaller than n≥n0 αn (M0 ) < M0 /2 ≤ m0 ≤ ri , i =
0, 1, 2, . . . , k and, as a chain intersects the boundary Ai , we can conclude that the chain
cannot intersect B(1−)ri (pi ). So, we obtain A?i ⊃ B(1−)ri (pi ).

3.4.3



Finding Zooming Returns

Let f : M → M be a zooming map. Since the set of points which has infinitely
many zooming times is full measure, we can consider such points. If a zooming time is
also a return time we call it a zooming return.
In order to prove the theorem we will need the following proposition.
Lemma 3.4.3 Given t ≤ 2δ and x ∈ M with infinitely many zooming times, there is
p0 ∈ M such that p0 has infinitely many zooming returns to the ball Bt (p0 ).
Proof. Let m1 < m2 < · · · < mk < . . . be the zooming times of x. Since M is
a compact set, there is a convergent subsequence of {f mk (x)}k . So, there are mk1 <
mk2 < · · · < mkj < . . . and y ∈ M such that lim f mkj (x) = y . It means that, given
j→∞

t ≤ 2δ , ∃ j0 ∈ N such that d(f

mkj

1

(x), f

mkj

2

(x)) < t, ∀ j1 , j2 ≥ j0 . Let p0 = f

mkj

0

(x)

and nj = mkj − mkj0 . We have that nj is a zooming time of p0 , ∀ j > j0 . Then,
∀ j > j0 , nj is both a zooming time of p0 and a return time to the ball Bt (p0 ). We got
infinitely many zooming returns to Bt (p0 ).



Remark 3.4.1 From the proof we observe that f mkj (x), ∀ j ≥ j0 has infinitely many
zooming returns to Bt (p0 ).

3.4.4

Constructing an Induced Scheme

Let r < 4δ such that A0 = Br (p0 ) ⊂ Bδ (f n0 (p0 )), where n0 is a zooming return for p0
and take Ai = Br (pi ), i = 1, 2, . . . , k, a finite pairwise disjoint collection of balls outside
A0 . By considering A0 , A1 , A2 , . . . , Ak with diameters diam(Ai = 2r, i = 0, 1, 2, . . . , k,
we can take M0 = 2r, in the Lemma 3.4.2 to obtain a zooming nested collection A0 =
{A00 , A01 , A02 , . . . , A0k } such that B r2 (pi ) ⊂ A0i ⊂ Br (pi ). We will construct the induced
full Markov map in A00 , respecting the hole H = ∪i=1 A0i . Note that diam(A0 ) ≤ 2δ and
∪ki=1 B r2 (pi ) ⊂ H ⊂ ∪ki=1 Br (pi ).

3.4 Induced Schemes Respecting Special Holes

35

Now, taking t = 2r < 8δ , we apply the Proposition 3.4.3 to the ball B r2 (p0 ) to find
infinitely many points in B r2 (p0 ) that have infinitely many zooming returns to B r2 (p0 ).
In particular, there are infinitely many points in A00 that have infinitely many zooming
returns to A00 .
Let EZ the collection of zooming pre-balls. Given x ∈ A00 , let Ω(x) be the collection
of all EZ -pre-images V of A00 such that x ∈ V .
Let h(x) = {f n (x) | n is a zooming time of x}. The set Ω(x) is not empty for every
x ∈ A00 that has a zooming return to A00 . Indeed, if x ∈ A00 and f n (x) ∈ A00 ∩ h(x),
then the ball Bδ (f n (x)) = f n (Vn (x)) ⊃ A00 (because diameter(A00 ) < 2δ ). Thus, for each
−n
(A00 ) of A00
h-return of a point x ∈ A00 we can associate the EZ - pre-image P = f|V
n (x)

with x ∈ P .
Definition 3.4.1 The inducing time on A00 associated to ”the first EZ -return to A00 ”
is the function R : Br (p) → N given by

R(x) =


 min{ord(V ) | V ∈ Ω(x)}, if Ω(x) 6= ∅


0

, if Ω(x) = ∅.

Definition 3.4.2 The induced map F associated to ”the first EZ -return to A00 ” is the
map given by F (x) = f R(x) (x), ∀x ∈ A00 .
Since the collection Ω(x) is totally ordered by inclusion it follows from the Corollary
3.3.1 that there is a unique I(x) ∈ Ω(x) such that ord(I(x)) = R(x) whenever Ω(x) 6= ∅.
Lemma 3.4.4 If Ω(x) 6= ∅ =
6 Ω(y) then either I(x) ∩ I(y) = ∅ or I(x) = I(y)
Definition 3.4.3 The Markov partition associated to ”the first EZ -return to A00 ” is
the collection of open sets P given by P = {I(x) | x ∈ A00 and Ω(x) 6= ∅}.
The following corollary guarantees that P is indeed a Markov partition of open sets.
Corollary 3.4.1 Let F, R, P be as above. If P =
6 ∅, then (F, P) is an induced full
Markov map for f on A00 .
We have an induced full Markov map defined on A00 . It remains to show that the
induced full Markov map defined on A00 , respects the hole H. In fact, given an element
P ∈ P and 0 ≤ j < R(P ) and suppose that we have f j (P ) = f j−R(P ) (A00 ) ∩ H 6= ∅.

3.4 Induced Schemes Respecting Special Holes

36

There is i ∈ {1, 2, . . . , k} such that f j (P ) = f j−R(P ) (A00 ) ∩ A0i 6= ∅. As the collection A0
is nested, we must have f j (P ) ⊂ A0i ⊂ H. This holds for all P ∈ P and 0 ≤ j < R(P ).
Then, the induced full Markov map respects the hole H. The Theorem is proved.

3.4.5

A Dense Partition

Now, we suppose that M is a differentiable manifold, the zooming set is dense and
the zooming contractions are exponential. We will show that we can find p0 such that
the partition P can be constructed in such a way that it is dense in A00 . In order to
do it, we will follow ideas of Theorem 7 in [25] by using that the set of points in M
that have infinitely many zooming times is dense. Now, by following the proof of the
theorem, we obtain that there exists a dense set D ⊂ A00 such that every point in D
has a zooming return. As a consequence, the partition is dense.
Proposition 3.4.1 For r > 0 sufficientely small, there exist p0 ∈ M and a set D ⊂
A00 = Br? (p0 ) of points with infinitely many zooming times dense in A00 .
Proof. Note that
W :=

∞ [
\
j=0 j≥n

[


Vn (x)

x∈Zn (α,δ,f )

is a residual set. Thus, the set of points x ∈ W that are transitive ω(x) = M is
also a residual set (because the set of transitive points is residual). Choose a transitive
point q ∈ W . As q ∈ W , there are sequences nk → ∞ and xk ∈ Hnk such that
q ∈ Vnk (xk ), ∀k ∈ N and limk→∞ f nk (xnk ) = p0 , for some p0 ∈ M . Of course that
xk → q, for d(xk , q) ≤ σ nk δ, ∀k.
Let α = {αn }n , where αn (r) = σ n r. As f is backward separated (because #f −1 (x) <
P
∞∀x ∈ M ) and as supr>0 n≥1 αn (r)/r < ∞, we can choose any r > 0 sufficiently
small and consider the zooming nested collection A0 = {Br? (pi ), i = 0, 1, 2, . . . , k} as in
Lemma 3.4.2.
We claim that there is D ⊂ Br? (p0 ) dense in Br? (p0 ) and such that every x ∈ D has
a zooming return to Br? (p0 ). Indeed, for each y ∈ Br? (p0 ) and γ > 0 one can find m ∈ N
such that d(f m (q), y) < γ/2. Taking k > m big enough so that d(f m (xk ), f m (q)) < γ/2,
it follows that d(f m (xk , y) < γ, f m (xk ) ∈ Hnk −m and f nk −m (xk ) ∈ Br? (p0 ).



3.5 Applications

37

As before, we can take any point in D to construct an element of the partition.
Since the set is dense, we conclude that the partition is also dense.

3.5

Applications

In this section, we give examples of a zooming maps. We begin by defining a
non-flat map.

3.5.1

Hyperbolic Times

The idea of hyperbolic times is a key notion on the study of non-uniformly hyperbolic dynamics and it was introduced by Alves et al. This is powerful to get expansion
in the context of non-uniform expansion. Here, we recall the basic definitions and results on hyperbolic times that we will use later on. We will see that this notion is an
example of a Zooming Time.
In the following, we give definitions taken from [1] and [25].
Definition 3.5.1 Let M be a compact Riemannian manifold of dimension d ≥ 1 and
f : M → M a continuous map defined on M . The map f is called non-flat if it is a
local C 1+α , (α > 0) diffeomorphism in the whole manifold except in a non-degenerate
set C ⊂ M . We say that C ⊂ M is a non-degenarate set if there exist β, B > 0 such
that the following two conditions hold.
(x)vk
• B1 d(x, C)β ≤ kDfkvk
≤ Bd(x, C)−β for all v ∈ Tx M .

For every x, y ∈ M \C with d(x, y) < d(x, C)/2 we have
B
• | log k Df (x)−1 k − log k Df (y)−1 k|≤ d(x,C)
β d(x, y).

In the following, we give the definition of a hyperbolic time [3], [25].
Definition 3.5.2 (Hyperbolic times). Let us fix 0 < b = 13 min{1, 1/β} < 12 min{1, 1/
β}. Given 0 < σ < 1 and  > 0, we will say that n is a (σ, )-hyperbolic time
for a point x ∈ M (with respect to the non-flat map f with a β-non-degenerate critical/singular set C) if for all 1 ≤ k ≤ n we have
n−1
Y
j=n−k

k(Df ◦ f j (x)−1 k ≤ σ k and dist (f n−k (x), C) ≥ σ bk .

3.5 Applications

38

We denote de set of points of M such that n ∈ N is a (σ, )-hyperbolic time by
Hn (σ, , f ).
Proposition 3.5.1 (Positive frequence). Given λ > 0 there exist θ > 0 and 0 > 0
such that, for every x ∈ U and  ∈ (0, 0 ],
#{1 ≤ j ≤ n; x ∈ Hj (e−λ/4 , , f )} ≥ θn,
whenever n1

Pn−1

i
−1 −1
≥ λ and n1
i=0 log k(Df (f (x))) k

Pn−1

λ
i=0 − log dist (x, C) ≤ 16β .

If f is non-uniformly expanding, it follows from the proposition that the points
of U have infinitely many moments with positive frequency of hyperbolic times. In
particular, they have infinitely many hyperbolic times.
The following proposition shows that the hyperbolic times are indeed zooming
times, where the zooming contraction is αk (r) = σ k/2 r.
Proposition 3.5.2 Given σ ∈ (0, 1) and  > 0, there is δ, ρ > 0, depending only on σ
and  and on the map f , such that if x ∈ Hn (σ, , f ) then there exists a neighbourhood
Vn (x) of x with the following properties:
(1) f n maps Vn (x) diffeomorphically onto the ball Bδ (f n (x));
(2) dist(f n−j (y), f n−j (z)) ≤ σ j/2 dist(f n (y), f n (z)), ∀y, z ∈ Vn (x) and 1 ≤ j < n.
n

Df (y)|
(3) log |det
≤ ρd(f n (y), f n (z)).
|det Df n (z)|

for all y, z ∈ Vn (x).
The sets Vn (x) are called hyperbolic pre-balls and their images f n (Vn (x)) = Bδ (f n (x)),
hyperbolic balls.
In the following, we give an example of such a map.

3.5.2

Viana maps

Example 3.5.1 (Viana maps) We recall the definition of the open class of maps with
critical sets in dimension 2, introduced by M. Viana in [34]. We skip the technical
points. It can be generalized for any dimension (See [1]).

3.5 Applications

39

Let a0 ∈ (1, 2) be such that the critical point x = 0 is pre-periodic for the quadratic
map Q(x) = a0 − x2 . Let S 1 = R/Z and b : S 1 → R a Morse function, for instance
b(θ) = sin(2πθ). For fixed small α > 0, consider the map
f0 : S 1 × R −→ S 1 × R
(θ, x) 7−→ (g(θ), q(θ, x))
where g is the uniformly expanding map of the circle defined by g(θ) = dθ(modZ)
for some d ≥ 16, and q(θ, x) = a(θ) − x2 with a(θ) = a0 + αb(θ). It is easy to check
that for α > 0 small enough there is an interval I ⊂ (−2, 2) for which f0 (S 1 × I) is
contained in the interior of S 1 × I. Thus, any map f sufficiently close to f0 in the
C 0 topology has S 1 × I as a forward invariant region. We consider from here on these
maps f close to f0 restricted to S 1 × I. Taking into account the expression of f0 it is
not difficult to check that for f0 (and any map f close to f0 in the C 2 topology) the
critical set is non-degenerate.
The main properties of f in a C 3 neighbourhood of f that we will use here are
summarized below (See [1], [6], [25]):
(1) f is non-uniformly expanding , that is, there exist λ > 0 and a Lebesgue full
measure set H ⊂ S 1 × I such that for all point p = (θ, x) ∈ H, the following
holds

n−1

1X
log k Df (f i (p))−1 k−1 < −λ.
lim sup
n→∞ n
i=0
(2) Its orbits have slow approximation to the critical set, that is, for every
 > 0 the exists δ > 0 such that for every point p = (θ, x) ∈ H ⊂ S 1 × I, the
following holds
n−1

1X
lim sup
− log distδ (p, C) < .
n→∞ n
i=0
where


 dist(p, C), if dist(p, C) < δ
distδ (p, C) =

1
if dist(p, C) ≥ δ

(3) f is topologically mixing;
(4) f is strongly topologically transitive;
(5) it has a unique ergodic absolutely continuous invariant (thus SRB) measure;

3.5 Applications

40

(6) the density of the SRB measure varies continuously in the L1 norm with f .

3.5.3

Benedicks-Carleson Maps

We study a class of non-hyperbolic maps of the interval with the condition of
exponential growth of the derivative at critical values, called Collet-Eckmann Condition. We also ask the map to be C 2 and topologically mixing and the critical points
to have critical order 2 ≤ α < ∞.
Given a critical point c ∈ I, the critical order of c is a number αc > 0 such that
f (x) = f (c) ± |gc (x)|αc , ∀x ∈ Uc where gc is a diffeomorphism gc : Uc → g(Uc ) and Uc
is a neighbourhood of c.
Let δ > 0 and denote C the set of critical points and Bδ = ∪c∈C (c − δ, c + δ). Given
x ∈ I, we suppose that
• (Expansion outside Bδ ). There exists κ > 1 and β > 0 such that, if xk =
f k (x) 6∈ Bδ , 0 ≤ k ≤ n − 1 then |Df n (x)| ≥ κδ (αmax −1) eβn , where αmax =
max{αc , c ∈ C}. Moreover, if x0 ∈ f (Bδ ) or xn ∈ Bδ then |Df n (x)| ≥ κeβn .
• (Collet-Eckmann Condition). There exists λ > 0 such that
|Df n (f (c))| ≥ eλn .
• (Slow Recurrence to C). There exists σ ∈ (0, λ/5) such that
dist(f k (x), C) ≥ e−σk .

3.5.4

Rovella Maps

There is a class of non-uniformly expanding maps known as Rovella Maps. They
are derived from the so-called Rovella Attractor, a variation of the Lorenz Attractor.
We proceed with a brief presentation. See [5] for details.
Contracting Lorenz Attractor
The geometric Lorenz attractor is the first example of a robust attractor for a flow
containing a hyperbolic singularity. The attractor is a transitive maximal invariant
set for a flow in three-dimensional space induced by a vector field having a singularity

3.5 Applications

41

at the origin for which the derivative of the vector field at the singularity has real
eigenvalues λ2 < λ3 < 0 < λ1 with λ1 + λ3 > 0. The singularity is accumulated by
regular orbits which prevent the attractor from being hyperbolic.
The geometric construction of the contracting Lorenz attractor (Rovella attractor)
is the same as the geometric Lorenz attractor. The only difference is the condition
(A1)(i) below that gives in particular λ1 + λ3 < 0. The initial smooth vector field X0
in R3 has the following properties:
(A1) X0 has a singularity at 0 for which the eigenvalues λ1 , λ2 , λ3 ∈ R of DX0 (0)
satisfy:
(i) 0 < λ1 < −λ3 < 0 < −λ2 ,
(ii) r > s + 3, where r = −λ2 /λ1 , s = −λ3 /λ1 ;
(A2) there is an open set U ⊂ R3 , which is positively invariant under the flow, containing the cube {(x, y, z) :| x |≤ 1, | y |≤ 1, | x |≤ 1} and supporting the Rovella
attractor
Λ0 =

\

X0t (U ).

t≥0

The top of the cube is a Poincaré section foliated by stable lines {x = const} ∩ Σ
which are invariant under Poincaré first return map P0 . The invariance of this
foliation uniquely defines a one-dimensional map f0 : I\{0} → I for which

f 0 ◦ π = π ◦ P0 ,
where I is the interval [−1, 1] and π is the canonical projection (x, y, z) 7→ x;
(A3) there is a small number ρ > 0 such that the contraction along the invariant
foliation of lines x =const in U is stronger than ρ.
See [5] for properties of the map f0 .
Rovella Parameters
The Rovella attractor is not robust. However, the chaotic attractor persists in
a measure theoretical sense: there exists a one-parameter family of positive Lebesgue

3.5 Applications

42

measure of C 3 close vector fields to X0 which have a transitive non-hyperbolic attractor.
In the proof of that result, Rovella showed that there is a set of parameters E ⊂ (0, a0 )
(that we call Rovella parameters) with a0 close to 0 and 0 a full density point of E, i.e.
| E ∩ (0, a) |
= 1,
a→0
a
lim

such that:
(C1) there is K1 , K2 > 0 such that for all a ∈ E and x ∈ I
K2 | x |s−1 ≤ fa0 (x) ≤ K1 | x |s−1 ,
where s = s(a). To simplify, we shall assume s fixed.
(C2) there is λc > 1 such that for all a ∈ E, the points 1 and −1 have Lyapunov
exponents greater than λc :
(fan )0 (±1) > λnc , ∀n ≥ 0;
(C3) there is α > 0 such that for all a ∈ E the basic assumption holds:
| fan−1 (±1) |> e−alphan , ∀n ≥ 1;
(C4) the forward orbits of the points ±1 under fa are dense in [−1, 1] for all a ∈ E.
Definition 3.5.3 We say that a map fa with a ∈ E is a Rovella Map.
Theorem 3.5.1 (Alves-Soufi [5]) Every Rovella map is non-uniformly expanding.

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[34] M.

Viana,

46
Multidimensional

NonHyperbolic

Attractors,

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