Strong Stochastic Stability for Non

Helder Soares Vilarinho
Strong Stochastic Stability for
Non-Uniformly Expanding Maps
DEPARTAMENTO MATEMATICA PURA
Helder Soares Vilarinho
Strong Stochastic Stability for
Non-Uniformly Expanding Maps
Tese submetida à Faculdade de Ciências da Universidade do Porto
para obtenção do grau de Doutor em Matemática
Agosto de 2009
DEPARTAMENTO MATEMATICA PURA
À minha mulher, Catarina.
Agradecimentos
Agradeço de um modo muito especial ao meu orientador, Professor José Ferreira
Alves, por toda a atenção que me despendeu. Pelas excelentes conversas, sem as
quais este trabalho não teria sido possı́vel. Pela amizade, sempre presente. Pela
confiança e pelas palavras de incentivo, verdadeiras âncoras naqueles momentos mais
difı́ceis.
Agradeço à minha mulher, Catarina, por estar sempre ao meu lado. Pela sua
compreensão e amizade.
Agradeço aos meus pais e à minha irmã pelo seu infindável apoio.
Agradeço a todas as pessoas que não tendo, imerecidamente, o seu nome aqui
registado são vitalmente importantes na minha vida e, em particular, em todo o
percurso que me conduziu aqui. São imensas.
Agradeço ao Departamento de Matemática da Universidade da Beira Interior,
ao Departamento de Matemática Pura da Faculdade de Ciências da Universidade
do Porto e ao Centro de Matemática da Universidade do Porto por todo o apoio.
Agradeço também à Fundação para a Ciência e Tecnologia pelo financiamento ao
longo deste trabalho (SFRH/BD/24353/2005).
v
Resumo
Este trabalho aborda a estabilidade estocástica forte de uma ampla classe de sistemas
dinâmicos discretos – as transformações não-uniformemente expansoras – quando
a transformação determinı́stica é sujeita a perturbações aleatórias. A estabilidade
estocástica foi estabelecida em [AAr03] para uma classe geral de transformações nãouniformemente expansoras, no sentido da convergência da medida fı́sica para a medida
de probabilidade SRB na topologia fraca∗ . Este resultado é aqui melhorado para a
estabilidade estocástica forte, i.e., prova-se a convergência da densidade da medida
fı́sica para a densidade da medida SRB na norma L1 , formulada num quadro mais
geral de perturbações aleatórias.
Como consequência, prova-se a estabilidade estocástica forte em dois exemplos de
transformações não-uniformemente expansoras. O primeiro exemplo refere-se a uma
classe de difeomorfismos locais introduzido em [ABV00] e o segundo às aplicações de
Viana, um exemplo em dimensões altas e com conjunto crı́tico introduzido em [V97].
vi
Abstract
In this work, we address the strong stochastic stability of a broad class of discretetime dynamical systems – non-uniformly expanding maps – when some random noise
is introduced in the deterministic dynamics. A weaker form of stochastic stability
was established in [AAr03] for a general class of non-uniformly expanding maps, in
the sense of convergence of the physical measure to the SRB probability measure
in the weak∗ topology. We improve this result obtaining strong stochastic stability,
i.e., the convergence of the density of the physical measure to the density of the
SRB probability measure in the L1 -norm, in a more general framework of random
perturbations.
Consequently, we are able to obtain the strong stochastic stability for two examples
of non-uniformly expanding maps. The first is related to an open class of local
diffeomorphisms introduced in [ABV00] and the second to Viana maps - a higher
dimensional example with critical set introduced in [V97].
vii
Contents
Introduction
1
1 Definitions and main results
6
1.1 Non-uniformly expanding maps . . . . . . . . . . . . . . . . . . . . .
6
1.2 Non-uniform expansion on random orbits . . . . . . . . . . . . . . . .
7
1.3 Strong Stochastic Stability . . . . . . . . . . . . . . . . . . . . . . . .
10
1.4 Strong Statistical Stability . . . . . . . . . . . . . . . . . . . . . . . .
11
2 Random induced schemes
13
2.1 Random Gibbs-Markov maps . . . . . . . . . . . . . . . . . . . . . .
13
2.2 Hyperbolic times and bounded distortion . . . . . . . . . . . . . . . .
15
2.3 Transitivity and growing to large scale . . . . . . . . . . . . . . . . .
27
2.4 The partitioning algorithm . . . . . . . . . . . . . . . . . . . . . . . .
30
2.5 Expansion and bounded distortion
. . . . . . . . . . . . . . . . . . .
34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.7 Uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.6 Metric estimates
3 Stochastic stability
43
3.1 Physical measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.1.1
Induced random measures . . . . . . . . . . . . . . . . . . . .
43
3.1.2
Sample and stationary measures . . . . . . . . . . . . . . . . .
47
3.1.3
Ergodicity and unicity . . . . . . . . . . . . . . . . . . . . . .
49
3.2 Strong stochastic stability . . . . . . . . . . . . . . . . . . . . . . . .
53
4 Examples
64
4.1 Local diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
64
ix
CONTENTS
4.2 Viana maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.2.1
Deterministic estimates . . . . . . . . . . . . . . . . . . . . . .
70
4.2.2
Estimates for random perturbations . . . . . . . . . . . . . . .
71
A Measure and Ergodic theory
73
A.1 Measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
A.2 SRB measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
B Random perturbations
79
B.1 The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
B.2
80
Stationary, Markov and Physical measures . . . . . . . . . . . . . . .
Bibliography
89
Introduction
The main ingredients of a Dynamical Systems are a phase space, where the action
takes place, and a law that governs the system and tell us how elements of the phase
space evolve in time, in a discrete or continuous manner. The phase space, that we
call generically M, is a set endowed with some structure, e.g. topological, differentiable or measurable, that is preserved by the dynamics. In the discrete case, time is
usually described by Z or N (for invertible time or not, respectively). The evolution
law is then an application f : M → M, and the dynamics in time corresponds to
successive iterations (compositions) of f applied to each element of M. It is quite
common in Dynamical Systems to denote the n-th iterate, n ≥ 1, by f n = f ◦ . . . ◦ f
(n times composition), f 0 = id, f −n = (f n )−1 if f is invertible, and call the sequence
(f n (x))n∈N or Z the orbit of x by f . In the continuous situation, time is usually described using R and R+ and the evolution is given by a flow φt that usually appears
as a solution of a differential equation that determines the evolution of the system for
and initial state x ∈ M. The orbit is in this case {φt (x)}t∈R or R+ .
The major goals of Dynamical Systems are to describe the behavior of typical
orbits, specially in asymptotic terms, and to understand how stable they are, that
is, how their behavior changes when the system is slightly modified, or it is exposed
to perturbations during time evolution. Despite the deterministic formulation of
dynamical systems, it is easy to find examples whose evolution law is of extremely
simple formulation and whose dynamics has a high level of complexity and sensitivity
to perturbations. The word chaos arises quite naturally associated to this phenomena
and has become a universal link among different areas of Mathematics, or between
this and many other sciences.
This work concerns stability of systems, in a sense that we shall precise later, in
a broad class of discrete-time dynamical systems – non-uniformly expanding maps –
1
CONTENTS
2
when some random noise is introduced in the deterministic dynamics.
Due to the already mentioned complexity on the time evolution of many dynamical systems, there are several ways of facing the study of Dynamical Systems
in general. An interesting approach is given by Ergodic Theory, which aims at a
statistical description of orbits in a measurable phase space. The existence of an
invariant measure for a given dynamics is an important fact in this context, specially
if we recall Birkhoff’s ergodic theorem, which describes time averages of observable
phenomena for typical points with respect to that measure. However, it may happen that an invariant measure lacks of physical meaning. Sinai-Ruelle-Bowen (SRB)
measures play a particularly important role in this context, since they provide information about the statistics of orbits for a large set of initial states. These are
invariant measures which are somewhat compatible with the reference volume measure, when this is not preserved. They are also often called physical measures, but
we shall use the term physical in another specific context. SRB measures were introduced in the 70’s by Sinai [Si72], Ruelle[R76] and Bowen [BowR75, Bow75] for
Anosov and Axiom A attractors, both in discrete and continuous time systems. See
also [KS69] for endomorphisms. The definition of SRB measures has known several
modifications, essentiality motivated by the development of the theory of Dynamical Systems and the appearance of new examples and subjects of interest, causing
even some ambiguity on definitions in different contemporary works. We refer e.g.
[Y02] for a compilation of related results and historical background, and references
therein. The classes of systems studied by Sinai, Ruelle and Bowen, exhibit uniform
expansion/contraction behavior in invariant sub-bundles of the tangent bundle of a
Riemannian manifold, and statistical properties of dynamical system with this properties were systematically addressed in subsequent work of many different authors; see
[Bow70, Bow75, BowR75, KS69, R76, Si68, Si72] for initials works. Systems exhibiting expansion only in asymptotic terms have been considered in [J81], where it was
established the existence of physical measures for many quadratic transformations of
the interval; see also [CE80, BeC85, BeY92]. Related to [BeC85] is the work [BeC91]
for Hénon maps exhibiting strange attractors. Results for multidimensional nonuniformly expanding systems appear in [V97, A00], and motivated by these results
[ABV00] stated general conclusions for systems exhibiting non-uniformly expanding
behavior.
3
CONTENTS
The introduction of random noise in non-uniformly expanding dynamical systems
was addressed in several works with slightly different means. The idea of random
perturbations is to consider at each iterate a map close to the original one, chosen
independently according to some probabilistic law θǫ , where ǫ > 0 is the noise level
(for instance, in an ǫ neighborhood of the original map). For the statistical analysis
of random perturbations we use a notion similar to invariant measure in deterministic
cases. We say that µǫ is a stationary measure if
Z Z
ϕ(ft (x)) dµǫ (x)dθǫ (t) =
Z
ϕ(x) dµǫ (x),
for every continuous function ϕ : M → R. We say that µǫ is a physical measure if for
a set with positive Lebesgue measure of initial points x ∈ M we have
n−1
1X
ϕ((fωj−1 ◦ · · · ◦ fω0 )(x)) =
lim
n→+∞ n
j=0
Z
ϕ(x) dµǫ (x),
for every continuous ϕ : M → R and almost all sequence (fω0 , fω1 , . . .) with respect
to the product measure (θǫ × θǫ × · · · ). Physical measures for random perturbations
are then the somewhat equivalent to SRB measures in the deterministic context. In
order to distinguish them in the deterministic and random perturbation contexts, we
shall refer to physical measures only in the random perturbation setting and to SRB
measures in the deterministic setting.
Stochastic stability is a rather vague notion, but it tries to reflect that the introduction of small random noise affects just slightly the statistical description of the
system. We call a system stochastically stable if the stationary physical measures converge in the weak∗ topology to related SRB measure, as ǫ goes to zero, and strongly
stochastically stable if the convergence is with respect to the densities of physical and
SRB measures in the L1 -norm. Stochastic stability was established in [AAr03] for a
general class of non-uniformly expanding maps in the weak sense described above.
The main goal of this work is to improve that result to strong stochastic stability
in a more general framework of random perturbations. Important results of stochastic stability for systems with non-uniform expansion were also obtained in [Y86] for
hyperbolic attractors and [BeY92] for certain one-dimensional maps.
We can also formulate the random perturbations in terms of Markov chains. A
CONTENTS
4
system will be stable if, for any one-parameter family the stationary measure converges weakly to the SRB measure preserved by an original map f , as the stochastic transition probabilities goes to δf , when vanishing the noise level. We refer to
[Ki86, Ki88] for a background and treatment of the topic. For stochastic stability in
both terms and types we described, see e.g. [BaY93, BaKS96, Ba97] for uniformly expanding maps, [BaV96] for a large class of non-uniformly hyperbolic unimodal maps
and [BeV06] for Hénon-like maps. For more related topics, see e.g. [CoY05] for an
analysis of SRB measures as zero-noise limits, and [Ar00] for important contributions
to the stochastic part of a conjecture by Palis in [Pa00].
The main result of this thesis is Theorem 1.3.1 stated in a general setting of nonuniformly expanding maps and whose proof relies on their intrinsic properties. This
enables us to use the result in several situations and examples, and can be a useful
tool in the analysis of stochastic properties of dynamical systems with non-uniform
expansion behavior. The work is structured in the following way. In Chapter 1
we present some definitions and establish the main result on the strong stochastic
stability for non-uniformly expanding maps allowing the presence of a critical set.
For help with the reading, as for the notation and clearness of definitions, there are
two complementary appendices. In Appendix A there are standard definitions and
results of Measure Theory and Ergodic Theory. All the subjects therein can be easily
found in several books and we just suggest a few references. In Appendix B are
introduced the random perturbations in the precise formulation we use along this
work. Due to the variety of formulations of this matter, it may be important for an
easy understanding of the subjects and notation. We compile some results of interest
to the work that are present in the literature, sometimes with different generality or
perspective. Chapters 2 and 3 are devoted to prove the main result. In Chapter 2,
based on the work of [ALP05] for deterministic non-uniformly expanding maps and
partially extended in [AAr03] to the random situation, we prove the existence of an
induced region with some Gibbs-Markov structure. In Chapter 3 we follow initially
some ideas of the random version of Young Towers from [BaBeM02] to construct
a stationary probability measure, and recover a result in [ABV00] to prove that,
indeed, this stationary probability measure will be ergodic, absolutely continuous,
and therefore unique. We finish this chapter proving the strong stochastic stability,
inspired in a proof at [AV02] were the strong statistical stability is achieved. In the
CONTENTS
5
last chapter we present two classes of examples that fit our assumptions and for which
we obtain the strong stochastic stability. The first example concerns to an open class
of local diffeomorphisms introduced in [ABV00] and the second to Viana maps, a
higher dimensional example with critical set introduced in [V97]. This improves the
weaker form of stochastic stability for both examples proved in [AAr03].
Chapter 1
Definitions and main results
In this chapter we introduce the fundamental definitions related to non-uniformly
expanding maps and random perturbations. We state the main result of this thesis
in Theorem 1.3.1, which expresses the strong stochastic stability for the class of
dynamical systems we address. The Chapters 2 and 3 are devoted to the proof of the
main theorem and, in Chapter 4, we present examples of dynamical systems for which
the theorem can be applied. For ease of reading, namely regarding the notations and
clearness of definitions involved, we refer to Appendices A and B, where we present
standard definitions and results on Measure Theory, Ergodic Theory and Random
Dynamical Systems.
1.1
Non-uniformly expanding maps
Let M be a compact boundaryless manifold endowed with the normalized Lebesgue
measure m. Let f : M → M be a continuous map which is a local diffeomorphism in
the whole manifold except, eventually, at a set C ⊂ M of critical points. This set C
may be taken as some set of points where the derivative of f fails to be invertible or
simply does not exist.
Definition 1.1.1. We say that C ⊂ M is a non-degenerate critical set if it has zero
Lebesgue measure and the following conditions hold:
1. There are constants B > 1 and β > 0 such that for every x ∈ M \ C
(c1 )
1
kDf (x)vk
dist(x, C)β ≤
≤ B dist(x, C)−β for all v ∈ Tx M \ {0}.
B
kvk
6
CHAPTER 1. DEFINITIONS AND MAIN RESULTS
7
2. For every x, y ∈ M \ C with dist(x, y) < dist(x, C)/2 we have
(c2 )
(c3 )
log kDf (x)−1k − log kDf (y)−1k ≤
B
dist(x, y);
dist(x, C)β
B
|log | det Df (x)| − log | det Df (y)| | ≤
dist(x, y).
dist(x, C)β
The first condition says that f behaves like a power of the distance to C and the
next two conditions say that the functions log | det Df | and log kDf −1k are locally
Lipschitz at points x ∈ M \ C, with Lipschitz constant depending on dist(x, C).
Given δ > 0 and x ∈ M \ C we define the δ-truncated distance from x to C
distδ (x, C) =
(
1,
if dist(x, C) ≥ δ
dist(x, C), otherwise.
Definition 1.1.2. Let f : M → M be a C 2 local diffeomorphism outside a non-
degenerate critical set C. We say that f is non-uniformly expanding on a set H ⊂ M
if the following conditions hold:
1. there is a0 > 0 such that for each x ∈ H
n−1
lim sup
n→+∞
1X
−1
log kDf (f j (x)) k < −a0 ;
n j=0
(1.1.1)
2. for every b0 > 0 there exists δ > 0 such that for each x ∈ H
n−1
1X
lim sup
− log distδ (f j (x), C) < b0 .
n→+∞ n
j=0
(1.1.2)
We will refer to the second condition above by saying that the orbits of points in
H have slow recurrence to C. The case C = ∅ may also be considered, and in such
case the definition reduces to the first condition. A map is said to be non-uniformly
expanding if it is non-uniformly expanding on a set of full Lebesgue measure.
1.2
Non-uniform expansion on random orbits
Given a non-uniformly expanding map f we introduce random perturbations in a
quite usual form, that we detail in the following. Apart our interest concerns to
8
CHAPTER 1. DEFINITIONS AND MAIN RESULTS
the forward random orbits behavior our techniques are developed considering also
the past of the orbits, in a two-sided random dynamical systems. As we point out
in Appendix B, in particular through Proposition B.2.3, finding an ergodic Markov
measure for the skew map (briefly, it is an invariant measure whose sample measures
only depend on the past) allows us to get a stationary measure and characterize a
random dynamical system in a statistical sense. Namely, the presence of an absolutely continuous ergodic stationary probability measure (and thus physical) means
that, even if we allow some randomness on the map we use at each iteration, there is
a relevant (in terms of the Lebesgue measure) set of initial points for which almost
all random orbits can be statistical characterized by this physical measure. Moreover, we discuss the uniqueness and robustness of the physical measure, that is, the
convergence of the physical measure to the SRB probability measure of the original
deterministic dynamical system when the noise level goes to zero.
We first introduce some definitions for random perturbations and refer for Appendix B for details. Let f : M → M be a non-uniformly expanding map and
Φ : T −→ C 2 (M, M)
t
7−→ Φ(t) = ft
a continuous map from a metric space T into the space of C 2 maps from M to M,
with f = ft∗ for some fixed t∗ ∈ T . For any element ω = (..., ω−1 , ω0 , ω1 , . . .) in the
infinite product space T Z we define
fωn (x) =
(
x
if
n=0
(fωn−1 ◦ · · · ◦ fω1 ◦ fω0 )(x)
if
n>0
.
In particular, fω = fω0 and fσl (ω) = fωl for every ω ∈ T Z and l ∈ Z, where σ : T Z → T Z
is the left shift on the elements ω ∈ T Z . Given x ∈ M and ω ∈ T Z we call the sequence
fωn (x) n∈N a random orbit of x. For ω ∗ = (..., t∗ , t∗ , t∗ , ...), the orbit fωn∗ (x) n∈N is
the unperturbed deterministic orbit of x given by the original dynamics f . We also
consider a family (θǫ )ǫ>0 of Borel probability measures on T , with non-empty supports,
such that
supp(θǫ ) → {t∗ } as ǫ → 0.
This provides a family of Borel product measure spaces (T Z , B(T )Z , θǫZ )ǫ>0 , where θǫZ
CHAPTER 1. DEFINITIONS AND MAIN RESULTS
9
is invariant for the shift map σ. The maps ft used in each iteration of a random orbit
are given by an independent identically distributed choice according to law (θǫ )∗ Φ on
Φ(T ) ⊂ C 2 (M, M). We will refer to such a pair {Φ, (θǫ )ǫ>0 } as above as a random
perturbation of f . We remark that no absolute continuity property will be required
for θǫ . So, if θǫ is the Dirac measure supported on t0 ∈ T then {Φ, (θǫ )ǫ>0 } reduces
to the deterministic dynamics ft0 .
Due to the presence of the critical set, we will restrict the class of perturbations
we are going to consider for maps with critical sets: we take all the maps ft with the
same critical set C by imposing that
Dft (x) = Df (x),
for every x ∈ M \ C and t ∈ T .
(1.2.1)
This may be implemented, for instance, in parallelizable manifolds (with an additive
group structure, e.g. tori Td (or cylinders Td−k × Rk ), by considering T = {t ∈ Rd :
ktk ≤ ǫ0 } for some ǫ0 > 0, and taking ft = f + t, that is, adding at each step a
random noise to the unperturbed dynamics.
Definition 1.2.1. We say that f is non-uniformly expanding on random orbits if
the following conditions hold, at least for small ǫ > 0:
1. there is a0 > 0 such that for θǫZ × m almost every (ω, x) ∈ T Z × M
n−1
1X
lim sup
log kDf (fωj (x))−1 k < −a0 ;
n→+∞ n
j=0
(1.2.2)
2. given any small b0 > 0 there is δ > 0 such that for θǫZ × m almost every
(ω, x) ∈ T Z × M
n−1
1X
lim sup
− log distδ (fωj (x), C) < b0 .
n→+∞ n
j=0
(1.2.3)
When C is equal to the empty set, then we naturally disregard the second condition
in the definition above. We set Ωǫ to be the θǫZ full measure subset of realizations
ω ∈ T Z for which conditions (1.2.2) and (1.2.3) are satisfied for all σ l (ω), l ∈ Z, and
Lebesgue almost every x ∈ M. Since f is itself a non-uniformly expanding map then
ω ∗ belongs to Ωǫ .
10
CHAPTER 1. DEFINITIONS AND MAIN RESULTS
For ω ∈ Ωǫ , the condition (1.2.2) implies that the expansion time function
(
n−1
1X
Eω (x) = min N ≥ 1 :
log kDf (fωj (x))−1 k ≤ −a0 , for all n ≥ N
n j=0
)
is defined and finite Lebesgue almost everywhere in M. We think of Eω (x) as the
waiting time before the exponential derivative growth along the random orbit kicks
in.
According to Remark 2.2.6, condition (1.2.3) is not needed in all its strength and
it is enough that it holds for suitable b0 > 0 and δ > 0, chosen in such a way that the
proof of Proposition 2.2.5 works. So, for ω ∈ Ωǫ we can define the recurrence time
function Lebesgue almost everywhere in M:
(
n−1
1X
Rω (x) = min N ≥ 1 :
− log distδ (fωj (x), C) ≤ b0 ,
n j=0
for all n ≥ N
)
.
This is again an asymptotic statement and we have no a-priori knowledge about how
fast this limit is approached or with what degree of uniformity for different points x
or different random orbits. We introduce the tail set (at time n)
Γnω = x : Eω (x) > n or Rω (x) > n .
(1.2.4)
This is the set of points in M whose random orbit at time n have not yet achieved
either the uniform exponential growth of derivative or the uniform subexponential
recurrence given by conditions (1.2.2) and (1.2.3). If the critical set is empty, we
simply ignore the recurrence time function and consider only the expansion time
function in the definition of Γnω .
1.3
Strong Stochastic Stability
It is known that a deterministic non-uniformly expanding map f admits a finite number of absolutely continuous ergodic probability measures (thus SRB measures) (see
[ABV00]). Moreover, if f is also topologically transitive, then the SRB probability
measure µf is unique (see [A03]). We state now our main result which asserts the
existence of a unique stationary ergodic probability measure and the strong stochastic
CHAPTER 1. DEFINITIONS AND MAIN RESULTS
11
stability for non-uniformly expanding maps. We mean, the convergence in the L1 norm of the density hǫ of the stationary measure µǫ to the density hf of the probability
measure µf .
Theorem 1.3.1. Let f be a transitive non-uniformly expanding map and non-uniformly
expanding on random orbits, for which exist p > 1, C > 0 such that m(Γnω ) < Cn−p
for every ω ∈ Ωǫ .
1. If ǫ is small enough then f admits a unique absolutely continuous ergodic stationary probability measure µǫ .
2. Set hǫ = dµǫ /dm. The map f is strongly stochastically stable:
lim khǫ − hf k1 = 0.
ǫ→0
This theorem improves the results in [AAr03] where it was established the stochastic convergence in the weak sense (the convergence of µǫ to µf in the weak∗ topology).
This strong form of stochastic convergence for higher dimensional non-uniformly expanding maps with critical sets examples is for first time stated here. As we will
show in Chapter 4, there are examples of non-uniformly expanding maps, in higher
dimensions and with or without critical set, that fits the hypothesis of this theorem. Consequently, we obtain this stronger version of stochastic stability for those
examples.
Theorem 1.3.1 will be proved throughout Chapters 2 and 3.
1.4
Strong Statistical Stability
Under the hypothesis of Theorem 1.3.1 we can formulate the strong statistical stability for such a non-uniformly expanding map, that is, the continuity of f 7→ dµf /dm,
with respect to the L1 −norm in the space of densities. Providing that supp(θǫ ) →
{t∗ }, as ǫ → 0, our arguments for stochastic stability can be carried out with no
extra assumptions on the probabilities θǫ , in particular no absolutely continuity with
respect to Lebesgue measure is needed. This is enough to obtain a statement about
the statistical stability as a consequence of the stochastic stability by a particular
CHAPTER 1. DEFINITIONS AND MAIN RESULTS
12
formulation of the random perturbations. The strong statistical stability for nonuniformly expanding maps was already obtained in [AV02].
2
Let T = F be some open set of non-uniformly expanding transitive maps in
C (M, M) satisfying (1.1.1) and (1.1.2), and Φ = idT . If F is a family of local diffeo-
morphisms we ignore condition (1.1.2), otherwise we assume condition (1.2.1) with respect to the critical set C of some f0 ∈ T . For f ∈ T we write f ∗ = (. . . , f, f, f, . . .) ∈
T Z so that we define Γf = Γf ∗ as in (1.2.4). Given any sequence (fn ) of maps in
T converging to f0 ∈ T we define θǫ to be the Dirac measure supported on fn for
1
, n1 ]. In this case, the stationary measure coincides with the unique absolutely
ǫ ∈ ( n+1
continuous invariant probability measure for the deterministic dynamics of the map
in the support of θǫ . As a corollary of Theorem 1.3.1 we get
Corollary 1.4.1. Let F be a family of non-uniformly expanding transitive maps
for which exist p > 1, C > 0 such that m(Γnf ) < Cn−p for every f ∈ F . Then
F ∋ f 7→ dµf /dm is continuous with respect to the L1 -norm in the space of densities.
Chapter 2
Random induced schemes
In a nutshell, this chapter establishes that for almost all random orbits we can induce
a random version of a piecewise expanding Gibbs-Markov map with some uniformity
on the constants. Most of the auxiliary results we use here can be obtained by
mimicking the deterministic ones in [ALP05], being that some of them have already
been extended to random perturbations in [AAr03]. Nevertheless, we describe in
detail the auxiliary results and their proofs, in order to easily track the extension to
random perturbations and monitor a certain uniformity on random orbits, which is
essential for our purposes. We must pay special attention to the algorithm for the
induced map, in order to avoid undesirable randomness and freedom on the choice of
elements for distinct (but related) partitions of the induced regions. We state at the
very beginning of this chapter the elements and results we are interested in, and we
devote the rest of the chapter to their proofs.
2.1
Random Gibbs-Markov maps
Definition 2.1.1. We say that ω ∈ T Z induces a piecewise expanding Gibbs-Markov
map Fω in a topological disk ∆ ⊂ M if there is a countable partition Pω of open sets
of a full Lebesgue measure subset V of ∆ and a return time function Rω : V → N,
R (x)
constant in each Uω ∈ Pω , such that the map Fω (x) = fω ω
(x) : ∆ → ∆ verifies:
1. Markov: Fω is a C 2 diffeomorphism from each Uω ∈ Pω onto ∆.
13
14
CHAPTER 2. RANDOM INDUCED SCHEMES
2. Expansion: there is 0 < κω < 1 such that for x in the interior of Uω ∈ Pω
kDFω (x)−1 k < κw .
3. Bounded distortion: there is some constant Kω > 0 such that for every Uω ∈ Pω
and x, y ∈ Uω
det DFω (x) ≤ Kω dist(Fω (x), Fω (y)).
log det DFω (y) For simplicity of notation we will write {Rω > n} for the set {x ∈ ∆ : Rω (x) > n}.
Theorem 2.1.2. Let f : M → M be a transitive non-uniformly expanding map and
non-uniformly expanding on random orbits.
1. The realization ω ∗ , associated to the dynamics f , induces a piecewise expanding
Gibbs-Markov map F : ∆ → ∆, for some ball ∆ ⊂ M.
2. If ǫ > 0 is small enough, then every ω ∈ Ωǫ induces a piecewise expanding
Gibbs-Markov map Fω in ∆ ⊂ M.
3. If there exists p > 0, C > 0 such that m(Γnω ) < Cn−p for every ω ∈ Ωǫ , then
there exist C ′ > 0 such that for every ω ∈ Ωǫ the return time function satisfies
m({Rω > n}) ≤ C ′ n−p .
(2.1.1)
Remark 2.1.3. The first item on the previous theorem refers only to the unperturbed
dynamics f and it was proved in [ALP05]. In this case, the integrability of the return
time function R with respect to m implies that the map F admits an absolutely
continuous invariant measure µF ; see e.g. [Y99]. This is one way to obtain an
absolutely continuous invariant probability measure µf for the map f , by normalizing
the measure
µ̃f =
+∞
X
j=0
f j ∗ (µF |{R > j}) ,
where (µ|A) means the conditional measure of µ with respect to the measurable set
A. The transitivity of f also implies the unicity of µf ; see [A03].
As we shall see later, the proof of the theorem above also provides the following
uniformity conditions:
15
CHAPTER 2. RANDOM INDUCED SCHEMES
(U1) For any given integer N > 1 and ̺ > 0, if ǫ > 0 is sufficiently small then for
j = 1, 2, ..., N
m {Rσ−j (ω) = j}△{Rσ−j (τ ) = j} ≤ ̺,
∀ω, τ ∈ Ωǫ ,
where △ is the symmetric difference between the sets.
(U2) If ǫ > 0 is sufficiently small, then for every ω ∈ Ωǫ , the constants Kω and κω for
the induced piecewise expanding Gibbs-Markov maps can be chosen uniformly.
We will refer to them as K > 0 and κ > 0, respectively.
2.2
Hyperbolic times and bounded distortion
The hyperbolic times were first introduced in [A00] for deterministic systems and
extended in [AAr03] to a random context. They are a powerful tool for the understanding of non-uniformly expanding dynamical systems. For the next definition we
fix B > 1 and β > 0 as in Definition 1.1.1, and take a constant b > 0 such that
2b < min{1, β −1}.
Definition 2.2.1. Given 0 < λ < 1 and δ > 0, we say that n ∈ N is a (λ, δ)-hyperbolic
time for (ω, x) ∈ T Z × M if, for every 1 ≤ k ≤ n,
n−1
Y
j=n−k
kDfσj (ω) (fωj (x))−1 k ≤ λk
and
distδ (fωn−k (x), C) ≥ λbk .
(2.2.1)
In the case of C = ∅ the definition of (λ, δ)-hyperbolic time reduces to the first
condition in (2.2.1) and we simply call it a λ-hyperbolic time.
We define, for ω ∈ T Z and n ≥ 1, the set
Hωn = {x ∈ M : n is a (λ, δ)-hyperbolic time for (ω, x) }.
For later use, let us observe that follows from the definition that if n is a (λ, δ)hyperbolic time for (ω, x) ∈ T Z × M, then (n − j) is a (λ, δ)-hyperbolic time for
(σ j (ω), fωj (x)), with 1 ≤ j < n.
CHAPTER 2. RANDOM INDUCED SCHEMES
16
Lemma 2.2.2 (Pliss). Given 0 < c ≤ A let ζ = c/A. Assume that a1 , . . . , aN are
real numbers satisfying aj ≤ A for every 1 ≤ j ≤ N and
N
X
j=1
aj ≥ cN.
Then there are ℓ ≥ ζN and 1 ≤ n1 < · · · < nℓ ≤ N so that
ni
X
j=n
aj ≥ 0
for every 1 ≤ n ≤ ni and 1 ≤ i ≤ ℓ.
Proof. Define for each 1 ≤ n ≤ N,
Sn =
n
X
aj ,
and also S0 = 0.
j=1
Then let 1 ≤ n1 < · · · < nℓ ≤ N be the maximal sequence such that Sni ≥ Sn for
every 0 ≤ n ≤ ni and 1 ≤ i ≤ ℓ. Note that ℓ ≥ 1, since SN > S0 . Moreover, by the
choice of Sni , for each 1 ≤ i ≤ ℓ we have
ni
X
j=n
aj ≥ 0,
for 1 ≤ n ≤ ni .
We are left to verify that ℓ > θN. Defining for convenience n0 = 0, we have by
definition of ni that for each 1 ≤ i ≤ ℓ
Sni −1 ≤ Sni−1 .
Adding ani to both sides and using that ani ≤ A, we easily deduce that
Sni − Sni−1 ≤ A.
Observing that Snℓ ≥ SN ≥ cN, we finally have
cN ≤ Snℓ =
ℓ
X
i=1
Sni − Sni−1 ≤ ℓA,
17
CHAPTER 2. RANDOM INDUCED SCHEMES
which completes the proof.
Definition 2.2.3. We say that the frequency of (λ, δ)-hyperbolic times for (ω, x) ∈
T Z × M is larger than ζ > 0 if, for large n ∈ N, there are ℓ ≥ ζn and integers
1 ≤ n1 < n2 · · · < nℓ ≤ n which are (λ, δ)-hyperbolic times for (ω, x).
Remark 2.2.4. Consider the following condition for non-uniformly expansion: there
is a0 > 0 such that for θǫZ × m almost every (ω, x) ∈ T Z × M
n−1
1X
−1
lim sup
log kDfσj (ω) (fωj (x)) k < −a0 .
n→+∞ n
j=0
(2.2.2)
Under the assumption (1.2.1) the condition (1.2.2) implies (2.2.2) (in fact they become
equivalent). However, if the critical set C is empty (e.g., if f is a local diffeomorphism
as we present in Section 4.1) it is enough for our purposes to assume condition (2.2.2),
so that we can even remove condition (1.2.1). Therefore, from now on, if the critical
set is empty we remove the assumption (1.2.1) for random perturbations and assume
the condition (2.2.2) instead (1.2.2).
Proposition 2.2.5. Assume that f is non-uniformly expanding on random orbits.
Then there are 0 < λ < 1, δ > 0 and ζ > 0 (depending only on a0 in (1.1.1) and
on the map f ) such that for all ω ∈ Ωǫ and Lebesgue almost every point x ∈ M, the
frequency of (λ, δ)-hyperbolic times for (ω, x) is larger than ζ.
Proof. The proof follows from using Lemma 2.2.2 twice, first for the sequence given by
aj = − log kDfωj−1 (fωj−1(x))−1 k (up to a cut off that makes it bounded from above in
the presence of critical set), and then with aj = log distδ (fωj−1 (x), C) for a convenient
choice of δ > 0. We prove that there exist many times ni for which the conclusion
of Lemma 2.2.2 holds, simultaneously, for both sequences. Then we check that any
such ni is a (λ, δ)-hyperbolic time for (ω, x).
Assuming that (2.2.2) and (1.2.3) holds for (ω, x), then for large N we have
N
X
j=1
− log kDfωj−1 (fωj−1 (x))−1 k ≥ a0 N .
18
CHAPTER 2. RANDOM INDUCED SCHEMES
If C = ∅, we just use Lemma 2.2.2 for the sequence
aj = − log kDfωj−1 (fωj−1(x))−1 k +
a0
,
2
with c = a0 /2 and A = maxω∈supp(θǫ ) maxx∈M {− log kDfω (x)−1 k + a0 /2}, we obtain
the result for ζ = a0 /(2A) and λ = e−a0 /2 (δ is not required in this case).
If C is not empty we recall assumption (1.2.1). Take B, β > 0 given by Defini-
tion 1.1.1. Condition (c1 ) implies that for large ρ > 0
log kDf (x)−1k ≤ ρ | log dist(x, C)|
(2.2.3)
for every x ∈ M \ C. Fix α1 > 0 so that ρα1 ≤ a0 /2. The condition (1.2.3) of slow
recurrence to C ensures that we may choose r1 > 0 so that for large N
N
X
j=1
log distr1 (fωj−1(x), C) ≥ −α1 N .
(2.2.4)
Fix any open neighborhood V of C and take Q ≥ ρ | log r1 | large enough so that it is
also an upper bound for − log kDf −1 k on M \ V . Then let
J = 1 ≤ j ≤ N : − log kDf (fωj−1(x))−1 k > Q .
Note that if j ∈ J, then fωj−1(x) ∈ V . Moreover, for each j ∈ J
ρ | log r1 | ≤ Q < − log kDf (fωj−1(x))−1 k < ρ | log dist(fωj−1 (x), C)|,
which shows that dist(fωj−1(x), C) < r1 for every j ∈ J. In particular,
distr1 (fωj−1(x), C) = dist(fωj−1(x), C) < r1 ,
∀j ∈ J.
Therefore, by (2.2.3) and (2.2.4),
X
j∈J
− log kDf (fωj−1(x))−1 k ≤ ρ
X
j∈J
| log dist(fωj−1 (x), C)| ≤ ρ α1 N ≤
a0
N.
2
19
CHAPTER 2. RANDOM INDUCED SCHEMES
Define
bj =
(
− log kDf (fωj−1(x))−1 k, if j ∈
/J
if j ∈ J.
0
By definition, bj ≤ Q for each 1 ≤ j ≤ N. As a consequence,
N
X
j=1
bj =
N
X
j=1
− log kDf (fωj−1(x))−1 k −
X
j∈J
− log kDf (fωj−1(x))−1 k ≥
a0
N.
2
Defining aj = bj − a0 /4, we have
N
X
j=1
aj ≥
a0
N.
4
Thus, we may apply Lemma 2.2.2 to a1 , . . . , aN , with c = a0 /4 and A = Q. The
lemma provides ζ1 > 0 and ℓ1 ≥ ζ1 N times 1 ≤ p1 < · · · < pℓ1 ≤ N such that for
every 0 ≤ n ≤ pi − 1 and 1 ≤ i ≤ ℓ1
pi
X
j=n+1
− log kDf (fωj−1(x))−1 k ≥
pi
X
bj =
j=n+1
pi X
aj +
j=n+1
a0 a0
≥ (pi − n)x.
4
4
(2.2.5)
Now fix α2 > 0 small enough so that α2 < ζ1 ba0 /4, and let r2 > 0 be such that
N
X
j=1
log distr2 (fωj−1(x), C) ≥ −α2 N .
Defining aj = log distr2 (fωj−1 (x), C) + ba0 /4 we have
N
X
j=1
aj ≥
bc0
− α2 N .
4
Applying now Lemma 2.2.2 to a1 , . . . , aN with c = ba0 /4 − α2 and A = ba0 /4, we
conclude that there are l2 ≥ ζ2 N times 1 ≤ q1 < · · · < qℓ2 ≤ N such that for every
0 ≤ n ≤ qi − 1 and 1 ≤ i ≤ ℓ2
qi
X
j=n+1
log distr2 (fωj−1(x), C) ≥ −
ba0
(qi − n).
4
(2.2.6)
20
CHAPTER 2. RANDOM INDUCED SCHEMES
Moreover,
ζ2 =
c
4α2
= 1−
.
A
ba0
Finally, our condition on α2 means that ζ1 + ζ2 > 1. Let ζ = ζ1 + ζ2 − 1. Then there
exist ℓ = (ℓ1 + ℓ2 − N) ≥ ζN times 1 ≤ n1 < · · · < nℓ ≤ N at which (2.2.5) and
(2.2.6) occur simultaneously:
n
i −1
X
j=n
and
n
i −1
X
j=n
− log kDf (fωj (x))−1 k ≥
log distr2 (fωj (x), C) ≥ −
a0
(ni − n)
4
ba0
(ni − n),
4
for every 0 ≤ n ≤ ni − 1 and 1 ≤ i ≤ ℓ. Letting λ = e−a0 /4 we easily obtain from the
inequalities above
nY
i −1
j=ni −k
kDf (fωj (x))−1 k ≤ λk
and
distr2 (fωni−k (x), C) ≥ λbk ,
for every 1 ≤ i ≤ ℓ and 1 ≤ k ≤ ni . In other words, all those ni are (λ, δ)-hyperbolic
times for (ω, x), with δ = r2 .
Remark 2.2.6. In the presence of critical set, one can sees that condition (1.2.3) is
not needed in all its strength in the proof of Proposition 2.2.5. Actually, it is enough
that (1.2.3) holds for some sufficiently small b0 > 0 (eg. b0 = min{α1 , α2 }) and some
convenient δ > 0 (eg. δ = max{r1 , r2 }).
Remark 2.2.7. Observe that the proof of Proposition 2.2.5 gives more precisely that
if for some (ω, x) ∈ T Z × M and N ∈ N
N
−1
X
j=0
− log kDfσj (ω) (fωj (x))−1 k
≥ a0 N
and
N
−1
X
j=0
log distδ (fωj (x), C) ≥ −b0 N
(where b0 and δ are chosen according to Remark 2.2.6), then there exist integers
0 < n1 < · · · < nl ≤ N with l ≥ ζN such that ni is a (λ, δ)-hyperbolic time for (ω, x),
for each 1 ≤ i ≤ l.
21
CHAPTER 2. RANDOM INDUCED SCHEMES
The next result give us property (m1 ) at Section 2.6, and is needed to ensure later
some metric estimates on the algorithm for the random induced partition.
Lemma 2.2.8. Let A ⊂ M be a set with positive Lebesgue measure, for whose points
x we have (ω, x) with frequency of (λ, δ)-hyperbolic times greater than ζ > 0, for all
ω ∈ Ωǫ . Then there is n0 ∈ N such that for n ≥ n0
n
1 X m(A ∩ Hωj )
≥ ζ.
n j=1
m(A)
Proof. Since we are assuming that points (ω, x) for which x is in A have frequency
of (λ, δ)-hyperbolic times greater than ζ > 0, there is n0 ∈ N such that for every
x ∈ A and n ≥ n0 there are (λ, δ)-hyperbolic times 0 < n1 < n2 < · · · < nℓ ≤ n for x
with ℓ ≥ ζn. Take n ≥ n0 and let #n be the measure in In = {1, . . . , n} defined by
#n (J) = #J/n, for each J ⊂ In . Then, putting χ(x, i) = 1 if x ∈ Hωi , and χ(x, i) = 0
otherwise, by Fubini’s Theorem
n
1X
m(A ∩ Hωj ) =
n j=1
=
Z Z
χ(x, i) dm(x) d#n (i)
A
Z Z
χ(x, i) d#n (i) dm(x).
A
Since for every x ∈ A and n ≥ n0 there are 0 < n1 < n2 < · · · < nℓ ≤ n with ℓ ≥ ζn
such that x ∈ Hωni for 1 ≤ i ≤ ℓ, then the integral with respect to d#n is larger
than ζ > 0 and the last expression in the formula above is bounded from below by
ζm(A).
Lemma 2.2.9. Given 0 < λ < 1 and δ > 0, there is δ1 > 0 such that if n is a
(λ, δ)-hyperbolic time for (ω, x) ∈ T Z × M, then
kDfω (y)−1k ≤ λ−1/2 kDfω (x)−1 k,
for any point y in the ball of radius δ1 λn/2 around x.
Proof. If C = ∅ and since f is a local diffeomorphism, for each x ∈ M there is a radius
δx > 0 such that f sends a neighborhood of x diffeomorphically onto B(f (x), δx ), the
ball of radius δx around f (x). By compactness of M we may choose a uniform radius
22
CHAPTER 2. RANDOM INDUCED SCHEMES
δ1 > 0. We choose δ1 > 0 small enough so that also
kDfω (y)−1k ≤ λ−1/2 kDfω (x)−1 k,
whenever
y ∈ B(x, δ1 λ1/2 ),
for every ω ∈ supp(θǫZ ).
In the case C =
6 ∅, if n is a (λ, δ)-hyperbolic time for (ω, x), then
distδ (x, C) ≥ λbn .
According to the definition of the truncated distance, this means that
dist(x, C) = distδ (x, C) ≥ λbn ,
or else
dist(x, C) ≥ δ.
In either case, considering 2δ1 < δ, we have for any point y in the ball of radius δ1 λn/2
around x
dist(y, x) <
1
dist(x, C),
2
because we haven chosen b < 1/2 and δ1 < δ/2 < 1/2. Therefore, we may use (c2 ) to
conclude that
log
kDf (y)−1k
dist(y, x)
δ1 λn/2
≤
B
≤
B
.
kDf (x)−1 k
dist(x, C)β
min{λbβn , δ β }
Since δ > 0, 0 < λ < 1 and we have taken bβ < 1/2, the term on the right hand side
is bounded by Bδ1 δ −β . Choosing δ1 > 0 small so that Bδ1 δ −β < log λ−1/2 we get the
conclusion.
We assume that given (λ, δ) as before we fix δ1 small so that Lemma 2.2.9 holds.
We further require δ1 small so that the exponential map is an isometry onto its image
in the ball of radius δ1 . This in particular implies that any point in the boundary of
a ball of radius δ1 can be joined to the center of the ball through a smooth curve of
minimal length (a geodesic arc).
Proposition 2.2.10. If n is (λ, δ)-hyperbolic time for (ω, x) ∈ T Z × M, then there
is a neighborhood Vωn (x) of x in M such that:
1. fωn maps Vωn (x) diffeomorphically onto B(fωn (x), δ1 );
23
CHAPTER 2. RANDOM INDUCED SCHEMES
2. for every y ∈ Vωn (x) and 1 ≤ k ≤ n we have kDfσkn−k (ω) (fωn−k (y))−1k ≤ λk/2 ,
Proof. We shall prove the result by induction on n. Let n = 1 be a (λ, δ)-hyperbolic
time for (ω, x) ∈ T Z × M. It follows from Lemma 2.2.9 and from the definition of
hyperbolic times that for any y in the ball B(x, δ1 λ1/2 ) ⊂ M of radius δ1 λ1/2 around x
kDfω (y)−1k ≤ λ−1/2 kDfω (x)−1 k ≤ λ1/2 .
(2.2.7)
This means that fω is a λ−1/2 -dilation in the ball B(x, δ1 λ1/2 ). Then, there exists some
neighborhood Vω1 (x) of x contained in B(x, δ1 λ1/2 ) which is mapped diffeomorphically
onto the ball B(fω (x), δ1 ) and for y ∈ Vω1 (x) condition (2.2.7) ensures the second
property kDfω (y)−1k ≤ λ1/2 .
Assume now that the conclusion holds for n ≥ 1. Let n + 1 be a (λ, δ)-hyperbolic
time for (ω, x) ∈ T Z × M. Take any z ∈ ∂B(fωn+1 (x), δ1 ), and let γ : [0, 1] → M be
a smooth curve of minimal length joining z to fωn+1(x). The curve γ necessarily lies
inside B(fωn+1 (x), δ1 ) by the choice of δ1 . Consider γn and γn+1 smooth curves which
are lifts of γ starting at fω (x) and x, respectively. This means that
n
γ = fσ(ω)
◦ γn
and γ = fωn+1 ◦ γn+1,
at least in the domains where the lifts make sense. Since n is a (λ, δ)-hyperbolic time
n
for (σ(ω), fω (x)), by induction hypothesis there is a neighborhood Vσ(ω)
(fω (x)) which
n
is sent diffeomorphically by fσ(ω)
onto the ball of radius δ1 around fωn+1 (x) with the
n
additional second condition property. One has that γn lies inside Vσ(ω)
(fω (x)).
j
n
Moreover, n−j is a (λ, δ)-hyperbolic time for (σ j+1 (ω), fωj+1(x)) and fσ(ω)
(Vσ(ω)
(fω (x)))
j+1
is a neighborhood Vσn−j
(x) which is mapped by fσn−j
j+1 (ω) of fω
j+1 (ω) diffeomorphically
onto B(fωn+1 (x), δ1 ) and also satisfies the second condition property, for 1 ≤ j ≤ n.
Claim. The curve γn+1 lies inside the ball of radius δ1 λ(n+1)/2 around x.
Assume, by contradiction, that γn+1 hits the boundary of B(x, δ1 λ(n+1)/2 ) before the
end time. Let 0 < t0 < 1 be the first moment in such conditions. One necessarily has
that γn+1 |[0, t0] is a curve inside the ball B(x, δ1 λ(n+1)/2 ) joining x to a point in the
boundary of that ball. Since n+1−j is a (λ, δ)-hyperbolic time for (σ j+1 (ω), fωj+1(x)),
24
CHAPTER 2. RANDOM INDUCED SCHEMES
by Lemma 2.2.9 that for each 0 ≤ t ≤ t0 and 0 ≤ j ≤ n
kDfσj (ω) (fωj (γn+1 (t)))−1 k ≤ λ−1/2 kDfσj (ω) (fωj (x))−1 k
j
and fωj (γn+1 ([0, t0 ])) ⊂ Vσn+1−j
j (ω) (fω (x)), which yields to
kDfωn+1 (γn+1(t))−1 k
≤
n
Y
j=0
kDfσn−j (ω) (γj+1 (t))−1 k
−1
= kDfω (γn+1 (t)) k ·
≤ λ−1/2 kDfω (x)−1 k ·
≤ λ
−1/2
λλ
−n/2 n
λ
n−1
Y
j=0
n−1
Y
j=0
kDfσn−j (ω) (γj+1(t))−1 k
λ−1/2 kDfσn−j (ω) (fωn−j (x))−1 k
= λ(n+1)/2 .
(2.2.8)
Hence
Z
t0
0
′
kγ (t)kdt =
≥
Z
t0
Z0 t0
0
′
kDfωn+1(γn+1 (t)) · γn+1
(t)kdt
′
λ−(n+1)/2 kγn+1
(t)kdt
= δ1
This gives a contradiction since t0 < 1, thus proving the claim.
Let us now finish the proof of the proposition. We simply consider the lifts by fωn+1
of the geodesics joining fωn+1 (x) to the points in the boundary of B(fωn+1 (x), δ1 ). This
defines a neighborhood Vωn+1 (x) of x which by (2.2.8) has the required properties.
We shall often refer to the sets Vωn as hyperbolic pre-balls and to their images
fωn (Vωn ) as hyperbolic balls. Notice that the latter are indeed balls of radius δ1 > 0.
Remark 2.2.11. It follows from the proof of the previous proposition that for every
x belonging to a hyperbolic pre-ball Vωn associated to a (λ, δ)-hyperbolic time n we
have kDfωn (x)−1 k ≤ λn/2 .
25
CHAPTER 2. RANDOM INDUCED SCHEMES
Lemma 2.2.12. If Vωn (x) is hyperbolic pre-ball, then for every y, z ∈ Vωn (x) and
1≤k≤n
dist(fωn−k (y), fωn−k (z)) ≤ λk/2 dist(fωn (y), fωn(z)).
Proof. Let γ be a curve of minimal length connecting fωn (z) to fωn (y). This curve γ
must be contained in B(fωn (x), δ1 ). For 1 ≤ k ≤ n, let γk be the (unique) curve in
fωn−k (Vωn (x)) joining fωn−k (z) to fωn−k (y) such that fσkn−k (ω) (γk ) = γ. For every n ≥ 1
length(γ) =
Z
Z
kγ ′ (t)kdt
kDfσkn−k (ω) (γk (t)) · γk′ (t)kdt
Z
− k2
kγk′ (t)kdt
≥ λ
=
k
= λ− 2 length(γk ).
As a consequence,
k
k
dist(fωn−k (y), fωn−k (z)) ≤ length(γk ) ≤ λ 2 length(γ) = λ 2 dist(fωn (y), fωn(z)).
Corollary 2.2.13 (Bounded Distortion). There exists C0 > 0 such that if n is a
(λ, δ)-hyperbolic time for (ω, x) ∈ T Z × M, then for every y, z ∈ Vωn (x),
log
| det Dfωn (y)|
≤ C0 dist(fωn (y), fωn(z)).
| det Dfωn (z)|
Proof. If C = ∅, since fω ∈ C 2 (M, M) then there is C0′ > 0 such that for all z, y ∈ M
and ω ∈ supp(θǫ ) we have | log | det Dfω (z)| − log | det Dfω (y)|| ≤ C0′ dist(z, y). And
for all z, y ∈ Vωn (x) we have
log
n−1
X
| det Dfσj (ω) (fωj (z))|
| det Dfwn (z)|
=
log
| det Dfωn (y)|
| det Dfσj (ω) (fωj (y))|
j=0
≤
≤
It is then enough to take C0 =
P∞
n−1
X
C0′ dist(fωj (z), fωj (y))
j=0
n−1
X
C0′ λn−j dist(fωn (z), fωn (y)).
j=0
′ k
k=0 C0 λ .
26
CHAPTER 2. RANDOM INDUCED SCHEMES
If C is not empty then let n be a (λ, δ)-hyperbolic time for (ω, x) ∈ T Z × M with
associated hyperbolic pre-ball Vωn . By Lemma 2.2.12 we have for each y, z ∈ Vωn and
each 0 ≤ k ≤ n − 1
dist(fωk (y), fωk (z)) ≤ δ1 λ(n−k)/2 .
On the other hand, since n is a hyperbolic time for (ω, x)
dist(fωk (y), C) ≥ dist(fωk (x), C) − dist(fωk (x), f k (y))
≥ λb(n−k) − δ1 λ(n−k)/2
1 b(n−k)
≥
λ
2
≥ 2δ1 λ(n−k)/2 ,
(2.2.9)
as long as δ1 < 1/4; recall that b < 1/2. Thus we have
dist(fωk (y), fωk (z)) ≤
1
dist(fωk (y), C),
2
and so we may use (c3 ) to obtain
log
| det Df (fωk (y))|
B
dist(fωk (y), fωk (z)).
≤
k
k
| det Df (fω (z))|
dist(fω (y), C)β
Hence, by (2.2.9) and Lemma 2.2.12
log
n−1
X
| det Dfωn (y)|
| det Df (fωk (y))|
log
=
| det Dfωn (z)|
| det Df (fωk (z))|
k=0
≤
It suffices to take C0 ≥
P+∞
k=1 2
β
n−1
X
k=0
2β B
λ(n−k)/2
dist(fωn (y), fωn(z)).
bβ(n−k)
λ
Bλ(1/2−bβ)k ; recall that bβ < 1/2.
It will be useful to have the following weaker forms of the previous corollary.
Corollary 2.2.14. There is a constant C1 > 0 such that if n is a (λ, δ)-hyperbolic
time for (ω, x) ∈ T Z × M and y, z ∈ Vωn (x), then
1
| det Dfωn (y)|
≤
≤ C1 .
C1
| det Dfωn (z)|
CHAPTER 2. RANDOM INDUCED SCHEMES
27
Proof. By Corollary 2.2.13 just have to consider C1 = exp(C0 δ1 ).
Corollary 2.2.15. There is a constant C2 > 0 such that for any hyperbolic pre-ball
Vωn (x) and any A1 , A2 ⊂ Vωn (x)
m(fωn (A1 ))
m(A1 )
1 m(A1 )
≤
≤
C
.
2
C2 m(A2 )
m(fωn (A2 ))
m(A2 )
Proof. By the change of variable formula for fωn we may write
R
| det Dfωn (z)| dm(z)
m(fωn (A1 ))
RA1
=
m(fωn (A2 ))
| det Dfωn (z)| dm(z)
A1
R det Df n (z) ω
| det Dfωn (z1 )| A1 det
dm(z)
Dfωn (z1 ) =
,
R det Dfωn (z) | det Dfωn (z2 )| A2 det
dm(z)
Dfωn (z2 )
with z1 and z2 choosen arbitrarily in A1 and A2 , respectively. From Corollary 2.2.14
we get the desired bounds.
2.3
Transitivity and growing to large scale
We do not need transitivity of f in all its strength. Before we tell what is the weaker
form of transitivity that is enough for our purposes, let us refer that given δ > 0,
a subset A of M is said to be δ-dense if any point in M is at a distance smaller
than δ from A. For our purposes it is enough that there is some point p ∈ M whose
pre-orbit does not hit the critical set of f and is δ-dense for some sufficiently small
δ > 0 (depending on the radius δ1 of hyperbolic balls for f ). As the lemma below
shows, in our setting of non-uniformly expanding maps this is a consequence of the
usual transitivity of f .
Lemma 2.3.1. Let f : M → M be a transitive non-uniformly expanding map. Given
−j
0
δ > 0 there is p ∈ M and N0 ∈ N such that ∪N
{p} is δ-dense in M and disjoint
j=0 f
from the critical set C.
Proof. Observe that the assumptions on f imply that the images and preimages of
sets with zero Lebesgue measure still have zero Lebesgue measure. Hence, the set
B = ∪n≥0 f −n (∪m≥0 f m (C)) has Lebesgue measure equal to zero. On the other hand,
CHAPTER 2. RANDOM INDUCED SCHEMES
28
since f is transitive, there is a unique f -invariant probability measure µ, which is
ergodic, absolutely continuous with respect to Lebesgue measure and whose support
is the whole manifold M; see [A03] . This implies that µ almost every point in M has
a dense orbit. Since µ is absolutely continuous with respect to Lebesgue, then there is
a positive Lebesgue measure subset of points in M with dense orbit. Thus there must
be some point q ∈ M \ B with dense orbit. Take N0 ∈ N for which q, f (q), . . . , f N0 (q)
is δ-dense. The point p = f N0 (q) satisfies the conclusions of the lemma.
Assuming that f is non-uniformly expanding and non-uniformly expanding on
random orbits, then by Proposition 2.2.5 there are λ, δ and ζ such that Lebesgue
almost every x ∈ M and θǫZ almost every ω, (ω, x) has frequency of (λ, δ)-hyperbolic
times greater than ζ. We fix once and for all p ∈ M and N0 ∈ N for which
−j
0
∪N
{p} is δ1 /3-dense in M and disjoint from C,
j=0 f
(2.3.1)
where δ1 > 0 is the radius of hyperbolic balls as for Proposition 2.2.10. Take constants
α > 0 and δ0 > 0 so that
√
2 δ 0 ≪ δ1
and 0 < α ≪ δ0 .
Lemma 2.3.2. There are constants K0 , D0 > 0 depending only on f , λ, δ1 and the
point p such that, if ǫ > 0 is sufficiently small, then for any ball B ⊂ M of radius δ1
and every ω ∈ supp(θǫZ ) there are an open set A ⊂ B and an integer 0 ≤ m ≤ N0 for
which:
√
1. fωm maps A diffeomorphically onto B(p, 2 δ 0 );
2. for each x, y ∈ A
det Dfωm (x) ≤ D0 dist(fωm (x), fωm (y));
log det Dfωm (y) √
and, moreover, for each 0 ≤ j ≤ N0 the j-preimages (fωj )−1 B(p, 2 δ 0 ) are all disjoint
from C, and for x belonging to any such j-preimage we have
1
≤ kDfωj (x)k ≤ K0 .
K0
CHAPTER 2. RANDOM INDUCED SCHEMES
29
−j
0
Proof. Since ∪N
{p} is δ1 /3 dense in M and disjoint from C, choosing δ0 > 0 suffij=0 f
√
ciently small we have that each connected component of the j-preimages f −j B(p, 2 δ 0 ),
with j ≤ N0 , are bounded away from the critical set C and contained in a ball of ra-
dius δ1 /3. Moreover, since we are dealing with a finite number of iterates, less than
N0 , and ft varies continuously with parameter t, if ǫ is sufficiently small then for
√
every ω ∈ supp(θǫZ ), each connected component of the j-preimages (fωj )−1 B(p, 2 δ 0 ),
with j ≤ N0 , is uniformly (on j and ω) bounded away from the critical set C and
contained in a ball of radius δ1 /3. This immediately implies that for ω ∈ supp(θǫZ ),
√
any ball B ⊂ M of radius δ1 contains a m-preimage A of B(p, 2 δ0 ) which is mapped
√
diffeomorphically by fωm onto B(p, 2 δ 0 ) for some m ≤ N0 . Since the number of
iterations and the distance to the critical region are uniformly bounded, the volume
distortion is uniformly bounded and moreover there is some constant K0 > 1 such
that for every ω ∈ supp(θǫZ )
1
≤ kDfωm (x)k ≤ K0 .
K0
√
for all 1 ≤ m ≤ N0 and x belonging to a m-preimage of B(p, 2 δ 0 ) by fωm .
Next we prove a useful and non-obvious consequence of the existence of hyperbolic
times, namely that if we start with some fixed given α > 0 then there exist some Nα
depending only on α such that, for ω ∈ Ωǫ , any ball on M of radius α has some subset
which grows to a fixed size with bounded distortion within Nα iterates.
Lemma 2.3.3. Given α > 0 there exists Nα > 0 such that if ǫ is sufficiently small,
then for every ω ∈ Ωǫ we have that any ball B ⊂ M of radius α contains a hyperbolic
pre-ball Vωn ⊂ B with n ≤ Nα .
Proof. Take any α > 0 and a ball B(z, α). By Lemma 2.2.12 we may choose nα ∈ N
large enough so that any hyperbolic pre-ball Vωn associated to a hyperbolic time
n ≥ nα will have diameter not exceeding α/2. Now notice that by Proposition 2.2.5 for
Lebesgue almost every x ∈ M, the point (ω ∗, x) has an infinite number of hyperbolic
times and therefore
S
m M \ nj=nα Hωj ∗ → 0
as n → +∞.
CHAPTER 2. RANDOM INDUCED SCHEMES
30
Hence, it is possible to choose Nα ∈ N such that
S α
j
m M\ N
H
< m(B(z, α/2)).
∗
j=nα ω
(2.3.2)
Observe that if n is a hyperbolic time for (ω ∗ , x) and ǫ is small enough, then for every
ω ∈ Ωǫ the natural n is also a hyperbolic time for (ω, x). Hence, if ǫ is small enough,
for given α > 0 we can take an integer Nα , only depending on α, λ and δ1 , such
that (2.3.2) holds for every ω ∈ Ωǫ in the place of ω ∗ . This ensures that, for every
ω ∈ Ωǫ , there is a point x̂ ∈ B(z, α/2) with a hyperbolic time n ≤ Nα and associated
hyperbolic pre-ball Vωn (x) contained in B(z, α).
2.4
The partitioning algorithm
We describe now the construction of the partition Pω (mod 0) of ∆0 = B(p, δ0 ), for
every ω ∈ Ωǫ . The basic intuition is that we wait for some iterate fωk (∆0 ) to cover
∆0 completely, and then define the subset U ⊂ ∆0 , for which fωk : U → ∆0 is a
diffeomorphism, as an element with return time k for the partitions corresponding to
′
all elements ω ′ ∈ Ωǫ with same first k coordinates as ω: ω0′ = ω0 , . . . , ωk−1
= ωk−1 .
After that, we continue to iterate the complement ∆0 \U until this complement covers
again ∆0 and repeat the same procedure to define more elements of the final partition
with higher return times. Using the fact that small regions eventually become large
due to the expansivity condition, it follows that this process can be continued and
that Lebesgue almost every point eventually belongs to some element of the partition.
Moreover, the return time function depends on the time that it takes small regions to
become large on average and this turns out to depend precisely on the measure of the
tail set. On the other hand, this process avoids undesirable randomness on the choice
of elements for distinct (but related) partitions of the induced regions. In particular,
for different realizations with similar initial nonnegative coordinates, the elements in
corresponding partitions with return times lower than the number of similar entries
are the same, as subsets of ∆0 .
31
CHAPTER 2. RANDOM INDUCED SCHEMES
Now we introduce neighborhoods of p
∆0 = B(p, δ0 ),
∆10 = B(p, 2δ0 ),
∆20 = B(p,
√
√
δ 0 ) and ∆30 = B(p, 2 δ 0 ).
For 0 < λ < 1 given by Proposition 2.2.5, let
Is = x ∈ ∆10 : δ0 (1 + λs/2 ) < dist(x, p) < δ0 (1 + λ(s−1)/2 ) ,
s ≥ 1,
be a partition (mod 0) into countably many rings of ∆10 \ ∆0 .
The construction of the partition Pω of ∆0 is inductive and we give the initial
and the general step of the induction. For the sake of a better visualization of the
process, and to motivate the definitions, we start with the first step. Define
[ω]k = {τ ∈ Ωǫ : ω0 = τ0 , ..., ωk−1 = τk−1 }.
k
The set Hτk is the same for any τ ∈ [ω]k and we will refer to this set as H[ω]
.
First step of the induction
Take R0 some large integer to be determined in Section 2.5 (can be taken independent
of ω); we ignore any dynamics occurring up to time R0 . For ω ∈ Ωǫ , let k ≥ R0 + 1 be
k
the first time that ∆0 ∩H[ω]
6= ∅. For j < k we define formally the objects Λjω , Ajω , Aj,α
ω ,
j
whose meaning will become clear in the next paragraphs, by Ajω = Aj,α
ω = Λω = ∆0 .
3
Let (Uk,j
)j be the connected components of Ak−1,α
∩ (fwk )−1 (∆30 ) contained in hyω
perbolic pre-balls Vωk−m , with k − N0 ≤ m ≤ k. This hyperbolic pre-balls Vωk−m
growth in k − m iterates to a hyperbolic ball of radius δ1 which fσmk−m (ω) maps diffeomorphically onto ∆30 . Now let
i
3
Uk,j
= Uk,j
∩ (fωk )−1 (∆i0 ),
i = 0, 1, 2,
0
and set Rω (x) = k for x ∈ Uk,j
. Take
Λkw = Λk−1
\ {Rω = k}.
w
where ∆00 = ∆0 ,
32
CHAPTER 2. RANDOM INDUCED SCHEMES
We define also a function tkω : Λkw → N by

s
k
tω (x) =
0
Let
1
if x ∈ Uk,j
and fωk (x) ∈ Is for some j;
otherwise.
Akω = {x ∈ Λkw : tkω (x) = 0},
Bωk = {x ∈ Λkw : tkω (x) > 0}.
We also define:
k
Ak,α
ω = Λw ∩
[
k+1
x∈Akω ∩Hω
(fωk+1|Vωk+1 (x) )−1 B(fωk+1 (x), α)
For all τ ∈ [ω]k and j ≤ k, we define the objects Ajτ , Bτj , Λkτ , {Rτ = j}, tkτ to be
the same as the corresponding ones as before. Moreover, for τ ∈ [ω]k+1 we also define
Ak,α
to be the same as Ak,α
τ
ω .
General step of the induction
The general inductive step of the construction now follows by repeating the arguments
above with minor modifications. More precisely we assume that the sets Λiτ , Aiτ , Bτi ,
{Rτ = i} and functions tiτ : Λiτ → N are defined for all i ≤ n − 1 and are exactly the
same for every τ ∈ [ω]n−1 . We also have defined Ai,α
τ , for i ≤ n − 1 to be the same
i
i
set for all τ ∈ [ω]n . For i ≤ R0 we just let Aiω = Ai,α
ω = Λw = ∆0 , Bω = {Rω = i} = ∅
3
and tiω ≡ 0. Now, let (Un,j
)j be the connected components of An−1,α
∩ (fωn )−1 (∆0 )
ω
contained in hyperbolic pre-balls Vωr , with n − N0 ≤ r ≤ n, which are mapped onto
∆30 by fωn . Take
i
3
Un,j
= Un,j
∩ (fωn )−1 (∆i0 ),
i = 0, 1, 2,
0
and set Rω (x) = n for x ∈ Un,j
. Take also
Λnw = Λn−1
\ {Rω = n}.
w
33
CHAPTER 2. RANDOM INDUCED SCHEMES
The definition of the function tnω : Λnw → N is slightly different in the general case:


s



tnω (x) = 0



tn−1 (x) − 1
ω
1
0
if x ∈ Un,j
\ Un,j
and fωn (x) ∈ Is for some j,
1
if x ∈ An−1
\ ∪j Un,j
,
ω
1
if x ∈ Bωn−1 \ ∪j Un,j
.
Finally let
Anω = {x ∈ Λnw : tnω (x) = 0},
and
n
An,α
ω = Λw ∩
[
n+1
x∈An
ω ∩Hω
Bωn = {x ∈ Λnw : tnω (x) > 0}.
(fωn+1|Vωn+1 (x) )−1 B(fωn+1 (x), α)
Once more, for all τ ∈ [ω]n we define the objects Anτ , Bτn , Λnτ , {Rτ = n}, tnτ to be,
respectively, Anω , Bωn , Λnω , {Rω = n}, tnω and for τ ∈ [ω]n+1 we also define An,α
as An,α
τ
ω .
0
Remark 2.4.1. Associated to each component Un−k
of {Rω = n − k}, for some k > 0,
1
0
we have a collar Un−k
\Un−k
around it; knowing that the new components of {Rω = n}
1
0
do not intersect “too much” Un−k
\ Un−k
is important for preventing overlaps on sets
of the partition.
In order to see that the sets we construct at each step do not intersect the previously constructed sets, it is enough to show that if α > 0 is sufficiently small, then
0
Un1 ∩ {tn−1
> 1} = ∅ for each component Un1 . Indeed, take some k > 0 and let Un−k
ω
1
be a component of {Rω = n − k} such that its collar Qk (the part of Un−k
that is
mapped by fωn−k onto Ik ) intersects Un1 . Recall that Qk is precisely the collar around
0
Un−k
on which tωn−1 takes the value 1. Letting q1 and q2 be any two points in distinct
components (inner and outer) of the boundary of Qk , we have by Lemma 2.2.12 and
Lemma 2.3.2
dist(fωn−k (q1 ), fωn−k (q2 )) ≤ K0 λ(k−N0 )/2 dist(fωn (q1 ), fωn (q2 )).
(2.4.1)
34
CHAPTER 2. RANDOM INDUCED SCHEMES
We also have
dist(fωn−k (q1 ), fωn−k (q2 )) ≥ δ0 (1 + λ(k−1)/2 ) − δ0 (1 + λk/2 )
= δ0 λk/2 (λ−1/2 − 1),
which combined with (2.4.1) gives
dist(fωn (q1 ), fωn (q2 )) ≥ K0−1 λN0 /2 δ0 (λ−1/2 − 1).
On the other hand, since Un1 ⊂ Aωn−1,α by construction of Un1 , taking
α < K0−1 λN0 /2 δ0 (λ−1/2 − 1)
(2.4.2)
we have Un1 ∩ {tωn−1 > 1} = ∅, for every ω ∈ Ωǫ .
2.5
Expansion and bounded distortion
The inductive construction we detailed before provides a family of topological balls
contained in ∆0 which, as we will see, define a Lebesgue modulo zero partition Pω of
∆0 . We start however, by checking conditions (1)-(3) in the definition of the induced
piecewise expanding Gibbs-Markov map in view to prove Theorem 2.1.2.
Recall that by construction, the return time Rω for an element U of the partition
Pω of ∆0 is formed by a certain number n of iterations given by the hyperbolic time
of a hyperbolic pre-ball Vωn ⊃ U, and a certain number m ≤ N0 of additional iterates
which is the time it takes to go from fωn (Vωn ), which could be anywhere in M, to
fωn+m (Vωn ) which covers ∆0 completely. The map Fω = fωRw : ∆ → ∆ is indeed a C 2
diffeomorphism from each component U onto ∆.
It follows from Remark 2.2.11 and Lemma 2.3.2 that
kDfωn+m (x)−1 k ≤ kDfσmn (ω) (fωn (x))−1 k · kDfωn (x)−1 k < K0 λn/2 ≤ K0 λ(R0 −N0 )/2 .
By taking R0 sufficiently large we can make this last expression smaller than some
κω , with 0 < κω < 1. Since K0 and N0 are independent of ω then R0 (and hence kw )
can be the same for all ω ∈ Ωǫ , proving part of property (U2).
CHAPTER 2. RANDOM INDUCED SCHEMES
35
For the bounded distortion estimate we need to show that there exists a constant
Kω > 0 such that for any x, y belonging to an element Uω ∈ Pω with return time Rω ,
we have
By the chain rule
det DfωRω (x) ≤ Kω dist(fωRω (x), fωRω (y)).
log R
ω
det Dfω (y) det Df Rnω −n (f n (x)) det DfωRω (x) det Dfωn (x) ω
σ
(ω)
= log .
log + log −n
det DfσRnω(ω)
det DfωRω (y) det Dfωn (y) (fωn (y)) For the first term in this last sum we observe that by Lemma 2.3.2 we have
det Df Rnω −n (f n (x)) ω
σ (ω)
log ≤ D0 dist(fωRω (x), fωRω (y)).
R
−n
ω
n
det Dfσn (ω) (fω (y)) For the second term in the sum above, we may apply Corollary 2.2.13 and obtain
det Dfωn (x) ≤ C0 dist(f n (x), f n (y)).
log ω
ω
n
det Dfω (y) Also by Lemma 2.3.2 we may write
dist(fωn (x), fωn (y)) ≤ K0 dist(fωRω (x), fωRω (y)).
We just have to take Kω = D0 + C0 K0 which, clearly, can be uniformly chosen on w,
completing property (U2).
2.6
Metric estimates
In this section we compute some estimates to show that the algorithm before indeed
produces a partition (Lebesgue mod 0) of ∆0 .
Estimates obtained from the construction
In this first part we obtain some estimates relating the Lebesgue measure of the
sets Anω , Bωn and {Rω > n} with the help of specific information extracted from the
inductive construction we performed in Section 2.4.
CHAPTER 2. RANDOM INDUCED SCHEMES
36
Lemma 2.6.1. There exists a constant c0 > 0 (not depending on δ0 ) such that for
every ω ∈ Ωǫ and n ≥ 1
m(Bωn−1 ∩ Anω ) ≥ c0 m(Bωn−1 ).
Proof. It is enough to see that this holds for each connected component of Bωn−1 at a
time. Let C be a component of Bωn−1 and Q be its outer ring corresponding to tn−1
= 1.
ω
Observe that by Remark 2.4.1 we have Q ⊂ C ∩ Anω . Moreover, there must be some
k < n and a component Uk0 of {Rω = k} such that fωk maps C diffeomorphically onto
D0 D
∪+∞
, where D is
i=k Ii and Q onto Ik , both with distortion bounded by C1 and e
the diameter of M; cf. Corollary 2.2.14 and Lemma 2.3.2. Thus, it is sufficient to
compare the Lebesgue measures of ∪+∞
i=k Ii and Ik . We have
m(Ik )
[δ0 (1 + λ(k−1)/2 )]d − [δ0 (1 + λk/2 )]d
≈
≈ 1 − λ1/2 .
(k−1)/2 )]d − δ d
[δ
(1
+
λ
m(∪+∞
I
)
0
0
i=k i
Clearly this proportion does not depend on δ0 neither on ω.
Lemma 2.6.2. There exist d0 , r0 > 0 with d0 + r0 < 1 such that for every ω ∈ Ωǫ
and n ≥ 1
1. m(An−1
∩ Bωn ) ≤ d0 m(An−1
ω
ω );
2. m(An−1
∩ {Rω = n}) ≤ r0 m(An−1
ω
ω ).
Moreover d0 → 0 and r0 → 0 as δ0 → 0.
Proof. It is enough to prove these estimates for each neighborhood of a component
Un0 of {Rω = n}. Observe that by construction we have Un3 ⊂ An−1,α
, which means
ω
√
2
n−1
that Un ⊂ Aω , because α < δ0 < δ 0 . Using the distortion bounds of fωn on Un3
given by Corollary 2.2.15 and Lemma 2.3.2 we obtain
m(Un1 \ Un0 )
m(∆10 \ ∆0 )
δ0d
≈
≈
≪ 1,
d/2
m(Un2 \ Un1 )
m(∆20 \ ∆10 )
δ0
which gives the first estimate. Moreover,
m(Un0 )
m(∆0 )
δ0d
≈
≈
≪ 1,
d/2
m(Un2 \ Un1 )
m(∆20 \ ∆10 )
δ0
37
CHAPTER 2. RANDOM INDUCED SCHEMES
and this gives the second one.
We state a useful lemma to prove the following proposition.
Lemma 2.6.3. There exists L, depending only on the manifold M, such that for every
finite Borel measure ϑ and every measurable subset G ⊂ M with compact closure there
is a finite subset I ⊂ G such that the balls B(z, δ41 ) around the points z ∈ Iare pairwise
disjoint, and
X δ1 ϑ B z,
∩ G ≥ Lϑ(G)
4
z∈I
Proof. See [A04], Lemma 4.9.
The next result asserts that a fixed proportion of An−1
∩ Hωn gives rise to new
ω
elements of the partition within a finite number of steps (not depending on n).
Proposition 2.6.4. There exist s0 > 0 and a positive integer N = N(α) such that
for every ω ∈ Ωǫ and n ≥ 1
!
N
[
m
Rω = n + i
≥ s0 m(An−1
∩ Hωn ).
ω
i=0
Proof. We use Lemma 2.6.3 with G = fωn (An−1
∩Hωn ) and ϑ = (fωn )∗ m, thus obtaining
ω
a finite subset I of points z ∈ fωn (An−1
∩ Hωn ) for which the conclusion of the lemma
ω
in particular implies
X
z∈I
δ1
n −1
n−1
n
m
(fω ) B z,
∩ Aω ∩ Hω ≥ Lm(An−1
∩ Hωn ).
ω
4
(2.6.1)
Fix now z ∈ I. Consider {Cj }j the set of connected components of (fωn )−1 B(z, δ1 /4),
which intersect An−1
∩ Hωn . Note that each Cj is contained in a hyperbolic preω
ball Vωn (xj ) associated to some point xj ∈ ((fωn )−1 B(z, δ1 /4)) ∩ An−1
∩ Hωn as in
ω
Proposition 2.2.10. In what follows, given A ⊂ B(zj , δ41 ), we will simply denote
δ1
n −1
(fωn |Vωn (xj ) )−1 (A) by (fωn )−1
j (A). Note that the sets {(fω )j B(z, 4 )}j are pairwise dis-
joint as long as that δ1 is sufficiently small (only depending on the manifold). In fact
fωn sends each of them onto B(z, δ41 ) and fwn is a diffeomorphism restricted to each
one of them. In particular, their union does not contain points of C.
38
CHAPTER 2. RANDOM INDUCED SCHEMES
n+kj
Claim 1. There is 0 ≤ kj ≤ Nα + N0 such that tω
is not identically 0 in
(fωn )−1
j B(z, α).
n+kj
Assume by contradiction that tω
n+kj ,α
implies that (fωn )−1
j B(z, α) ⊂ Aω
|(fωn )−1
= 0 for all 0 ≤ kj ≤ Nα + N0 . This
j B(z,α)
for all 0 ≤ kj ≤ Nα + N0 . Using Lemma 2.3.3
n+kj
we may find a hyperbolic pre-ball Vσmn (ω) ⊂ B(z, α) ⊂ Aω
with m ≤ Nα . Now, since
fσmn (ω) (Vσmn (ω) ) is a ball of radius δ1 it follows from Lemma 2.3.2 that there is some
′
V ⊂ fσmn (ω) (Vσmn (ω) ) and m′ ≤ N0 with fσmn+m (ω) (V ) = ∆0 . Thus, taking kj = m+m′ we
m
have that 0 ≤ kj ≤ Nα + N0 and (fωn )−1
j (Vσn (ω) ) contains an element of {Rω = n + kj }
n+kj
inside (fωn )−1
j B(z, α). This contradicts the fact that tω
0 ≤ kj ≤ Nα + N0 .
|(fωn )−1 B(zj ,α) = 0 for all
j
Claim 2. (fωn )−1
j B(z, δ1 /4) contains a component {Rw = n + kj }, with 0 ≤ kj ≤
Nα + N0 .
n+kj
Let kj be the smallest integer 0 ≤ kj ≤ Nα + N0 for which tω
is not identically
n−1,α
zero in (fωn )j−1 B(z, α). Since (fωn )−1
⊂ {tn−1
≤ 1}, there must be
ω
j B(z, α) ⊂ Aω
0
1
some element Un+k
of {Rω = n + kj } for which ((fωn )−1
j B(z, α)) ∩ Un+kj 6= ∅. Recall
j
n+k
1
that by definition fω j sends Un+k
diffeomorphically onto ∆10 , the ball of radius 2δ0
j
around p. From time n to n + kj we may have some final “bad” period of length at
most N0 where the derivative of f may contract, however being bounded from below
1
by 1/K0 in each step. Thus, the diameter of fωn (Un+k
) is at most 4δ0 K0N0 . Since
j
0
1
B(z, α) intersects fωn (Un+k
) and α < δ0 < δ0 K0N0 , we have (fωn )−1
j B(z, δ1 /4) ⊃ Un+kj ,
j
as long as we take δ0 > 0 small enough so that
5δ0 K0N0 <
δ1
.
4
(2.6.2)
Thus, we have shown that (fωn )−1
j B(z, δ1 /4) contains some component of {Rω = n+kj }
with 0 ≤ kj ≤ Nα + N0 , and proved the claim.
Since n is a hyperbolic time for each xj , we have by the distortion control given
by Corollary 2.2.15
m((fωn )−1
m(B(z, δ1 /4))
j B(z, δ1 /4))
≤
C
.
2
0
0
m(Un+k
)
m(fωn (Un+k
))
j
j
(2.6.3)
From time n to time n + kj we have at most kj = m1 + m2 iterates with m1 ≤ Nα ,
39
CHAPTER 2. RANDOM INDUCED SCHEMES
0
m2 ≤ N0 and fωn (Un+k
) containing some point yj ∈ Hσmn1(ω) . By the definition of
j
(λ, δ)-hyperbolic time we have distδ (fσi n (ω) (x), C) ≥ λbNα for every 0 ≤ i ≤ m1 , which
implies that there is some constant D1 = D1 (α) > 0 such that | det(Dfσi n (ω) (x))| ≤ D1
0
for 0 ≤ i ≤ m1 and x ∈ fωn (Un+k
). On the other hand, since the first N0 preimages
j
of ∆0 are uniformly bounded away from C we also have some D2 > 0 such that
| det(Dfσi n+m1 (ω) (x))| ≤ D2 for every 0 ≤ i ≤ m2 and x belonging to an i-preimage
(fσi n+kj −i (ω) )−1 ∆0 of ∆0 . Hence,
0
m(fωn (Un+k
)) ≥
j
1
m(∆0 ),
D1 D2
which combined with (2.6.3) gives
0
m (fωn )−1
j B (zj , δ1 /4) ≤ Dm(Un+kj ),
with D only depending on C2 , D1 , D2 , δ0 and δ1 . Moreover, if ǫ is small enough, D1
and D2 can be taken uniform over ω.
We are now able to compare the Lebesgue measures of ∪N
i=0 Rω = n + i and
An−1
∩ Hωn . Using (2.6.1) we get
ω
X
n−1
m (fωn )−1
∩ Hωn
j B (zj , δ1 /4)) ∩ Aω
z∈I X
X
−1
≤L
m (fωn )−1
j B (zj , δ1 /4)
m(An−1
∩ Hωn ) ≤ L−1
ω
z∈I j
XX
−1
≤ DL
z∈I
0
m(Un+k
)
j
j
0
One should mention that the sets Un+k
also depend on z ∈ I. They are disjoint for
j
different values of z ∈ I. Hence, putting N = N0 + Nα , we have
m(Aωn−1 ∩ Hωn ) ≤ DL−1 m ∪N
.
i=0 Rω = n + i
To finish the proof we only have to take s0 = DL−1 .
Remark 2.6.5. It follows from the choice of the constants D1 and D2 and D that the
constant s0 only depends on the constants λ, b, Nα , N0 , C2 , δ0 and δ1 . Recall that L
is an absolute constant only depending on M.
CHAPTER 2. RANDOM INDUCED SCHEMES
40
Independent metric estimates
We have taken a disk ∆0 of radius δ0 > 0 around a point p ∈ M with certain properties
and, for every ω ∈ Ωǫ , we defined inductively the subsets Anω , Bωn , {Rω = n} and Λnω
which are related in the following way:
˙ ωn .
Λnω = ∆0 \ {Rω ≤ n} = Anω ∪B
Since we are dealing with a non-uniformly expanding on random orbits system, for
each ω ∈ Ωǫ and each n ∈ N we also have defined the set Hωn ⊂ M of points that have
n as a (λ, δ)-hyperbolic time, and the tail set Γnω as in (1.2.4). From the definition of
Γnω , Remark 2.2.7 and Lemma 2.2.8 we deduce that for every ω ∈ Ωǫ :
(m1 ) there is ζ > 0 such that for every ω ∈ Ωǫ , n ≥ 1 and every A ⊂ M \ Γnω with
m(A) > 0
n
1 X m(A ∩ Hωj )
≥ ζ.
n j=1
m(A)
Moreover, we have proved in Lemma 2.6.1, Lemma 2.6.2 and Proposition 2.6.4 that
the following metric relations also hold:
(m2 ) there is c0 > 0 (not depending on δ0 ) such that for every ω ∈ Ωǫ and n ≥ 1
m(Bωn−1 ∩ Anω ) ≥ c0 m(Bωn−1 );
(m3 ) there are d0 , r0 > 0 with d0 + r0 < 1 and d0 , r0 → 0 as δ0 → 0, such that for
every ω ∈ Ωǫ and n ≥ 1
m(Aωn−1 ∩ Bωn )
≤ d0
m(An−1
ω )
and
m(An−1
∩ {Rω = n})
ω
≤ r0 ;
m(An−1
ω )
(m4 ) there is s0 > 0 and an integer N ≥ 0 such that for every ω ∈ Ωǫ and n ≥ 1
!
N
[
m
Rω = n + i
≥ s0 m(An−1
∩ Hωn ).
ω
i=0
In the inductive process of construction of the sets Anω , Bωn , {Rω = n} and Λnω we
have fixed some large integer R0 , this being the first step at which the construction
41
CHAPTER 2. RANDOM INDUCED SCHEMES
began. Recall that Anω = Λnω = ∆0 and Bωn = {Rω = n} = ∅ for n ≤ R0 . We will
assume that
R0 > max {2(N + 1), 12/ζ} .
(2.6.4)
Note that since N and ζ do not depend on R0 this is always possible, so we can follow
Section 4.5.2 at [A04] to conclude that for every ω ∈ Ωǫ this process indeed produces
a partition Pω = {Rω = n}n of ∆0 . Moreover, it also follows from there that, if there
exist C, p > 0 such that for every ω ∈ Ωǫ we have m(Γnω ) ≤ Cn−p then there exists
C ′ such that for every ω ∈ Ωǫ the return time function satisfies
m(Rω > n) ≤ C ′ n−p .
(2.6.5)
It is possible to check that constant C ′ depend ultimately on the constants B, β and
b0 associated to the non-uniform expanding condition in Definition 1.2.1. This implies
that C ′ can be considered the same for every ω ∈ Ωǫ (in particular for every σ l (ω),
l ∈ Z). So, once we are considering p > 1, we conclude that the sum
+∞
X
n=N
m({Rσ−n (ω) > n}) ≤
+∞
X
C ′ n−p
n=N
can be made arbitrarily small if we take N large enough.
Remark 2.6.6. Throughout this independent metric estimates we have to consider δ0
small enough (depending on ζ) so that the estimates in [A04] work.
2.7
Uniformity
The condition (U2) was proved throughout Section 2.5. Note, moreover, that providing ǫ is sufficiently small, Lemma 2.3.2 ensures that the domains coming from
∆ and turned meanwhile into balls of radius δ1 - that can be anywhere in M - will
return in finite (and uniform bounded) iterates for all ω ∈ Ωǫ and cover completely
√
B(p, 2 δ 0 ). This ensures that we can locate the random Gibbs-Markov structures in
the ball ∆ = ∆0 = B(p, δ0 ), for every ω ∈ Ωǫ .
For the uniformity condition (U1), given N > 1 and ̺ > 0 we define, for every
ω ∈ Ωǫ , the sets {Rω = j}, Aωj and Bjω , with j ≤ N, as described in Section
42
CHAPTER 2. RANDOM INDUCED SCHEMES
2.4. The process that leads to the construction of these sets is based on the fact
that small domains in ∆0 became large (in balls of radius δ1 ) by fωk , for some 0 ≤
k ≤ Nα , and then by fσi k (ω) , with 0 ≤ i ≤ N0 , they cover completely the ball
√
B(p, 2 δ0 ) ⊃ ∆0 . Hence, just by the continuity of Φ, associated to the random
perturbation {Φ, {θǫ }ǫ>0 }, we guarantee that, for any two realizations ω, ω ′ in Ωǫ the
Lebesgue measure of the symmetric difference of respective sets {Rω = j}, Ajω and
Bj , for j ≤ N, is smaller than ̺, as long as we take ǫ sufficiently small. In particular,
this holds for {Rσ−j (ω) = j} and {Rσ−j (ω′ ) = j}, for every j = 1, 2, ..., N.
For a better understanding of the dependencies of the constants we expose in
the following a table, using the principle that no constant depends on a constant
from a line below, and that all the constants on the first column can be considered
independent of a realization ω, provided ǫ is small enough.
Constant
Dependence
Reference
λ, δ, ζ
a0
Proposition 2.2.5
δ1
B, β, λ, δ
Lemma 2.2.9
N0
δ1
(2.3.1)
D0 , K 0
λ, δ1
Lemma 2.3.2
C0
B, β, b, λ
Corollary 2.2.13
C1 , C2
C0
Corollary 2.2.14
δ0
N0 , K0 , δ1 , ζ
Lemma 2.3.2, (2.6.2), Remark 2.6.6
α
K0 , N0 , δ0 , λ
(2.4.2)
Nα
α, λ, δ1
Lemma 2.3.3
c0
λ, C1 , D0
Lemma 2.6.1
d 0 , r0
C1 , D0 , δ0
Lemma 2.6.2
s0
λ, b, Nα , C2 , N0
Remark 2.6.5
N
N0 , Nα
Proposition 2.6.4
R0
K0 , λ, N0 , N, ζ
Section 2.5, (2.6.4)
Chapter 3
Stochastic stability
This chapter is comprised of two main parts. In the first one we obtain the unique absolutely continuous stationary probability measure and, in the second part, we prove
the strong stochastic stability for non-uniformly expanding maps. In the following
we assume the hypothesis of Theorem 1.3.1 and we consider ǫ > 0 sufficiently small
so that we are able to use Theorem 2.1.2 and conditions (U1)-(U2) hold.
3.1
Physical measure
In order to obtain a physical measure we use the induced Gibbs-Markov structure
from previous Chapter. Using this structure we construct some suitable induced
measures and project them so that we construct a family of finite measures on M,
with the additional property that they only depend on the past. They allow us to
construct an absolutely continuous stationary finite measure, which we can normalize
to obtain a physical probability measure.
3.1.1
Induced random measures
The results in this particular Section can be obtained starting from a general axiomatic assumptions containing conditions (U1) and (U2), the conclusions of Theorem
2.1.2 and some particular conditions on the elements of the induced partitions, in way
we can ensure the only past dependence property for the sample measures. For this,
the algorithm in Section 2.4 guarantees that for all ω ∈ Ωǫ the sets {Rσ−n (ω) = j},
43
CHAPTER 3. STOCHASTIC STABILITY
44
for 1 ≤ j ≤ n, does not depend on the future ω0+ = (ω0 , ω1 , . . .) of ω.
We start defining a new random induced dynamical system. Let us consider
disjoint copies ∆ω of ∆, associated to the corresponding ω ∈ Ωǫ , as the partitions
R (x)
Pω . For x ∈ ∆ω we put Fω (x) = fω ω
(x) and the dynamics consist in hopping
from x ∈ ∆ω to Fω (x) ∈ ∆σRω (x) (ω) . However, we also can keep regarding this as
a dynamical system over ∆. We refine recursively Pω on ∆ω with the partitions
associated to the images of each element of Pω :
Pw(n) =
n
_
_
j=0 k∈Lω,j
(Fωj )−1 Pσk (ω)
where Lω,j = {k ∈ N0 : Fωj (∆ω ) ∩ ∆σk (ω) 6= ∅}.
Our next aim is to prove that for each ω ∈ Ωǫ there is some absolutely continuous
measure νω defined on ∆ with some interesting invariance property. Moreover, the
density ρω of νω with respect to the Lebesgue measure will belong to a Hölder-type
space:
H = {ϕ : ∆ → R| ∃Kϕ > 0, |ϕ(x) − ϕ(y)| ≤ Kϕ d(x, y) ∀x, y ∈ ∆} .
Consider any measurable set E ⊂ ∆ω . We define
(F −1 )ω (E) =
G
x ∈ ∆σ−n (ω) : Rσ−n (ω) (x) = n and Fσ−n (ω) (x) ∈ E
n∈N
and [(F j )−1 ]ω (E) by induction. Moreover, given a family ν̂ω = {νσ−n (ω) }n∈N of meaF
sures on n∈N ∆σ−n (ω) we set
∗
(F j )ω ν̂ω (E) =
X
n∈N
νσ−n (ω) ([(F j )−1 ]ω (E) ∩ ∆σ−n (ω) )
Here we use ∗ superscript to distinguish this push-forward from the usual one for deterministic systems, whose notation is usually ∗ subscript. The next theorem follows
closely Theorem 3.2 in [BaBeM02].
Theorem 3.1.1. For every ω ∈ Ωǫ there is an absolutely continuous finite measure
νω on ∆ such that (F )ω ∗ {νσ−n (ω) }n∈N = νω and ρω = dνω /dm ∈ H. Moreover, there
is a constant K1 > 0 such that K1 −1 ≤ ρω ≤ K1 for all ω ∈ Ωǫ .
45
CHAPTER 3. STOCHASTIC STABILITY
For each ω ∈ Ωǫ , the Hölder constant Kρω for ρω ∈ H will depend only on K and
κ given by Theorem 2.1.2. By (U2), considering ǫ small enough, the constants Kρω
can be taken the same for all ω ∈ Ωǫ , which we will refer as K2 > 0.
Proof. Let m0 be the probability measure (m|∆)/m(∆) on ∆ and set m̂0 as the family
F
of measures on k∈N ∆σ−k (ω) so that m0 is the measure on each ∆σ−k (ω) .
F
(j)
Let A ⊂ k∈N ∆σ−k (ω) , with A ⊂ [(F j )ω ]−1 (∆ω ) and A ∈ Pσ−n (ω) , for some n ∈ N,
and set on ∆ω the function
ρj,A
ω =
d
(Fσj−n (ω) )∗ (m0 |A).
dm0
Let x, y ∈ ∆ω be arbitrary points, and let x′ , y ′ ∈ A such that x′ ∈ [(F j )−1 ]ω (x) and
y ′ ∈ [(F j )−1 ]ω (y), so that x′ , y ′ ∈ ∆σ−n (ω) . For the decreasing sequence n = n0 > ... >
nj = 0 given by
′
nl = nl−1 − Rσnl−1 (ω) (Fσl−1
−n (ω) (x )),
for 1 ≤ l ≤ j,
we find that Fσl −n (ω) (x′ ), Fσl −n (ω) (y ′) lies in the same element Uσ−nl (ω) of Pσ−nl (ω) , for
0 ≤ l < j. By Theorem 2.1.2 (recall items 2. and 3. in Definition 2.1.1 and (U2))
j
j−1
′
det DF −nl (F l −n (x′ )) X
|
det
DF
−n (ω) (x )|
ρj,A
(y)
σ
(ω)
σ
σ
(ω)
ω
log j,A
= log
=
log j
l
′
det DFσ−nl (ω) (Fσ−n (ω) (y )) | det DFσ−n (ω) (y ′)|
ρω (x)
l=0
≤
≤
≤
j−1
X
l=0
j−1
X
l=0
j−1
X
K dist Fσ−nl (ω) (Fσl −n (ω) (x′ )), Fσ−nl (ω) (Fσl −n (ω) (y ′))
l+1
′
l+1
′
K dist Fσ−n (ω) (x ), Fσ−n (ω) (y )
Kκj−l−1 dist(x, y)
l=0
≤ K2 dist(x, y)
1
with K2 = K 1−κ
, which is uniform in ω, j and A. The sequence
ρω,n
d
=
dm0
n−1
1X j ∗
(F )ω m̂0
n j=0
!
46
CHAPTER 3. STOCHASTIC STABILITY
is a linear combination of terms as ρj,A
ω so that one has ρω,n (x) ≤ exp(K2 2δ0 )ρω,n (y)
for all x, y in ∆, since ∆ has diameter 2δ0 . In particular there exists K1 > 0 such
that K1 −1 ≤ ρω,n ≤ K1 . Moreover,
ρω,n (x)
ρω,n (x) |ρω,n (x) − ρω,n (y)| ≤ ||ρω,n ||∞ − 1 ≤ K1 C log
≤ C ′ d(x, y).
ρω,n (y)
ρω,n (y) By Ascoli-Arzela theorem, the sequence (ρω,n )n is relatively compact in L∞ (∆, m0 )
and has some subsequence (ρω,nωi )i converging to some ρω . By the construction process
we have K1 −1 ≤ ρω ≤ K1 and ρω ∈ H. The measure νω = ρω dm is finite because
νw (∆) ≤ K1 m(∆) < ∞.
We will now choose a convenient family {νσl (ω) }l∈Z of such finite measures. We
consider the sequence (nωi )i as before and, by repeating the procedure, we obtain
σ−1 (ω)
a subsequence (ni
)i of (nωi )i such that (ρω,nσ−1 (ω) )i converges to some ρσ−1 (ω) .
i
σ(ω)
We do it also to obtain
)i and proceed
inductively
so thatwe have convenient
i
n (n
o
σ−l (ω)
family of sequences (ni
)i : l ∈ Z for which ρ −l
converges to some
σ −l (ω)
σ
(ω),ni
i
ρσ−l (ω) , for all l ∈ Z. We use a diagonalization arguments to obtain a sequence (ki )
such that (ρσl (ω),ki )i converges to ρσl (ω) , for all l ∈ Z. Starting from a represen-
tational element of {σ l (ω)}l∈Z we obtain a family {νσl (ω) }l∈Z of finite measures on
F
∗
l∈Z ∆σl (ω) . We check now the quasi-invariance property (F )ω {νσ−n (ω) }n∈N = νω .
LetA ⊂ ∆ω be any measurable
set. Write νσ−n (ω),i for the measure with density
P
k
−1
d
1
i
j
∗
j=0 (F )σ−n (ω) m̂0 . We have
dm ki
νσ−n (ω) (A) =
Z
χA ρσ−n (ω) dm =
Z
χA lim ρσ−n (ω),ki dm =
i→+∞
Z
lim χA ρσ−n (ω),ki dm,
i→+∞
and by dominated convergence theorem,
Z
lim χA ρσ−n (ω),ki dm = lim
i→+∞
so that νσ−n (ω) (A) = lim νσ−n (ω),i (A).
i→+∞
i→+∞
Z
χA ρσ−n (ω),ki dm,
47
CHAPTER 3. STOCHASTIC STABILITY
We finally have
(F )ω ∗ ν̂ω (A) =
=
+∞
X
νσ−n (ω) ((F −1 )ω (A) ∩ ∆σ−n (ω) )
n=1
+∞ X
n=1
Fσ−n (ω) |{Rσ−n (ω) =n}
∗ νσ−n (ω)
"
+∞ X
=
Fσ−n (ω) |{Rσ−n (ω) =n} ∗
n=1
=
=
=
=
=
(A)
1
ki
lim
i→+∞
kX
i −1
(F j )σ−n (ω) ∗ m̂0
j=0
!!
(A)
!
ki −1 1 X
lim
Fσ−n (ω) |{Rσ−n (ω) =n} ∗ (F j )σ−n (ω) ∗ m̂0 (A)
i→+∞
k
i j=0
n=1
!
!
ki −1 X
+∞
1 X
Fσ−n (ω) |{Rσ−n (ω) =n} ∗ (F j )σ−n (ω) ∗ m̂0 (A)
lim
i→+∞
ki j=0 n=1
!
ki −1
1 X
lim
(F j+1 )ω ∗ m̂0 (A)
i→+∞
ki j=0
!
ki −1
1 X
lim
(F j )ω ∗ m̂0 (A) − m0 (A) + (F ki )ω ∗ m̂0 (A)
i→+∞ ki
j=0
+∞
X
#
lim νω,i (A)
i→+∞
= νω (A)
3.1.2
Sample and stationary measures
We define the family {µ̃ω }ω of finite Borel measures in M by
+∞
X
µ̃ω =
(fσj−j (ω) )∗ (νσ−j (ω) |{Rσ−j (ω) > j}),
j=0
where the measures νσ−j (ω) are given by Theorem 3.1.1. Since
µ̃ω (M) =
+∞
X
j=0
νσ−j (ω) ({Rσ−j (ω) > j}) ≤ K1
+∞
X
m({Rσ−j (ω) > j}),
j=0
the hypothesis on the decay of m(Γnω ) and Theorem 2.1.2 provide that those are finite
measures. We could, at this time, normalize it in order to obtain a family {µω }ω∈Ωǫ
48
CHAPTER 3. STOCHASTIC STABILITY
of sample probability measures.
Proposition 3.1.2. The measures of the family {µ̃ω }ω∈Ωǫ are absolutely continuous
measures and satisfies fω ∗ µ̃w = µ̃σ(ω) .
Proof. The absolute continuity follows from absolute continuity of measures {νw }ω∈Ωǫ .
For the invariance property let A ⊂ M be a measurable set. Then
fω ∗ µ̃ω (A) =
=
+∞
X
j=0
+∞
X
j=0
=
+∞
X
j=0
+
(fω )∗ (fσj−j (ω) )∗ (νσ−j (ω) |{Rσ−j (ω) > j})(A)
(j+1)
(νσ−j (ω) |{Rσ−j (ω) > j})((fσ−j (ω) )−1 (A))
(j+1)
(νσ−j (ω) |{Rσ−j (ω) = j + 1})((fσ−j (ω) )−1 (A))
+∞
X
j=0
=
+∞
X
(j+1)
(νσ−j (ω) |{Rσ−j (ω) > j + 1})((fσ−j (ω) )−1 (A))
νσ−l (σ(ω)) ((F −1 )σ(ω) (A) ∩ ∆σ−l (σ(ω)) )
l=1
+∞
X
+
l=1
νσ−l (σ(ω)) |{Rσ−l (σ(ω)) > l})((fσl −l (σ(ω)) )−1 (A))
∗
= (F )σ(ω) ν̂σ(ω) (A) +
+∞
X
l=0
(νσ−l (σ(ω)) |{Rσ−l (σ(ω)) > l})((fσl −l (σ(ω)) )−1 (A))
−(νσ(ω) |{Rσ(ω) > 0})(A)
+∞
X
= νσ(ω) (A) +
(fσl −l (σ(ω)) )∗ (νσ−l (σ(ω)) |{Rσ−l (σ(ω)) > l})(A) − νσ(ω) (A)
l=0
= νσ(ω) (A) + µ̃σ(ω) (A) − νσ(ω) (A) = µ̃σ(ω) (A).
Remark 3.1.3. By construction, all the measures in the family {νσ−n (ω) }n∈N depend
only in the past ω − = (..., ω−2 , ω−1 ) of ω. Moreover, for ω, τ ∈ T Z with the same
past (ωi = τi , for i ≤ −1), we have that the sets {Rσ−j (ω) = j} and {Rσ−j (τ ) = j}, for
j ≥ 1, are exactly the same (as subsets of ∆ ⊂ M). The measure µ̃ω involve sums of
49
CHAPTER 3. STOCHASTIC STABILITY
the type (fσj−j (ω) )∗ (νσ−j (ω) |{Rσ−j (ω) > j}), and since
m({Rσ−j (ω) > j}) = m ∆ \
(
j
[
{Rσ−k (ω) = k}
k=1
)!
and νσ−j (ω) ≪ m, the measures µ̃ω depend only on the past ω − of ω.
Starting from the induced region ∆ we constructed a family of Borel induced
random measures {νω }ω on ∆ and project them to a family of finite measures {µ̃ω }ω on
M that almost surely depends on the past. Proceeding as in the proof of Proposition
R
B.2.3, we conclude that the measure µ̃ǫ = µ̃ω dθǫZ is a finite stationary measure,
and is absolutely continuous due to the absolute continuously of the measures {µ̃ω }ω .
For the finiteness we have, under the hypothesis of Theorem 1.3.1, that exists C > 0
and p > 1 such that
µ̃ǫ (M) =
Z
µ̃ω (M) dθǫZ (ω)
=
Z X
+∞
νσ−n (ω) ({Rσ−n (ω) >
n=0
n})dθǫZ (ω)
≤ K1
+∞
X
Cn−p ,
n=0
which is finite. We can normalize µ̃ǫ and define an absolutely continuous stationary
probability measure µǫ = µ̃ǫ /µ̃ǫ (M) with density hǫ . We prove in the next Section
that µǫ is the unique absolutely continuous ergodic stationary probability measure
(and thus physical).
3.1.3
Ergodicity and unicity
We say that A ⊂ M is a random forward invariant set if for µǫ almost every x ∈ A
we have ft (x) ∈ A for θǫ almost every t. Let δ1 > 0 be given by Lemma 2.2.9.
Proposition 3.1.4. Given any random forward invariant set A ⊂ M with µǫ (A) > 0,
there is a ball B of radius δ1 /4 such that m(B \ A) = 0.
Proof. It is enough to prove that there exist disks of radius δ1 /4 where the relative
measure of A is arbitrarily close to one. For n ≥ 1 let An be the set of points x ∈ A
for which fωn (x) ∈ A, for θǫZ almost every ω, and Ã = ∩+∞
n=1 An . Since A is random
forward invariant then µǫ (A \ An ) = 0 for all n and thus µǫ (A \ Ã) = 0. We have
µǫ (Ã) = µǫ (A) > 0 which implies m(Ã) > 0, since µǫ ≪ m, and for θǫZ almost every
ω we have fωk (Ã) ⊂ A. Since the set of points x ∈ M for which for θǫZ almost every
50
CHAPTER 3. STOCHASTIC STABILITY
ω there are infinitely many hyperbolic pre-balls Vωn (x) is random forward invariant,
we may assume, with no loss of generality, that every point in A has infinitely many
hyperbolic pre-balls Vωn (x) for θǫZ almost every ω. Recall that the hyperbolic pre-balls
Vωn (x) are sent diffeomorphically by fωn onto hyperbolic balls with radius δ1 , that is
fωn (Vωn (x)) = B(fωn (x), δ1 ).
Let γ > 0 be some small number. By regularity of m, there is a compact set
Ãc ⊂ Ã and an open set Ão ⊃ Ã such that
m(Ão \ Ãc ) < γm(Ã).
(3.1.1)
Assume that n0 is large enough so that, for every x ∈ Ãc and θǫZ almost every
ω, any hyperbolic preball Vωn (x), associated with the (λ, δ)-hyperbolic time n for
(ω, x), with n ≥ n0 , is contained in Ão . Let Wωn (x) be the part of Vωn (x) which is
sent diffeomorphically by fωn onto the ball B(fωn (x), δ1 /4). By compactness there are
x1 , ..., xr ∈ Ãc and n(x1 ), . . . , n(xr ) ≥ n0 such that
Ãc ⊂ Wωn(x1 ) (x1 ) ∪ ... ∪ Wωn(xr ) (xr ).
(3.1.2)
For the sake of notational simplicity, for each 1 ≤ i ≤ r we shall write
Vωi = Vωn(xi ) (xi ),
Wωi = Wωn(xi ) (xi ) and ni = n(xi ).
Assume that
{n1 , ..., nr } = {n∗1 , ..., n∗s },
with n∗1 < n∗2 < ... < n∗s .
Let I1 ⊂ N be a maximal subset of {1, ..., r} such that for each i ∈ I1 both ni = n∗1 ,
and Wωi ∩ Wωj = ∅ for every j ∈ I1 with j 6= i. Inductively, we define Ik for 2 ≤ k ≤ s
as follows: supposing that I1 , . . . , Ik−1 have already been defined, let Ik be a maximal
set of {1, . . . , r} such that for each i ∈ Ik both ni = n∗k , and Wωi ∩ Wωj = ∅ for every
j ∈ I1 ∪ ... ∪ Ik with i 6= j.
Define I = I1 ∪ · · · ∪ Is . By construction we have that {Wωi }i∈I is a family of
pairwise disjoint sets. We claim that {Vωi }i∈I is a covering of Ãc . To see this, recall
that by construction, given any Wωj with 1 ≤ j ≤ r, there is some i ∈ I with
51
CHAPTER 3. STOCHASTIC STABILITY
x
n(xi )
n(xi ) ≤ n(xj ) such that Wω j ∩ Wωxi 6= ∅. Taking images by fω
we have
fωn(xi ) (Wj ) ∩ B(fωn(xi ) (xi ), δ1 /4) 6= ∅.
It follows from Lemma 2.2.12 that
δ1 (n(xj )−n(xi ))/2 δ1
λ
≤ ,
2
2
diam(fωn(xi ) (Wωj )) ≤
and so
fωn(xi ) (Wωj ) ⊂ B(fωn(xi ) (xi ), δ1 ).
This gives that Wωj ⊂ Vωi . We have proved that given any Wωj with 1 ≤ j ≤ r, there
is i ∈ I so that Wωj ⊂ Vωi . Taking into account (3.1.2), this means that {Vωi }i∈I is a
covering of Ãc .
By Corollary 2.2.15 one may find τ > 0 such that
m(Wωi ) ≥ τ m(Vωi ),
for all i ∈ I.
Hence,
m
[
i∈I
Wωi
!
=
X
m(Wωi )
i∈I
≥ τ
X
m(Vωi )
i∈I
≥ τm
[
Vωi
i∈I
!
≥ τ m(Ãc ).
From (3.1.1) one easily deduces that m(Ãc ) > (1−γ)m(Ã). Noting that the constant τ
does not depend on γ, choosing γ > 0 small enough we may have
m
[
i∈I
Wωi
!
>
τ
m(Ã).
2
(3.1.3)
52
CHAPTER 3. STOCHASTIC STABILITY
We are going to prove that
2γ
m(Wωi \ Ã)
<
,
i
m(Wω )
τ
for some i ∈ I.
n(xi )
This is enough for our purpose. First, since fω
Wωi we have
(3.1.4)
n(xi )
(Ã) ⊂ A and fω
is injective on
m(fωn(xi ) (Wωi ) \ A) ≤ m(fωn(xi ) (Wωi ) \ fωn(xi ) (Ã))
= m(fωn(xi ) (Wωi \ Ã)).
n(xi )
Therefore, by Corollary 2.2.15, taking B = fω
(Wωi ) as a ball of radius δ1 /4 we have
m(B \ A)
m(f n(xi ) (Wωi \ Ã))
m(Wωi \ Ã)
2C2 γ
≤
C
≤
=
,
2
n(x
)
i
i
m(B)
m(Wω )
τ
m(fω (Wωi ))
which can obviously be made arbitrarily small, setting γ → 0. From this one easily deduces that there are disks of radius δ1 /4 where the relative measure of A is
arbitrarily close to one.
Finally, let us prove (3.1.4). Assume, by contradiction, that it does not hold.
Then, using (3.1.1) and (3.1.3)
γm(Ã) > m(Ão \ Ãc )
!
!
[
≥ m
Wωi \ Ã
i∈I
[
2γ
≥
m
Wωi
τ
i∈I
!
> γm(Ã).
This gives a contradiction.
Proposition 3.1.5. The stationary measure µǫ is the unique absolutely continuous
ergodic stationary probability measure.
Proof. We prove first that there is a finite partition H1 , . . . Hn of a full Lebesgue
measure set in M such that the normalized restrictions of µǫ to each Hi , i = 1, . . . , n
is ergodic. Then, we use the topological transitivity of f to ensure the unicity.
CHAPTER 3. STOCHASTIC STABILITY
53
Suppose µǫ is is not ergodic. Then we may decompose M into two disjoint random
invariant sets H1 and H2 (= M \ H1 ) both with positive µǫ -measure. In particular,
both H1 and H2 have positive Lebesgue measure. Let µ1ǫ and µ2ǫ be the normalized
restrictions of µǫ to H1 and H2 , respectively. They are also absolutely continuous
stationary measures. If they are not ergodic, we continue decomposing them, in the
same way as we did for µǫ .
On the other hand, by Proposition 3.1.4, each one of the random invariant sets we
find in this decomposition has full Lebesgue measure in some disk with fixed radius.
Since these disks must be disjoint, and M is compact, there can only be finitely
many of them. So, the decomposition must stop after a finite number of steps, giving
Pp
i
that µǫ can be written µǫ =
i=1 µǫ (Hi )µǫ where H1 , . . . , Hp is a partition of M
into random invariant sets with positive measure and each µiǫ = (µǫ |Hi )/µǫ (Hi ) is an
ergodic stationary probability measure.
For the unicity assume that there are two distinct ergodic absolutely continuous
stationary measures µ1ǫ and µ2ǫ . Since B(µ1ǫ ) and B(µ2ǫ ) are random forward invariant
sets, then by Proposition 3.1.4 there are disks ∆1 = B(p1 , δ1 /4) and ∆2 = B(p2 , δ1 /4)
such that m(∆i \ B(µiǫ )) = 0 for i = 1, 2. The topological transitivity of f , the
continuity of Φ and the random invariance of B(µ1 ) and B(µ2 ) imply that, if ǫ is
small enough, then m(B(µ1ǫ ) ∩ B(µ2ǫ )) > 0. Consider any point x in this intersection.
P
j
For every continuous function ϕ : M → R and a θǫZ full subset of T Z , n1 n−1
j=0 ϕ(fω (x))
R
R
converges to ϕ dµ1ǫ , as n goes to infinity and, similarly, to ϕ dµ2ǫ . The unicity of
R
R
the limit implies ϕ dµ1ǫ = ϕ dµ2ǫ so that µ1ǫ = µ2ǫ .
If we consider any stationary probability measure µ̂ǫ on M, we can do the same
procedure as before, and get a finite decomposition of µ̂ǫ in ergodic components,
containing (Lebesgue mod 0) disks of a fixed radius. As we saw, by the topological
transitivity of f one should have µ̂ǫ = µǫ .
3.2
Strong stochastic stability
We mean for strong stochastic stability for a non-uniformly expanding maps f by
the convergence of the density hǫ of the unique absolutely continuous stationary
probability measure µǫ for {Φ, (θǫ )ǫ>0} to the density hf of the unique f -invariant
absolutely continuous measure µf , in the L1 topology. The strategy is to use the
54
CHAPTER 3. STOCHASTIC STABILITY
induced random measures νω defined on ∆, constructed previously and within get a
measure νǫ on ∆ with density ρǫ , by averaging over νω . This family of densities (ρǫ )ǫ>0
has an accumulation point ρ∞ in L1 (∆), as ǫ goes to 0, which give us a measure ν∞
on ∆. We can project this measure to a probability measure µ∞ on M, using the
dynamics of f , in such a way that we can compare the densities of the measures µǫ
and µ∞ on M, attesting their convergence. To conclude we prove in Proposition 3.2.3
that µ∞ is f - invariant and, by unicity (see Remark 2.1.3), is actually equal to µf .
For ω ∈ Ωǫ , the measure νω given by Theorem 3.1.1 has density ρw ∈ H, bounded
away and below with K1 −1 ≤ ρw ≤ K1 and, for x, y ∈ ∆, |ρω (x) − ρω (y)| ≤ K2 d(x, y).
We define an absolutely continuous measure νǫ on ∆, with density ρǫ , by
νǫ (x) =
Z
νω (x) dθǫZ (ω),
and where ρǫ (x) =
Z
ρω (x) dθǫZ (ω).
Consider any sequence (ǫn )n , with ǫn > 0 and ǫn → 0, as n → +∞. The family {ρǫn }n
is uniformly bounded and equicontinuous. In fact, K1 −1 ≤ ρǫn ≤ K1 and, for x, y ∈ ∆
we have |ρǫn (x) − ρǫn (y)| ≤ K2 d(x, y). By Ascoli-Arzela theorem this implies that
(ρǫ n )n has a converging subsequence {ρǫ′n } to some ρ∞ , with K1 −1 ≤ ρ∞ ≤ K1 , in the
complete space L1 (∆). Without loss of generality we will assume that all sequence
converge (see Remark 3.2.4). This means that for given γ > 0 exists δ such that if
ǫn < δ then
||ρǫn − ρ∞ k1 < γ.
(3.2.1)
We write ν∞ to be the Borel measure on ∆ with density ρ∞ and define an absolutely
continuous Borel measure µ̃∞ with density h̃∞ by
µ̃∞ =
∞
X
j=0
(f j )∗ (ν∞ |{R > j}).
Since the unperturbed map f is non-uniformly expanding, the decay of the return
time and the bound ρ∞ ≤ K1 implies that µ̃∞ is finite. We can normalize it to obtain
an absolutely continuous probability measure µ∞ with density h∞ .
We introduce an operator acting on L1 (∆)
Ljω ϕ(x) =
X
y=(fωj )−1 (x)
ϕ(y)
| det Dfωj (y)|
55
CHAPTER 3. STOCHASTIC STABILITY
and, to simplify the notation, we write Lj ϕ(x) for Ljω∗ ϕ(x). For this operator we have
Z
(Ljω ϕ)ψdm
=
Z
ϕ(ψ ◦ fωj )dm
whenever the integrals make sense. We will use the fact that this operator never
expands:
Z
for every ϕ ∈ L1 (∆).
j
Lω ϕ(x) dm ≤
Z
Ljω
|ϕ(x)| dm =
Z
|ϕ| dm
Lemma 3.2.1. For given γ > 0 and j ∈ N there is αj > 0 such that if ǫ < αj then
for every ω, τ ∈ Ωǫ we have
Z
j
L ϕ(x) − Lj ϕ(x) dm ≤ γ
ω
τ
for every ϕ in H such that kϕk∞ < K1 , where K1 is given by Theorem 3.1.1.
Proof. Let ϕ in H such that kϕk∞ < K1 . Our assumptions on the critical set imply
that the critical set C intersects ∆ in a zero Lebesgue measure set. Given any γ1 > 0,
define C(γ1 ) as the γ1 -neighborhood of this intersection. For every ω ∈ Ωǫ , we have
m(fωj (C(γ1 ))) ≤ const m(C(γ1 )) for some constant that may be taken uniform over ω
if ǫ < αj for some small αj .
By the continuity of fωj with respect to ω, we may fix γ1 = γ1 (j) small enough so
that if ǫ < αj < γ1 then
m
(fωj )−1 (fτj (C(γ1 ))
1
≤
2
γ
8K1
,
(3.2.2)
for every ω, τ in Ωǫ , where K1 is the bound constant given by Theorem 3.1.1.
We decompose ∆ \ C(γ1 ) into a finite collection D(ω) of domains of injectivity
of fωj . We may define a corresponding collection D(τ ) of domains of injectivity for
fτj in ∆ \ C(γ1 ), and there is a natural bijection associating to each Dω ∈ D(ω)
a unique Dτ ∈ D(τ ) such that the Lebesgue measure of Dτ △Dω is small, where
Dτ △Dω denotes the symmetric difference of the two sets Dτ and Dω . Observe that
Ljτ is supported in
fτj (∆) = fτj (C(γ1 )) ∪
[
Dτ ∈D(τ )
fτj (Dτ ),
56
CHAPTER 3. STOCHASTIC STABILITY
and analogously for Ljω . So,
Z
|Ljτ ϕ
−
Ljω ϕ| dm
≤
+
Z
fωj (C(γ1 ))∪fτj (C(γ1 ))
X Z
Dω ∈D(ω)
+
|Ljτ ϕ| + |Ljω ϕ| dm
X Z
Dω ∈D(ω)
fωj (Dω )∩fτj (Dτ )
fωj (Dω )△fτj (Dτ )
|Ljτ ϕ − Ljω ϕ| dm
|Ljτ ϕ| + |Ljω ϕ| dm,
(3.2.3)
(3.2.4)
(3.2.5)
where Dτ always denotes the element of D(τ ) associated to each Dω ∈ D(ω). Let us
now estimate the expressions on the right hand side of this inequality. We start with
(3.2.3). For notational simplicity, we write E = fωj (C(γ1 )) ∪ fτj (C(γ1 )). Then
Z
E
|Ljτ ϕ| dm
≤
Z
χE Ljτ |ϕ|
dm =
Z
(χE ◦ fτj )|ϕ| dm.
It follows from Hölder’s inequality and (3.2.2) that
Z
γ
γ
K1 = .
(χE ◦ fτj )|ϕ| dm ≤ m (fτj )−1 E kϕk∞ ≤
8K1
8
The case associated to fωj gives a similar bound for the second term in (3.2.3). So,
Z
fωj (C(γ1 ))∪fτj (C(γ1 ))
γ
|Ljτ ϕ| + |Ljω ϕ| dm ≤ .
4
(3.2.6)
Making the change of variables y = fωj (x) in (3.2.4), we may rewrite it as
ϕ
ϕ
j −1
j
j
◦
((f
)
◦
f
)
−
τ
ω
j · | det Dfω | dm,
| det Df j |
d
|
det
Df
|
Dω
τ
ω
Z
cω = (f j )−1 f j (Dω ) ∩ f j (Dτ ) = Dω ∩ (f j )−1 ◦ f j (Dτ ). For notational
where D
ω
ω
τ
ω
τ
simplicity, we introduce g = (fτj )−1 ◦ fωj . The previous expression is bounded by
Z
bω
D
|ϕ ◦ g − ϕ| ·
| det Dfωj |
| det Dfωj |
+ |ϕ| · − 1 dm.
j
j
| det Dfτ | ◦ g
| det Dfτ | ◦ g
57
CHAPTER 3. STOCHASTIC STABILITY
Choosing αj > 0 sufficiently small, the assumption ǫ < αj implies
| det Dfωj |
≤ γ ,
−
1
| det Df j | ◦ g
8K1
τ
and so
| det Dfωj |
≤2
| det Dfτj | ◦ g
b ω ). Hence, since ϕ belongs to H,
on ∆ \ C(γ1 ) (which contains D
Z
fωj (Dω )∩fτj (Dτ )
|Ljτ ϕ
−
Z
γ
≤ 2
|ϕ ◦ g − ϕ| dm +
8K1
bω
D
Z
γ
≤ 2K2
kg − id∆ k0 dm +
8
bω
D
Ljω ϕ| dm
Reducing αj > 0, we can make kg − id∆ k0 smaller than
Z
fωj (Dω )∩fτj (Dτ )
γ
,
16K2 m(∆)
|Ljτ ϕ − Ljω ϕ| dm ≤
γ
.
4
Z
|ϕ| dm
so that
(3.2.7)
We estimate the terms in (3.2.5) in much the same way as we did for (3.2.3). For
each Dω let E now be fωj (Dω )△fτj (Dτ ). The properties of the operator, followed by
Hölder’s inequality, yield
Z
E
|Ljτ ϕ|dm
≤
Z
(χE ◦ fτj )|ϕ|dm ≤ m (fτj )−1 E kϕk∞ ≤ K1 m (fτj )−1 E
Fix γ2 > 0 such that #D(ω)4γ2 < γ. Taking αj sufficiently small, we may ensure
that the Lebesgue measure of all the sets
(fτj )−1 E = (fτj )−1 (fωj (Dω )△fτj (Dτ ))
is small enough so that
K1 m (fτj )−1 E < γ2 .
In this way we get
Z
fωj (Dω )△fτj (Dτ )
|Ljω ϕ| + |Ljτ ϕ| dm ≤ 2γ2
(3.2.8)
(the second term on the left is estimated in the same way as the first one). Putting
58
CHAPTER 3. STOCHASTIC STABILITY
(3.2.6), (3.2.7), (3.2.8) together, we obtain
Z
|Ljτ ϕ − Ljω ϕ| dm ≤
γ
2
+ #D(ω)2γ2 < γ
which concludes the proof.
Proposition 3.2.2. Let (ǫn )n be a sequence such that ǫn > 0 and ǫn → 0, as n → ∞.
The density hǫn converges to h∞ in the L1 -norm.
Proof. For simplicity we prove that kh̃ǫn − h̃∞ k1 converges to zero as ǫn goes to zero,
where h̃ǫn = dµ̃ǫn /dm, which implies the desired result. For given γ > 0 we are
looking for α > 0 such that if ǫn < α then kh̃ǫn − h̃∞ k1 < γ.
By (2.1.1) there is an integer N ≥ 1 for which
+∞
X
m({Rσ−j (ω) > j}) <
j=N +1
γ
,
4K1
∀ω ∈ Ωǫn .
We split the following sums
µ̃∞ =
N
X
j=0
ξ∞,j + η∞,N
and µ̃ǫn =
N
X
j=0
ξǫn ,j + η∞,N ,
where, for every j = 0, 1, 2, ..., N, we have
ξ∞,j = (f j )∗ (ν∞ |{R > j}),
Z
ξǫn ,j =
(fσj−j (ω) )∗ (νσ−j (ω) |{Rσ−j (ω) > j}) dθǫZn (ω)
and the remaining sums are
η∞,N =
ηǫn ,N =
∞
X
(f j )∗ (ν∞ |{R > j})
j=N +1
Z
+∞
X
(fσj−j (ω) )∗ (νσ−j (ω) |{Rσ−j (ω) > j}) dθǫZn (ω).
j=N +1
(3.2.9)
CHAPTER 3. STOCHASTIC STABILITY
59
Recall that the realization ω ∗ , which reproduces the original deterministic dynamical system, belongs to Ωǫn . By (3.2.9) we have
η∞,N (M) =
=
+∞
X
(f j )∗ (ν∞ |{R > j})(M)
j=N +1
+∞
X
j=N +1
=
ν∞ ({R > j})
+∞ Z
X
ρ∞ χ{R>j} dm
j=N +1
≤ K1
≤
γ
.
4
+∞
X
m({R > j})
j=N +1
Similarly,
ηǫn ,N (M) =
=
=
Z
Z
Z
+∞
X
j=N +1
+∞
X
+∞ Z
X
j=N +1
≤ K1
γ
.
4
νσ−j (ω) ({Rσ−j (ω) > j}) dθǫZn
j=N +1
≤ K1
≤
(fσj−j (ω) )∗ (νσ−j (ω) |{Rσ−j (ω) > j})(M) dθǫZn
Z
Z
+∞
X
ρσ−j (ω) χ{Rσ−j (ω) >j} dmdθǫZn
m({Rσ−j (ω) > j}) dθǫZn
j=N +1
γ
dθZ
4K1 ǫn
Altogether this gives us
dη∞,N
dηǫn ,N γ
dm − dm ≤ η∞,N (M) + ηǫn ,N (M) ≤ 2 .
1
(3.2.10)
60
CHAPTER 3. STOCHASTIC STABILITY
By (3.2.1) there is some α0 > 0 such that ǫn < α0 implies
dξǫn,0 dξ∞,0 γ
dm − dm = kρǫn − ρ∞ k1 < 4 .
1
(3.2.11)
On the other hand, for every j = 1, 2, ..., N
Z
dξǫn,j
dξ∞,j j
Z
j
−j (ω) χ{R
−
=
L
(ρ
)
dθ
−
L
(ρ
χ
)
−j
>j}
∞
{R>j}
σ
ǫn
σ (ω)
dm
σ −j (ω)
dm 1
1
≤ Aj + Bj ,
where
Aj
Bj
Z
j
j
Z = Lσ−j (ω) (ρσ−j (ω) χ{Rσ−j (ω) >j}) − L (ρσ−j (ω) χ{Rσ−j (ω) >j} ) dθǫn 1
Z
j
j
Z = L (ρσ−j (ω) χ{Rσ−j (ω) >j}) − L (ρ∞ χ{R>j} ) dθǫn .
1
For the first term, Lemma 3.2.1 implies that there exists some αj > 0 such that if
ǫn < αj then
ZZ j
Aj ≤
Lσ−j (ω) (ρσ−j (ω) χ{Rσ−j (ω) >j}) − Lj (ρσ−j (ω) χ{Rσ−j (ω) >j} ) dmdθǫZn ≤
We split term Bj as Bj ≤ Bj1 + Bj2 , where
Bj1
Bj2
Z
j
Z = L (ρσ−j (ω) χ{Rσ−j (ω) >j} − ρσ−j (ω) χ{R>j} ) dθǫn 1
Z
j
Z = L (ρσ−j (ω) χ{R>j} − ρ∞ χ{R>j} ) dθǫn .
1
We consider αj small enough so that, by condition (U1), if ǫn < αj then
m {Rσ−j (ω) = l}△{R = l} ≤
for l = 1, 2, . . . , j and ω ∈ Ωǫn .
γ
,
12jNK1
γ
.
12N
61
CHAPTER 3. STOCHASTIC STABILITY
Since
m({Rσ−j (ω) > j}△{R > j}) = m
j
≤
≤
X
l=1
j
X
l=1
∆\
j
[
l=1
!
{Rσ−j (ω) = l} △ ∆ \
m {Rσ−j (ω) = l}△{R = l}
γ
γ
=
12jNK1
12NK1
j
[
{R = l}
l=1
we have then
Bj1
ZZ j
≤
L ρσ−j (ω) (χ{Rσ−j (ω) >j} − χ{R>j} ) dmdθǫZn
ZZ
≤
ρσ−j (ω) χ{Rσ−j (ω) >j} − χ{R>j} dmdθǫZn
Z
≤ K1 m {Rσ−j (ω) > j}△{R > j} dθǫZn
≤
γ
.
12N
If α0 is small enough, for ǫn < α0 we also have
Bj2
=
=
≤
=
≤
=
Z
j
Z
L (ρσ−j (ω) (x)χ{R>j} (x) − ρ∞ (x)χ{R>j} (x)) dθ (ω)
ǫ
n
1
Z Z
Lj [(ρσ−j (ω) (x) − ρ∞ (x))χ{R>j} (x)] dθǫZ (ω) dm(x)
n
Z Z
Lj (ρσ−j (ω) (x) − ρ∞ (x)) dθZ (ω) dm(x)
ǫn
Z Z X
ρσ−j (ω) (y) − ρ∞ (y) Z
dm(x)
dθ
(ω)
ǫn
j (y)|
|det
Df
y=f −j (x)
Z
Z X
1
ρσ−j (ω) (y) − ρ∞ (y) dθZ (ω) dm(x)
ǫn
|det Df j (y)| y=f −j (x)
Z X
1
|ρǫn (y) − ρ∞ (y)| dm(x)
j (y)|
|det
Df
−j
y=f
(x)
= kρǫn (x) − ρ∞ (x)k1
γ
≤
.
12N
!!
62
CHAPTER 3. STOCHASTIC STABILITY
Altogether, considering ǫn < minj∈{0,1,...,N } {αj } we get Aj + Bj <
γ
,
4N
for 1 ≤ j ≤ N.
The sum over all these j’s added to the superior limit at (3.2.11) is less than
γ
.
2
Together with (3.2.10) completes the proof.
Proposition 3.2.3. The measure µ∞ is f -invariant.
Proof. Take any continuous map ϕ : M → R. Since
Z
Z
ϕ dµǫn − ϕ dµ∞ ≤ kϕk∞ khǫn − h∞ k1
and khǫn − h∞ k1 → 0, as ǫn → 0, then µǫn converges to µ∞ in weak∗ sense and
Z
ϕ dµǫn →
Z
ϕ dµ∞ .
However, since µǫn is a stationary measure we have whenever ǫn → 0 that
Z
ϕ(x) dµǫn (x) =
It suffices then to prove that
ZZ
ZZ
(ϕ ◦ fs )(x) dµǫn (x)dθǫn (s).
(ϕ ◦ fs ) dµǫn dθǫn →
Z
(ϕ ◦ f ) dµ∞. So,
Z Z
Z
(ϕ
◦
f
)
dµ
dθ
−
(ϕ
◦
f
)
dµ
s
ǫ
ǫ
∞
n
n
Z
Z Z
Z
Z
≤ (ϕ ◦ fs ) dµǫn dθǫn − (ϕ ◦ f ) dµǫn + (ϕ ◦ f ) dµǫn − (ϕ ◦ f ) dµ∞ .
For ǫn sufficiently small, (ϕ ◦ fs − ϕ ◦ f ) is uniformly close to 0, for every s ∈ supp θǫn .
The second term is smaller than kϕk∞ khǫn − h∞ k1 , which is close to zero if ǫn is
small enough.
Remark 3.2.4. The unicity of an SRB measure µf for f ensures that in the previous
arguments we can consider all the sequence ǫn instead just a subsequence of it. To
see this, for every subsequence of ǫn we can repeat the previous process and obtain a
new subsequence ǫ′n for which the corresponding sequence of densities (hǫ′n )n has limit
h′∞ , and µ′f = h′∞ dm is also a f -invariant SRB measure (thus equal to µf ). On the
other hand, one knows that if all subsequences of a sequence admits a subsequence
converging to the same limit then the all sequence converge to that limit.
CHAPTER 3. STOCHASTIC STABILITY
This finishes the proof of Theorem 1.3.1.
63
Chapter 4
Examples
4.1
Local diffeomorphisms
One example of transformations that fits our hypothesis was introduced in [ABV00]
and consists on robust (C 1 open) classes of local diffeomorphisms (with no critical
sets) that are non-uniformly expanding. The existence and unicity of SRB probability
measures for this maps was proved in [ABV00] and [A03]. Random perturbations for
this maps were considered in [AAr03], where it was proved a weak form of stochastic
stability - the convergence in the weak∗ topology of the density of the unique stationary probability measure to the density of the unique SRB probability measure.
We improve it to the strong version of stochastic stability. As a corollary we also
obtain the strong statistical stability, proved in [A04]. We follow closely the constructions and results in [ABV00] and [AAr03] and introduce some extras in view to
have required transitivity.
This class of maps can be obtained, e.g. through deformation of a uniformly
expanding map by isotopy inside some small region. In general, these maps are not
uniformly expanding: deformation can be made in such way that the new map has
periodic saddles.
Let M be the d-dimensional torus Td , for some d ≥ 2, and m the normalized
Riemannian volume form. Let f0 : M → M be a uniformly expanding map and
V ⊂ M be a small neighborhood of a fixed point p of f0 so that the restriction of f0
to V is injective. Consider a C 1 -neighborhood U of f0 sufficiently small so that any
map f ∈ U satisfies:
64
65
CHAPTER 4. EXAMPLES
i) f is expanding outside V : there exists λ0 < 1 such that
kDf (x)−1 k < λ0
for every x ∈ M \ V ;
ii) f is volume expanding everywhere: there exists λ1 > 1 such that
| det Df (x)| > λ1
for every x ∈ M;
iii) f is not too contracting on V : there is some small γ > 0 such that
kDf (x)−1 k < 1 + γ
for every x ∈ V ,
and constants λ0 , λ1 and γ are the same for all f ∈ U. Moreover, for f ∈ U we intro-
duce the random perturbations {Φ, (θǫ )ǫ>0 }. In particular, we consider a continuous
map
Φ : T −→ C 2 (M, M)
t
7−→ ft
where T is a metric space, Φ(T ) ⊂ U. We set f ≡ ft∗ for some t∗ ∈ T and consider a
family (θǫ )ǫ>0 of probability measures on T such that their supports are non-empty
and satisfies supp(θǫ ) → {t∗ }, when ǫ → 0. We are going to prove that we can choose
appropriately the constants λ0 , λ1 and γ so that every map f ∈ U is non-uniformly
expanding on all random orbits:
Proposition 4.1.1. Consider f0 , U, f ∈ U and {Φ, (θǫ )ǫ>0 } as before. There exists
a0 > 0 such that for every ω ∈ supp(θǫZ ) and Lebesgue almost every x ∈ M
n−1
1X
lim sup
log kDfσj (ω) (fωj (x))−1 k ≤ −a0 .
n→+∞ n
j=0
(4.1.1)
In particular, (4.1.1) hold for ω ∗ = (. . . , t∗ , t∗ , t∗ , . . .) and Lebesgue almost every
x ∈ M, so that f is a non-uniformly expanding map. This allows us to conclude
Theorem 4.1.2. There are open sets U ⊂ C 2 (M, M) such that every f ∈ U is
non-uniformly expanding and non-uniformly expanding on all random orbits.
66
CHAPTER 4. EXAMPLES
Thus we are going to prove simultaneously that there are open sets U ⊂ C 2 (M, M)
of non-uniformly expanding maps, constructed in the way we detailed, and that ele-
ments in U are non-uniformly expanding on (all) random orbits. To prove Proposition
4.1.1 we start with the following result:
Lemma 4.1.3. Let B1 , . . . , Bp , Bp+1 = V be any partition of M into domains such
that f is injective on Bj , for 1 ≤ j ≤ p + 1. Then there is ζ > 0 such that for every
ω ∈ supp(θǫZ ) and m-almost every x ∈ M and large enough n ≥ 1
(
# 0 ≤ j < n : fωj (x) ∈
p
[
Bk
k=1
)
≥ ζn.
(4.1.2)
Moreover, there is 0 < τ < 1 such that for each ω ∈ supp(θǫZ ) the set In (ω) of points
x ∈ M for which (4.1.2) does not hold satisfies m(In (ω)) ≤ τ n , for n ≥ 1.
Proof. Let us fix n ≥ 1 and ω ∈ supp(θǫZ ). Given i = (i0 , . . . , in−1 ) ∈ {1, . . . , p + 1}n
we write
[iω ] =
n−1
\
(fwk )−1 (Bik )
k=0
and define g(i) = #{0 ≤ j < n : ij ≤ p}. We start by observing that for ζ > 0 the
number of sequences i such that g(i) < ζn is bounded by
X n
X n
k
p ≤
pζn .
k
k
k<ζn
k≤ζn
Using Stirling’s formula (cf. [BV00, Section 6.3]) the expression on the right hand
side is bounded by eγn pζn , where γ > 0 depends only on ζ and goes to zero when ζ
goes to zero.
λ
On the other hand, assumptions ii), iii) on the maps ft ensure that m([i]ω ) ≤
−(1−ζ)n
for all ω ∈ supp(θǫZ ) (recall that we are assuming m(M) = 1). Hence, given
ω ∈ supp(θǫZ ) the measure of the union In (ω) of all the sets [i]ω with g(i) < ζn is
bounded by
λ−(1−ζ)n eγn pζn .
−(1−ζ)
Since λ1 > 1 we may fix ζ so small that τ := eγ pζ λ1
< 1. Then m(In (ω)) ≤ τ n
for n ≥ 1. Note that τ do not depend on ω ∈ supp(θǫZ ). This proves the second part
67
CHAPTER 4. EXAMPLES
of the lemma. For the first parte, since for each ω ∈ supp(θǫZ ) we have
X
n≥1
m(In (ω)) < ∞,
then Borel-Cantelli’s Lemma implies that
m (∩n≥1 ∪k≥n Ik (ω)) = 0,
and this means that for all ω ∈ supp(θǫZ ), m almost every x ∈ M belongs in only
finitely many sets In (ω), and thus satisfies (4.1.2).
Let ζ > 0 be the constant provided by Lemma 4.1.3. We fix γ > 0 sufficiently
small so that λζ0 (1 + γ) ≤ e−a0 holds for some a0 > 0, and take ω ∈ supp(θǫZ ) and
x ∈ M satisfying (4.1.2). The assumptions on ft for t ∈ T imply
n−1
Y
j=0
kDfσj (ω) (fωj (x))−1 k ≤ σ0ζn (1 + γ)(1−ζ)n ≤ e−a0 n
for large enough n. This implies that f is non-uniformly expanding on all random
orbits:
n−1
lim sup
n→+∞
1X
log kDfσj (ω) (fωj (x))−1 k ≤ −a0
n j=0
for every ω ∈ supp(θǫZ ) and m almost every x ∈ M. Moreover, defining Γnω as in
(1.2.4) ignoring the recurrence time function, from the second part of Lemma 4.1.3
we have
m x ∈ M : Γnω > n ≤ τ n ,
(4.1.3)
for n ≥ 1 and every ω ∈ supp(θǫZ ).
We introduce now some modifications to ensure the transivity in view to apply
Theorem 1.3.1. Let us consider a map f¯: M → M (in the boundary of the set of
uniformly expanding maps) which satisfies (i), (ii) and (iii) as the cartesian product
of one-dimensional maps ϕ1 × · · · × ϕd , with ϕ1 , . . . , ϕd−1 uniformly expanding in S 1 ,
and ϕd the intermittent map in S 1 , which in a neighborhood of 0 can be written as
ϕd (x) = x + x1+α ,
for some 0 < α < 1,
CHAPTER 4. EXAMPLES
68
and ϕ′d (x) > 1 for every x ∈ S 1 \ {0}. One clearly has that any f in a sufficiently
small C 1 -neighborhood Ū of f¯ satisfies (i), (ii) and (iii) for a convenient choice of
constants λ0 , λ1 , γ, and a neighborhood V of the fixed point p = 0 ∈ Td . Next lemma
ensures that f¯ is topologically mixing, and thus topologically transitive, so that we
are able to apply Theorem 1.3.1 to f¯. Nevertheless, we show moreover that if Ū is
sufficiently small, then all the maps in Ū satisfy the hypothesis of Theorem 1.3.1. We
obtain this way an open class of maps that fits our hypothesis, and for which we can
apply the results in previous chapters.
Lemma 4.1.4. Given α > 0 there is Nα ≥ 1 such that f¯Nα (B(x, α)) = Td for any
x ∈ M.
Proof. This is an immediate consequence of the fact that a similar conclusion holds
for the maps ϕ1 , . . . , ϕd in S 1 . This is standard for the uniformly expanding maps
ϕ1 , . . . ϕd−1 , and also for the intermittent map ϕd as it is topologically conjugate to a
uniformly expanding map of the same degree.
Let us now obtain a similar conclusion for any map f in Ū . This cannot be done
by a simple continuity argument, since for smaller radii α in principle we need to
diminish the size of the C 1 -neighborhood. However, a continuity argument works if
one just needs to consider balls of some fixed radius. By Proposition 4.1.1 any map
f ∈ Ū is non-uniformly expanding and, if we consider a random perturbation of f
as before, then f is also non-uniform expanding on (all) random orbits (naturally,
for sufficiently small noise level), with the Lebesgue measure of the tail set estimated
in (4.1.3). By Propositions 4.1.1 and 2.2.5, Lebesgue almost every point x ∈ M has
infinitely many λ-hyperbolic times and, moreover, we may take λ = e−a0 /2 . Lemmas
2.2.9 and 2.2.10 implies that there exists δ1 > 0 (uniform over all the maps in Ū) such
that Lebesgue almost every point in Td has arbitrarily small neighborhoods which
are sent onto balls of radius δ1 > 0. Taking α = δ1 /2 in Lemma 4.1.4, there is some
positive integer N for which every ball of radius δ1 /2 is sent onto M by f¯N . Then,
just by continuity, one has that any ball of radius δ1 is sent onto M by f N for any
f ∈ Ū, provided this C 1 -neighborhood is sufficiently small. Then, in particular, each
f ∈ Ū is topologically transitive. The next theorem is now a direct application of
Theorem 1.3.1.
69
CHAPTER 4. EXAMPLES
Theorem 4.1.5. Every map in the open set Ū ⊂ C 2 (M, M) is strongly stochastically
stable.
Let µf be the unique ergodic absolutely continuous invariant probability measure
for f ∈ Ū. Applying Corollary 1.4.1 one get the strong statistical stability, already
established in [A04].
Corollary 4.1.6. The map Ū ∋ f 7→ dµf /dm is continuous with respect to the
L1 -norm in the space of densities.
4.2
Viana maps
We consider now an important open class of non-uniformly expanding maps with
critical sets in higher dimensions introduced in [V97]. This example features the hypothesis of Theorem 1.3.1 resulting on the proof of their strong stochastic stability.
The existence of a unique ergodic absolutely continuous invariant probability measure
and the strong statistical stability were proved in [AV02]. A weaker form of stochastic stability (weak∗ convergence of the stationary measure to µf ) was established in
[AAr03]. In order to check the hypothesis of Theorem 1.3.1 we use essentially the
results in [AAr03] about the non-uniform expansion, slow recurrence to the critical
set and uniform decay of the Lebesgue measure of the tail set, both for deterministic
and random case. Without loss of generality we discuss the two-dimensional case and
we refer [V97], [AV02] and [AAr03] for details.
Let p0 ∈ (1, 2) be such that the critical point x = 0 is pre-periodic for the quadratic
map Q(x) = p0 − x2 . Let S 1 = R/Z and b : S 1 → R be a Morse function, for instance,
b(s) = sin(2πs). For fixed small α > 0, consider the map
fˆ : S 1 × R −→
(s, x)
7−→
S1 × R
ĝ(s), q̂(s, x)
where ĝ is the uniformly expanding map of the circle defined by ĝ(s) = ds (mod Z)
for some d ≥ 16, and q̂(s, x) = a(s) − x2 with a(s) = p0 + αb(s). As it is shown in
[AAr03], it is no restriction to assume that C = {(s, x) ∈ S 1 × I : x = 0} is the critical
set of fˆ and we do so. If α > 0 is small enough there is an interval I ⊂ (−2, 2) for
70
CHAPTER 4. EXAMPLES
which fˆ(S 1 × I) is contained in the interior of S 1 × I. Any map f sufficiently close to
fˆ in the C 3 topology has S 1 × I as a forward invariant region (in fact, here it suffices
to be C 1 close). We consider a small C 3 neighborhood V of fˆ as before and will refer
to maps in V as Viana maps. Thus, any Viana map f ∈ V has S 1 × I as a forward
invariant region, and so an attractor inside it, which is precisely
Λ=
\
n≥0
f n (S 1 × I).
We introduce the random perturbations {Φ, (θǫ )ǫ } for this maps. We set T ⊂ V
to be a C 3 neighborhood of fˆ consisting in maps f restricted to the forward invariant
region S 1 × I for which Df (x) = D fˆ(x) if x ∈
/ C, the map Φ to be the identity map at
T and (θǫ )ǫ a family of Borel measures on T such that their supports are non-empty
and satisfies supp(θǫ ) → {t∗ }, when ǫ → 0. Let hfˆ to be the density of the unique
absolutely continuous invariant probability measure µ ˆ for fˆ. We will show that the
f
Viana maps fˆ ∈ V satisfies the hypothesis of Theorem 1.3.1 so that we can conclude
Theorem 4.2.1. Let fˆ be a Viana map.
1. If ǫ is small enough then fˆ admits a unique absolutely continuous ergodic stationary
probability measure µǫ .
2. Let hǫ = dµǫ /dm. The map fˆ is strongly stochastically stable:
lim khǫ − hfˆk1 = 0.
ǫ→0
Applying Corollary 1.4.1 we obtain the strong statistical stability for Viana maps.
This result was already established in [AV02].
Corollary 4.2.2. Let V be a family of Viana maps. Then V ∋ fˆ →
7 dµfˆ/dm is
continuous with respect to the L1 -norm in the space of densities.
4.2.1
Deterministic estimates
The results in [V97] show that if V is sufficiently small (in the C 3 topology) then
f ∈ V has two positive Lyapunov exponents almost everywhere: there is a constant
η > 0 for which
lim inf
n→+∞
1
log kDf n (s, x)vk ≥ η
n
71
CHAPTER 4. EXAMPLES
for Lebesgue almost every (s, x) ∈ S 1 × I and every non-zero v ∈ T(s,x) (S 1 × I). This
does not necessarily imply that f is non-uniformly expanding. However, as it was
shown in [AAr03], a slightly deeper use of Viana’s arguments enables us to prove the
non-uniform expansion and the slow recurrence to the critical set of any C 2 map f
such that
kf − fˆkC 2 < α.
In particular they proved that there exist C, ζ > 0 such that for f as before and n ≥ 1
there is a set Γn ⊂ S 1 × I for which
m(Γn ) ≤ Ce−ζ
√
n
,
and such that for each (s, x) ∈
/ Γn we have
1. there is a0 > 0 such that
k−1
1X
log kDf (f j (s, x))−1 k ≤ −a0
k j=0
for all k ≥ n, and
2. for given small b0 > 0 there is δ > 0 such that
k−1
1X
− log distδ (f j (s, x), C) ≤ b0
k j=0
for all k ≥ n.
Moreover, the constants ζ, a0 and δ only depend on the quadratic map Q and α > 0.
In [AV02] it was also proved a topological mixing property:
Theorem 4.2.3. For every f ∈ V and every open set A ⊂ S 1 × I there is some
nA ∈ N for which f nA (A) = Λ.
Proof. [AV02], Theorem C.
This implies the required transitivity for fˆ.
4.2.2
Estimates for random perturbations
Let f be close to fˆ in the C 3 topology and fix a random perturbation {Φ, (θǫ )ǫ>0 }
as before. We want to show that if ǫ > 0 is small enough then f is non-uniformly
72
CHAPTER 4. EXAMPLES
expanding on random orbits and Γnω decays sufficiently fast and uniformly on ω:
∃ C > 0, p > 1 : m(Γnω ) < Cn−p , ∀ω ∈ Ωǫ .
The estimates in [AAr03] for log distδ (f j (s, x), C) and log kDf (f j (s, x))−1 k over the
orbit of a given point (s, x) ∈ S 1 × I can easily be done replacing the iterates f j (s, x)
by random iterates fωj (s, x). Briefly, those estimates rely on a delicate decomposition
of the orbit of the point (s, x) from time 0 until time n into finite pieces according
√ √
to its returns to the neighborhood S 1 × (− α, α) of the critical set. The main
tools for this estimates were [V97, Lemma 2.4] and [V97, Lemma 2.5] whose proofs
may easily be mimicked for random orbits. Indeed, the important fact in the proof
of the referred lemmas is that orbits of points in the central direction stay close to
orbits of the quadratic map Q for long periods, as long as α > 0 is taken sufficiently
small. Hence, such results can easily be obtained for random orbits as long as we
take ǫ > 0 with ǫ ≪ α and ω ∈ supp(θǫZ ). It was also proved in [AAr03] that exists
C > 0, ζ > 0 such that m(Γnω ) < Ce−ζ
√
n
, for almost every ω ∈ supp(θǫZ ), which
clearly is enough for our purposes. Moreover, the constants for the estimates on the
the tail set, non-uniform expansion and slow recurrence remains depending only on
the quadratic map Q and α. In particular, they are uniform over ω.
Appendix A
Measure and Ergodic theory
A.1
Measure spaces
This appendix contains some elementary concepts and results from Measure Theory
and Ergodic Theory. They are relevant for a good understanding of this thesis, for the
clearness of some definitions and to fix notation. All the results in this brief section
are someway standards and they are mentioned and proved in several books. We just
mention, for instance, [W82] and [Ma87].
Let X be set and B be a collection of subsets of X. We say that B is a σ-algebra
on X if the following conditions hold:
1. X ∈ B;
2. if B ∈ B then X\B ∈ B;
3. if B1 , B2 , B3 · · · ∈ B then
S+∞
i=1
Bi ∈ B.
We will refer to a pair (X, B) where B is a σ-algebra on X as a measurable space.
A measure on (X, B) is a function µ : B → R+
0 ∪ {+∞} satisfying:
1. µ(∅) = 0;
Pn
2. if B1 , B2 , · · · ∈ B are pairwise disjoint, then µ ∪+∞
i=1 Bi =
i=1 µ(Bi ).
We say (X, B, µ) is a measure space if (X, B) is a measurable space and µ is a measure
on (X, B). If µ(X) < ∞ then we say that µ is a finite measure and if µ(X) = 1 then
73
APPENDIX A. MEASURE AND ERGODIC THEORY
74
we say that µ is a probability measure and that (X, B, µ) is a probability space. The
collection P(X) of all subsets of X is a σ-algebra and any intersection of σ-algebras is
again a σ-algebra. So given a collection Q of subsets of X we can define the σ-algebra
B(Q) generated by Q as the smallest (in terms of inclusion) σ-algebra that contains
all elements of Q.
We give well known examples, also used on the thesis.
Example A.1.1. Let (X, P(X)) be a measurable space, and for each A ⊂ X we write
#(A) for the number of elements of A (+∞ if A infinite). The function # : P(X) →
N0 ∪ {+∞} defines a measure on P(X) that will be called the counting measure on
X.
Example A.1.2. Let X be a set and x ∈ X. We define δx : P(X) → {0, 1} by
putting, for every A ⊂ X, δx (A) = 1 if x ∈ A, and δx (A) = 0 otherwise. δx defines a
probability measure on X that will be called the Dirac measure supported on x.
Assume, for instance, that X is a metric space. We define B(X), the Borel σ-
algebra on X, as the σ-algebra generated by the open sets of X, i.e. the smallest (in
terms of inclusion) σ-algebra that contains the open sets of X. A measure defined on
the Borel σ-algebra of a metric space is said to be a Borel measure.
Example A.1.3. There is a unique borel measure m defined on B(Rd ) such that for
intervals I1 , · · · , Id ⊂ R one has
m
d
Y
j=1
Ij
!
= |I1 | × · · · × |Id |,
where |Ij | means the length of the interval |Ij |. This is called the Lebesgue measure
on Rd . Using a volume form we introduce the Lebesgue measure on Riemannian
manifolds in a similar way.
Let (X, B, µ) be measure space. We say that A ⊂ X has null measure if there is
B ∈ B such that A ⊂ B and µ(B) = 0. We say that some property on the elements
of X holds almost everywhere, or even for µ almost every point if the set of points for
which that property does not hold has null measure.
The support of a Borel measure µ, which is denoted by supp(µ), is defined as
supp(µ) = {x ∈ X : µ(U) > 0 for each neighborhood U of x} .
APPENDIX A. MEASURE AND ERGODIC THEORY
75
A Borel probability measure µ on a compact metric space X is said to be regular if
for all A ∈ B(X) and ǫ > 0 there are a closed set Fǫ ⊂ A and an open set Uǫ ⊃ A
such that µ(Uǫ \Fǫ ) < ǫ.
Theorem A.1.4. Every Borel probability measure of a compact metric space is regular.
Proof. [W82], Theorem 6.1.
Let S be a collection of sets of X. We call S a semi algebra if i) ∅ ∈ S; ii) if
A, B ∈ S, then A ∩ B ∈ S and iii) if A ∈ S, then X \ A = ∪ni=1 Ei , where Ei are
pairwise disjoint subsets of X and each Ei ∈ S. A collection of sets A is an algebra
if satisfies i) and ii) as before and iii) if A ∈ A then X \ A ∈ A or, equivalently, iii’)
is A1 , . . . , An ∈ A then ∪ni=1 Ai ∈ A.
The colection P(X) of all subsets of X is an algebra and the intersection of any
family of algebras still be an algebra. In this way, we can define the algebra A(Q)
generated by the collection Q of subsets of X, that is, the smallest (in terms of
inclusion) algebra that contains all the elements of Q. A sub-algebra A′ of A is an
algebra such that A ∈ A′ implies A ∈ A.
Consider a semi algebra S and a function λ : S → R+
0 . We call λ finitely additive
Pn
n
if λ(∅) = 0 and λ(∪i=1 Ei ) = i=1 λ(Ei ) whenever Ei are pairwise disjoint members
of S and ∪ni=1 Ei ∈ S. We say that λ is countably additive if the last condition
Pn
+∞
is replaced by λ(∪+∞
i=1 Ei ) =
i=1 λ(Ei ) whenever {Ei }i=1 is a sequence of pairwise
disjoint members of S and ∪+∞
i=1 Ei ∈ S.
Theorem A.1.5. If S is a semi algebra of subsets of X and λ0 : S → R+
0 is finitely
additive then there is a unique finitely additive function λ : A(S) → R+
0 which is an
extension of λ0 (that is, λ = λ0 on S). If λ0 is countably additive then so it is λ.
Proof. [W82], Theorem 0.2.
The next theorem discuss the extension from an algebra A to the σ-algebra B(A).
Theorem A.1.6. Let A be an algebra of subsets of X and let λ : A → R+
0 be countably
additive with λ(X) = 1. Then there is a unique probability measure λ∗ on B(A) which
extends λ.
Proof. [W82], Theorem 0.3.
APPENDIX A. MEASURE AND ERGODIC THEORY
76
Combining Theorems A.1.5 and A.1.6 we see that a countably additive function λ0
on a semi algebra S can be uniquely extended to a probability measure on (X, B(S))
P
if ni=1 λ0 (Ei ) = 1 when X = ∪ni=1 Ei is a disjoint union of members of S. To start
from a finitely additive function we have the following theorem, useful to prove that
the finitely additive extension λ on A is countably additive.
Theorem A.1.7. Let A be an algebra of subsets of X and let λ : A → R+
0 be finitely
additive with λ(X) = 1. Then λ will be countably additive if for every decreasing
sequence E1 ⊃ E2 ⊃ . . . of members of A with ∩+∞
n=1 En = ∅ we have lim λ(En ) = 0.
Proof. [W82], Theorem 0.4.
Not rare case we do not have explicit knowledge of the elements on a σ-algebra B,
but we do on a algebra A that generates B. Denote by A△B the symmetric difference
between two subsets A and B of X, that is, A△B = (B\A) ∪ (A\B).
Theorem A.1.8. Let (X, B, µ) be a probability space and A be an algebra of subsets
of X such that B(A) = B. Then, for each α > 0 and each B ∈ B there is some A ∈ A
such that µ(A△B) < α
Proof. [W82], Theorem 0.7.
A.2
SRB measures
Let (X, B, µ) be a measure space. We say that f : X → X is a measurable transfor-
mation if f −1 (B) ∈ B for each B ∈ B. The measure µ is said to be invariant by f (or
f preserves µ) if µ(f −1 (B)) = µ(B) for all B ∈ B. We may associate to a measurable
transformation f and a measure µ a new measure that we denote by f∗ µ and call the
push-forward of the measure µ by f , and is defined by f∗ µ(B) = µ(f −1 (B)) for each
B ∈ B. Note that µ is invariant by f if and only if f∗ µ = µ.
Sometimes we do not have explicit or enough knowledge on all members of B, but
we often do have in the semi algebra S such that B(S) = B. Then the following result
is important to checking the invariance on a measure.
Theorem A.2.1. Suppose (X, B, µ) is a probability space and f : X → X a transformation. Let S be a semi algebra that generates B. If, for each E ∈ S we have
f −1 (E) ∈ B and µ(f −1 (E) = µ(E) then f preserves µ.
77
APPENDIX A. MEASURE AND ERGODIC THEORY
Proof. [W82], Theorem 1.1.
Let again X be a compact metric space and denote by P(X) the space of probability measures defined on the Borel σ-algebra of X. Note that P(X) is a convex
space. We introduce the weak* topology on P(X) in the following way: a sequence
(µn )n in P(X) converges to µ ∈ P(X) if and only if
Z
ϕ dµn →
Z
ϕ dµ,
for each continuous ϕ : X → R.
Since we are taking X a compact metric space, then C(X) is separable, and so we
may find a sequence (ψn )n dense in C(X). The function
Z
Z
+∞
X
1 ψn dµ − ψn dν dP (µ, ν) =
n
2
k=1
defines a metric on P(X) which gives the weak∗ topology.
Let µ be a finite measure invariant by f : X → X. We say that µ is an ergodic
measure if the phase space cannot be decomposed into invariant regions that are
relevant in terms of the measure µ, i.e. if A ∈ A satisfies f −1 (A) = A, then µ(A)µ(X \
A) = 0.
Theorem A.2.2 (Birkhoff). Let f : X → X preserve a probability measure µ. Given
any ϕ ∈ L1 (µ) there exists ϕ∗ ∈ L1 (µ) with ϕ∗ ◦ f = ϕ∗ such that
n−1
1X
lim
ϕ(f n (x)) = ϕ∗ (x)
n→+∞ n
j=0
for µ almost every x ∈ X. Moreover, if µ is ergodic, then ϕ∗ =
where.
(A.2.1)
R
ϕ dµ almost every-
Proof. [W82], Theorem 1.14.
Definition A.2.3. We define B(µ), the basin of µ, as the set of those points x ∈ M
for which
n−1
1X
lim
ϕ(f j (x)) =
n→+∞ n
j=0
Z
ϕ dµ for any continuous ϕ : M → R.
APPENDIX A. MEASURE AND ERGODIC THEORY
78
Definition A.2.4. Let µ be a probability measure invariant by f . We say that µ is
a Sinai-Ruelle-Bowen (SRB) measure if m(B(µ)) > 0.
The definition is equivalent to say that for a positive Lebesgue set of initial points
x ∈ X, the average of Dirac measures along the orbit of x converges to µ in the
weak∗ topology. The SRB measures characterize statistically a dynamical system for
a relevant (positive Lebesgue measure) set of initial points in X. Note that the total
measure set of points for which (A.2.1) hold can depend on the function ϕ. However,
for continuous dynamical systems we can found a subset B of the phase space with
µ full measure for whose points (A.2.1) holds for every continuous ϕ.
Theorem A.2.5. Let X a compact metric space, f : X → X continuous and µ an
invariant and ergodic probability measure. Then there is Y ⊂ B(X), with µ(Y ) = 1
such that
n−1
1X
ϕ(f j (x)) =
lim
n→+∞ n
j=0
Z
ϕ dµ ∀x ∈ Y, for any continuous ϕ : M → R.
Proof. See [W82], Lemma 6.13.
Definition A.2.6. Let µ and ν be finite measures defined on a same σ-algebra B.
We say that ν is absolutely continuous with respect to µ, and write ν ≪ µ, if ν(A) = 0
whenever µ(A) = 0. The measures µ and ν are said to be equivalent if both µ ≪ ν
and ν ≪ µ.
The next result points out the interest of studying absolutely continuous ergodic
measures.
Theorem A.2.7. Suppose f is continuous and µ is an ergodic probability measure
which is absolutely continuous with respect to the Lebesgue measure. Then µ is an
SRB measure.
Proof. By Theorem A.2.5, B(µ) has full µ measure. Since µ is absolutely continuous
with respect to the Lebesgue measure, the basin of µ cannot have zero Lebesgue
measure.
Appendix B
Random perturbations
B.1
The setup
Consider a metric space X and a map f : X → X in the topological space C 2 (X, X)
of the C 2 maps from X to itself. The idea is to replace the original obits given
by iterations with f by random orbits generated by an independent and identically
distributed random choice of a map in C 2 (X, X) at each iteration, made according
some given distribution law. To be more precise, consider a continuous map
Φ : T −→ C 2 (X, X)
t
7−→ Φ(t) = ft
from a metric space T into C 2 (X, X), with f = ft∗ for some fixed t∗ ∈ T . For any
element ω = (..., ω−1 , ω0 , ω1, . . .) in the infinite product space T Z , we define
fωn (x) =
(
x
if
n=0
(fωn−1 ◦ · · · ◦ fω1 ◦ fω0 )(x)
if
n>0
.
In particular, fω = fω0 and fσl (ω) = fωl for every ω ∈ T Z and l ∈ Z, where σ : T Z → T Z
is the left shift, that is σ(. . . , ω−1 , ω0, ω1 , . . .) = (. . . , ω0 , ω1 , ω2 , . . .). The element
ω ∗ = (..., t∗ , t∗ , t∗ , ...) gives rise to the unperturbed deterministic dynamic of f . Given
x ∈ X and ω ∈ T Z we call the sequence fωn (x) n∈N a random orbit of x. Sometimes
we refer for ω as a realization.
We also consider a family (θǫ )ǫ>0 of Borel probability measures on T . This provides
79
80
APPENDIX B. RANDOM PERTURBATIONS
a family of Borel product measure spaces (T Z , B(T )Z , θǫZ )ǫ>0 , where θǫZ is invariant for
the shift map σ. We will refer to such a pair {Φ, (θǫ )ǫ>0 } as above as a random
perturbation of f .
B.2
Stationary, Markov and Physical measures
We start with the notion of physical measures which plays a role in this random
perturbation setting similar to that played by SRB measures in deterministic systems.
Definition B.2.1. We say that a measure µǫ is a physical measure if, for a positive
Lebesgue measure set of points x ∈ X,
n−1
1X
lim
ϕ fωj (x) =
n→+∞ n
j=0
Z
ϕ(x) dµǫ (x)
(B.2.1)
for all continuous ϕ : X → R and θǫZ almost every ω ∈ T Z . The set of points x ∈ X
for which (B.2.1) holds for all continuous ϕ and θǫZ almost every ω ∈ T Z is denoted
by B(µǫ ) and called the basin of µǫ .
The SRB measures are frequently called physical measures in several places. Here
we use this last term only in the context of random perturbations of a map, in order
to distinguish them from the deterministic SRB measures. The usual invariance
of a measure with respect to a transformation can be interpreted, in the random
perturbations scheme, as an invariance in average:
Definition B.2.2. A measure µǫ on the Borel sets of X is called a stationary measure
for {Φ, (θǫ )ǫ>0 } if
ZZ
(ϕ ◦ ft )(x) dµǫ (x)dθǫ (t) =
Z
ϕ(x) dµǫ (x),
for all ϕ : X → R continuous.
If there is no confusion we will refer such a measure µǫ as a stationary measure
for f . We introduce the two-sided skew-product map
S : T Z × X −→
(ω, z)
7−→
TZ × X
σ(ω), fω0 (z) .
81
APPENDIX B. RANDOM PERTURBATIONS
In what concerns the σ-algebra of the measurable space (T Z × X, B(T )Z × B(X)) one
has B(T )Z × B(X) = B(T Z × X). An invariant measure µ∗ for the skew-product
S (in the usual deterministic sense), defined on B(T Z × X), is characterized by an
essentiality unique disintegration dµ∗ (ω, x) = dµω (x)dθǫZ (ω) given by a family {µω }ω
of sample measures on X satisfying: i) for all B ∈ B(X), ω 7→ µω (B) is θǫZ -measurable,
ii) for θǫZ almost every ω, B 7→ µω (B) is a probability measure on (X, B(X)) and iii)
fω ∗ µω = µσ(ω) for θǫZ almost every ω. The relation between µ∗ and the family of
sample measures can be expressed as
∗
µ (A) =
Z
µω (Aω ) dθǫZ (ω),
where A is a Borel subset of T Z × X and Aω = {x ∈ X : (ω, x) ∈ A}. Given
ω = (. . . , ω−1 , ω0, ω1 , . . .) we define the future of ω as ω0+ = (ω0 , ω1 , . . .) and the past
of ω as ω − = (. . . , ω−2 , ω−1 ). We moreover introduce the projection map
π : TZ × X → TN × X
π(ω, x) 7→ (ω0+ , x).
We say that a Borel measure µ∗ on T Z ×X is a Markov measure if for the corresponding
family of sample measures {µω }ω , θǫZ almost every sample measure µw depends only
on the past ω − of ω. The next Proposition relates stationary measures and Markov
measures. The proof can be found at the bibliography (e.g. [Arn98] and [LQ95]),
albeit sometimes with a somewhat different approach, generality or just parts of it.
Proposition B.2.3. The set of stationary probabilities µǫ for {Φ, (θǫ )ǫ>0 } corresponds
in a one-to-one way to the set of S-invariant Markov probabilities µ∗ , with the correspondence being given by
∗
µ 7→ µǫ :=
Z
µω dθǫZ (ω)
and
µǫ 7→ µ∗ := lim S∗n (θǫZ × µǫ ).
n→+∞
Moreover, for given stationary probability measure µǫ , the corresponding µ∗ can be
recognized as the unique S-invariant probability measure such that π∗ µ∗ = θǫN × µǫ .
Proof. Consider {µω }ω to be a family of sample measures of a Markov measure µ∗ .
R
We are going to prove that µǫ = µω dθǫZ (ω) is a stationary probability measure. For
82
APPENDIX B. RANDOM PERTURBATIONS
ω = (. . . , ω−1 , ω0 , ω1 , . . .) ∈ T Z we set ω + = (ω1 , ω2 , . . .) ∈ T N . Since
µǫ =
ZZ
+
−
µω dθǫZ0 (ω0+ )dθǫZ (ω − )
=
Z
−
µω dθǫZ (ω − ),
for every continuous map ϕ : X → R we have
Z
ϕ(x) dµǫ (x) =
=
ZZ
ZZ
ϕ(x) dµω (x)dθǫZ (ω)
ϕ(x) dµσ(ω) (x)dθǫZ (ω), by the θǫ - invariance of σ
ZZ
(ϕ ◦ fω )(x) dµω (x)dθǫZ (ω)
ZZZZ
−
+
=
(ϕ ◦ fω0 )(x) dµω (x)dθǫZ (ω − )dθǫ (ω0 )dθǫZ (ω + )
ZZZ
+
=
(ϕ ◦ fω0 )(x) dµǫ (x)dθǫ (ω0 )dθǫZ (ω + )
ZZ
=
(ϕ ◦ fω0 )(x) dµǫ (x)dθǫ (ω0 )
=
which gives the desired result. Consider now any stationary probability measure µǫ .
We want to prove that (S n )∗ (θǫZ × µǫ ) converges weakly to a Markov measure µ∗ , and
this is the unique S-invariant probability measure such that π∗ µ∗ = θǫN × µǫ . Let Bn
denote the σ-algebra

−(n+1)
Y

−∞
T × C : C ∈ B(
+∞
Y
−n
T × X)



on T Z × X and µ∗n = (S n )∗ (θǫZ × µǫ )|Bn a probability measure, for every n ≥ 0. We
will show that
µ∗n+1 |Bn = µ∗n |Bn
∀n ≥ 0.
(B.2.2)
Since Bn is generated by the semi algebra



j
+∞
 −(n+1)

Y
Y
Y


Sn =
T×
Ai ×
T × B : Ai ∈ B(T ), −n ≤ i ≤ j, j ≥ −n, B ∈ B(X) ,
 −∞

i=−n
j+1
by Theorems A.1.5, A.1.6 and A.1.7, we only have to show that for all E ∈ Sn we
83
APPENDIX B. RANDOM PERTURBATIONS
have µn+1∗ (E) = µ∗n (E). Let

E=
−(n+1)
Y
−∞
j
T×
Y
i=−n
Ai ×
+∞
Y
j+1

T × B
be an element of Sn . Since µǫ is stationary, we have
µ∗n+1 (E) = θǫZ × µǫ (S −(n+1) (E))
Z
=
Q
Q
=
=
=
=
Z
Z
Z
Z
σ−(n+1)
Q−(n+1)
−∞
T×
T×
Qj
i=−n
i=−n
= µ∗n (E)
Ai ×
Ai ×
T×
Qj
i=−n
i=−n Ai ×
Qj
Qj
−(n+1)
−∞
Ai ×
Q+∞
j+1 T
Q+∞
j+1
Q+∞
j+1
T
T
j
i=−n
Z
T
Q
Ai × +∞
j+1 T
Q+∞
j+1
T
µǫ ((fωn+1)−1 (B)) dθǫZ
−1
µǫ ((fσn+1
(B)) dθǫZ
−(n+1) (ω) )

−1
µǫ ((fσn+1
(B)) d 
−(n+1) (ω) )
+∞
Y
−(n+1)
µǫ ((fω−(n+1) )
−1

θǫ 
((fσn−n (ω) )−1 (B))) dθǫ (ω−(n+1) )d
µǫ ((fσn−n (ω) )−1 (B)) d
+∞
Y
−n
θǫ
!
+∞
Y
θǫ
−n
!
Z
Z
The algebra B∞ = ∪+∞
n=0 Bn is a sub-algebra of B(T ×X) and generates B(T ×X).
By (B.2.2) we can define a measure µ∗∞ on B∞ just setting µ∗∞ |Bn = µ∗n , for n ≥ 0.
To show that µ∗∞ is indeed a measured on B∞ we use Theorem A.1.7 showing that,
for every decreasing sequence E0 ⊃ E1 ⊃ E2 ⊃ . . . of members of B∞ such that
∗
∩+∞
n=0 En = ∅, we have lim µ∞ (En ) = 0. Suppose there is some decreasing sequence
E0 ⊃ E1 ⊃ E2 ⊃ . . . of members of B∞ such that ∩+∞
n=0 En = ∅ and there is δ0 such
that for every n ≥ 0 we have µ∗∞ (En ) ≥ δ0 . So one can take a subsequence {Bin }in ≥0
such that En ∈ Bin for all n ≥ 0. By Theorem A.1.4 for each n ≥ 0 on can find a
Q
Q−in −1
compact subset Kn of +∞
T × Kn we have
−in T × X such that writing Wn =
−∞
Wn ⊂ En and
µ∗in (En \ Wn ) < 3−(n+1) δ0 .
For Fn = ∩nj=0 Wj we have µin (Fn ) ≥ δ0 /3, and hence Fn 6= ∅, for all n ≥ 0.
By a diagonalization process we can choose a sequence (ω (n) , x(n) )n≥1 , with (ω (n) , x(n) ) ∈
84
APPENDIX B. RANDOM PERTURBATIONS
+∞
Fn and (ω (n) , x(n) ) → (ω (0) , x(0) ) ∈ ∩+∞
n=0 Fn ⊂ ∩n=0 En , contradicting the fact that
∗
∗
∩+∞
n=0 En = ∅. Therefore, µ∞ is a measure on B∞ . By Theorem A.1.6 µ∞ can be
uniquely extended to a Borel probability measure µ∗ on T Z ×X. Let π2 : T Z ×M → M
be the projection on the second coordinate: π2 (ω, x) = x. It follows from the construction process that
π∗ µ∗ = θǫN × µǫ and (π2 )∗ µ∗ = µǫ .
Moreover, for each E ∈ B∞ we have S∗ µ∗ (E) = µ∗ (E) and since B∞ generates
B(T Z × X), by theorem A.2.1 we have S∗ µ∗ = µ∗ . It also becomes clear that µ∗ is the
weak∗ limit for S∗n (θǫZ × µǫ ). Moreover, the sample measures for µ∗ can be obtained
as µω = lim fσn−n (ω) µǫ so we can easily see that µ∗ is a Markov measure.
∗
It remains to show that if we start with a Markov measure µ∗ on B(T Z × X), get
the corresponding stationary measure µǫ on B(X) and starting from µǫ we construct
a Markov measure ν ∗ as before then µ∗ = ν ∗ . Indeed, let νn∗ = S∗n (θǫZ × µǫ )|Bn and,
Q−(n+1)
Q
Q
for each measurable rectangle E =
T × ji=−n Ai × +∞
−∞
j+1 T × B ∈ Sn , we
have
S∗n µ∗ (E)
∗
Z
= µ (S (E)) = µω ((S −n (E))ω ) dθǫZ (ω)
Z
=
µω ((fωn )−1 (B)) dθǫZ (ω)
Q−n−1
Qj
Q+∞
−n
σ ( −∞ T × i=−n Ai × j+1 T )
Z
= Q
µω ((fωn )−1 (B)) dθǫZ (ω)
Q
Q
=
=
=
Z
Z
−n
−1
−∞
T×
Qj
i=−n
Qj
i=−n
S∗n (θǫZ
j
i=−n
Ai ×
Ai
Ai ×
Q+∞
j+n+1
T
+∞
Zj+n+1
Q−1
T
−∞
−
T
+
µω ((fω0 ◦ . . . ◦ fωn−1 )−1 (B)) dθǫZ (ω − )dθǫZ0 (ω0+ )
µǫ ((fω0 ◦ . . . ◦ fωn−1 )−1 (B)) dθǫn (ω0 , . . . , ωn−1)
× µǫ )(E).
So that
νn∗ = µ∗ |Bn .
While the left-hand side converges to ν ∗ , the right-hand side converges to µ∗ |B(T Z ×X) =
µ∗ , and so ν ∗ = µ∗ .
85
APPENDIX B. RANDOM PERTURBATIONS
From now on, we refer for µǫ and µ∗ to be the corresponding stationary and
Markov probability measures, respectively.
We define the one-sided skew-product map by
S + : T N × X −→
(ω0+ , z)
7−→
TN × X
σ(ω0+ ), fω0 (z) ,
where σ here means the one-side shift map: σ(ω0 , ω1 , . . .) = (ω1 , ω2 , . . .). It is easy to
see that S + ◦ π = π ◦ S.
Proposition B.2.4. The following conditions are equivalent:
i) µǫ is a stationary probability measure.
ii) µ∗ is S-invariant.
iii) θǫN × µǫ is S + -invariant.
Proof. The equivalence between i) and ii) is Proposition B.2.3. We prove the equivalence between i) and iii). Suppose first that θǫN × µǫ is S + -invariant and for a Borel
set B ⊂ M consider U = T N × B. Then,
θǫN × µǫ ((S + )−1 (U)) = θǫN × µǫ
= θǫ × µǫ
=
ZZ
[
t∈T
[
t∈T
{t} × T N × (ft )−1 (B)
{t} × B
!
!
χB (ft (x)) dµǫ (x)dθǫ (t)
and by the S + -invariance this is equal to θǫN × µǫ (U) = µǫ (B), so that
ZZ
χB (ft (x)) dµǫ (x)dθǫ (t) = µǫ (B) =
Z
χB (x)dµǫ (x).
Since simple functions are dense in L1 (M, R) and C(M, R) ⊂ L1 (M, R), by linearity
we have that
ZZ
ϕ(ft (x)) dµǫ (x)dθǫ (t) =
Z
ϕ(x) dµǫ(x)
for every ϕ : M → R continuous, and thus we have i).
(B.2.3)
86
APPENDIX B. RANDOM PERTURBATIONS
Conversely, if (B.2.3) holds for every ϕ continuous, then it holds for every element
in L1 (M, R), since f˜(ω + , x) = fω+ (x) is continuous and µǫ , θǫ are Borel measures. In
particular it holds for ϕ = χB for any Borel subset B ⊂ M. Then if A ⊂ T Z is also a
Borel subset we will show that θǫN × µǫ ((S + )−1 (A × B)) = θǫN × µǫ (A × B). Indeed,
θǫN
+ −1
× µǫ ((S ) (A × B)) =
θǫN
[
× µǫ
t∈T
{t} × A × (ft ) (B)
= θN (A) θǫ × µǫ
N
= θ (A)
= θN (A)
ZZ
Z
−1
[
t∈T
{t} × B
!!
!
χB (ft (x)) dµǫ (x)dθǫ (t)
χB (x) dµǫ (x)
= θǫN × µǫ (A × B).
Definition B.2.5. A set A ⊂ X is random invariant if for µǫ almost every x ∈ M
we have
x∈A
x∈X \A
=⇒
=⇒
ft (x) ∈ A, for θǫ almost every t;
ft (x) ∈ X \ A, for θǫ almost every t.
Definition B.2.6. A stationary measure µǫ is ergodic if for every random invariant
set A we have µǫ (A) = 0 or µǫ (A) = µǫ (X).
Proposition B.2.7. The following conditions are equivalent:
i) µǫ is ergodic.
ii) θǫN × µǫ is S + -ergodic.
iii) µ∗ is S-ergodic.
Proof. The equivalence between i) and ii) can be found in [Ki86]. The equivalence
between ii) and iii) is given in [LQ95] in a context of random diffeomorphisms, which
we reproduce for the clearness of the result in our setting.
Suppose that θǫN × µǫ is S + -ergodic and let E ⊂ T Z × X be some measurable set
satisfying S −1 (E) = E. By Theorem A.1.8, for given α > 0 there is some measurable
87
APPENDIX B. RANDOM PERTURBATIONS
set W ⊂ T N × X such that
µ∗ (E △ π −1 (W )) < α
(B.2.4)
θǫN × µǫ (W △ (S + )−n (W )) < α, ∀n ≥ 0.
(B.2.5)
and
Since θǫN × µǫ is S + -ergodic, by Birkhoff’s Ergodic Theorem
n−1
1X
χW ◦ (S + )j = θǫN × µǫ (W ), θǫN × µǫ almost everywhere.
n→∞ n
j=0
lim
(B.2.6)
From (B.2.5) and (B.2.6) it follows that
Z
thus
χW − θN × µǫ (W ) dθN × µǫ < α,
ǫ
ǫ
θǫN × µǫ (W ) >
1√
1 − 2α
2
θǫN × µǫ (W ) <
1√
1 + 2α
2
or
Since α ir arbitrary, it follows from (B.2.4) that µ∗ (E) = 0 or µ∗ (E) = 1 and hence
µ∗ is S-ergodic.
Assuming now that µ∗ is S-ergodic, let B ⊂ T N ×X be a measurable set, invariant
by S + . Since S + ◦ π = π ◦ S and π∗ µ∗ = θǫN × µǫ , then π −1 (B) = π −1 ◦ (S + )−1 (B) =
S −1 ◦ π −1 (B), that is, π −1 (B) is S-invariant. By the ergodicity of µ∗ , then we must
have either µ∗ (π −1 (B)) = 0, which gives θǫN × µǫ (B) = 0, or µ∗ (π −1 (B)) = 1, which
is the same as θǫN × µǫ (B) = 1. Then θǫN × µǫ is S + -ergodic.
Assume from now on that µǫ is an ergodic stationary probability measure. Then
θǫN × µǫ is S + -ergodic and, by Theorem A.2.5,
n−1
1X
lim
ψ((S + )j (ω + , x)) =
n→+∞ n
j=0
Z
ψ(ω + , x) dθǫN × µǫ ,
(B.2.7)
for θǫN × µǫ -almost every (ω + , x) and every ψ : T N × X → R continuous. In particular,
88
APPENDIX B. RANDOM PERTURBATIONS
we have
n−1
1X
ϕ(fωj + (x)) =
lim
n→+∞ n
j=0
Z
ϕ(x) dµǫ ,
(B.2.8)
for θǫN × µǫ almost every (ω + , x) and any ϕ : X → R continuous, just by considering
ψ = ϕ ◦ π2+ in (B.2.7), where π2+ is the projection of T N ×M on the second coordinate,
i.e. π2+ (ω + , x) = x, and fωn+ = fωn ◦ . . . ◦ fω1 . Let N be the θǫN × µǫ total measure
subset of T N × X for whose points (ω + , x) the equality (B.2.8) holds. By Fubini’s
R
theorem, θǫN × µǫ (N) = θǫN (Nx ) dµǫ(x), where Nx = {ω + ∈ T N : (ω + , x) ∈ N}. Since
µǫ is a probability measure and 0 ≤ θǫN (Nx ) ≤ 1 then, θǫN (Nx ) = 1 for µǫ almost
every x ∈ X. It follows that if µǫ is an ergodic stationary probability measure then
µǫ (B(µǫ )) = 1 and if also µǫ ≪ m then µǫ is a physical measure.
On the other hand, condition (B.2.8) is equivalent to
n−1
1X
lim
ϕ(fωj (x)) =
n→+∞ n
j=0
Z
ϕ(x) dµǫ ,
(B.2.9)
for θǫZ × µǫ almost every (ω, x) and any ϕ : X → R continuous. Moreover, the ergod-
icity of µǫ implies that µ∗ is S-ergodic and applying Birkhoff’s Ergodic Theorem to
S we have
n−1
1X
ψ(S j (ω, x)) =
lim
n→+∞ n
j=0
Z
ψ(ω, x) dµ∗,
for µ∗ almost every (ω, x) and every ψ : T Z × M → R continuous. Hence we have
(B.2.9) for µ∗ almost every (ω, x) and any continuous function ϕ : M → R, just
considering ψ = ϕ ◦ π, since π∗ µ∗ = θǫN × µǫ and condition (B.2.8) is equivalent to
(B.2.9). However, in general, µ∗ is not a product measure. This happens if and only
if for θǫ almost every t ∈ T we have ft ∗ µǫ = µǫ , that is, almost all maps ft leaves the
measure µǫ invariant.
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