P esin’s F orm ula for T ranslation In v arian t
Random Dynamical Systems
.
.
v orgelegt v on
M. Sc.
Vitalii Senin
.
v on der F akult¨ at I I – Mathematik und Naturwissensc haften
der T ec hnisc hen Univ ersit¨ at Berlin
zur Erlangung des ak ademisc hen Grades
.
Doktor der Naturwissensc haften
Dr.rer.nat.
.
genehmigte Dissertation
.
Promotionsaussc h uss:
.
V orsitzender: Prof. Dr. Martin Skutella
Gutac h ter: Prof. Dr. Mic hael Sc heutzo w
Gutac h ter: Prof. Dr. Marc Keßeb¨ ohmer
.
T ag der wissensc haftlic hen Aussprac he: 24.09.2019
Berlin 2019

T o A nastasiia
2

Ac kno wledgemen t
I thank Prof. Dr. Mic hael Sc heutzo w for sup ervising this thesis, Prof. Dr. Marc
Keßeb¨ ohmer for accepting to b e a co-examiner and Prof. Dr. Martin Skutella for
accepting to c hair the Ph.D. examination. I also thank Alex Blumen thal from
the Univ ersit y of Maryland for sev eral useful discussions and suggestions. Finan-
cial supp ort from the In ternational Researc h T raining Group Sto chastic A nalysis
with Applic ations in Biolo gy, Financ e and Physics funded b y the German Re-
searc h Council (DF G) and from Berlin Mathematical Sc ho ol (BMS) is gratefully
ac kno wledged.
3

Zusammenfassung
P esins F ormel b esagt, dass die metrisc he En tropie eines dynamisc hen Systems
gleic h der Summe seiner p ositiv en Ly apuno v Exp onen ten ist, w ob ei die metrisc he
En tropie die Chaotizit¨ at des Systems b esc hreibt und Ly apuno v Exp onen ten die
asymptotisc he exp onen tielle Rate der T renn ung b enac h barter T ra jektorien messen.
Es ist b ek ann t, dass diese F ormel f ¨ ur dynamisc he Systeme auf einer k ompakten
Riemannsc hen Mannigfaltigk eit mit in v arian tem W ahrsc heinlic hk eitsmaß gilt.
T ranslationsin v arian te Bro wnsc he Fl ¨ usse sind eine sp ezifisc he Klasse sto c hastis-
c her Fl ¨ usse auf R d mit unabh¨ angigen station¨ aren Inkremen ten und einer V erteilung,
die im Bezug auf T ranslationen im R d un v er¨ anderlic h ist. Sie hab en ein Ly apuno v
Sp ektrum, ab er k ein in v arian tes W ahrsc heinlic hk eitsmaß. Wir repr¨ asen tieren
translationsin v arian te Bro wnsche Fl¨ usse als zuf¨ allige dynamisc he Systemen im
Sinne v on [18] und [25]. Außerdem definieren wir die En tropie f ¨ ur translationsin-
v arian te (in der V erteilung gegen ¨ ub er T ranslationen im R d ) zuf¨ allige dynamisc he
Systeme, w ob ei die Definition auf den Einheitsw ¨ urfel b esc hr¨ ankt wird. Es stellt
sic h heraus, dass diese Definition aufgrund der T ranslationin v arianz der Systeme
sinn v oll ist. Danac h zeigen wir, dass f ¨ ur translationsin v arian te zuf¨ allige dynamis-
c he Systeme die definierte En tropie kleiner o der gleic h der Summe ihrer p ositiv en
Ly apuno v Exp onen ten ist. Außerdem legen wir P esins F ormel f ¨ ur den F all fest,
w enn das System das V olumen b eib eh¨ alt. Dies impliziert auc h die jew eiligen
Ergebnisse f ¨ ur translationsin v arian te Bro wnsc he Fl ¨ usse.
Wir diskutieren auc h einen alternativ en Ansatz zur Definition v on En tropie.
Wir definieren die En tropie f ¨ ur zuf¨ allige dynamisc he Systeme mit festem Ur-
sprung mit Ideen v on Brin und Katok, siehe [9]. Danac h b ew eisen wir Ruelles
Ungleic h ung mit dieser Definition, d.h. wir sc h¨ atzen v on ob en her die definierte
En tropie durc h die Summe der p ositiv en Ly apuno v Exp onenten der Systeme ab.
Dies impliziert das jew eilige Ergebnis f ¨ ur translationsin v arian te zuf¨ allige dynamis-
c he Systeme und translationsin v arian te Bro wnsc he Fl ¨ usse.
4

Abstract
P esin’s form ula asserts that metric en trop y of a dynamical system is equal to
the sum of its p ositiv e Ly apuno v exp onen ts, where metric en trop y measures the
c haoticit y of the system, whereas Ly apuno v exp onen ts measure the asymptotic
exp onen tial rate of separation of nearb y tra jectories. It is w ell kno wn, that this
form ula holds for dynamical systems on a compact Riemannian manifold with an
in v arian t probabilit y measure.
T ranslation in v arian t Bro wnian flo ws is a sp ecific class of sto chastic flo ws on
R d with indep enden t and stationary incremen ts and with a distribution, whic h is
in v arian t with resp ect to translations in R d . They ha v e a Ly apuno v sp ectrum but
do not ha v e an in v arian t probabilit y measure. W e represen t translation in v ari-
an t Bro wnian flo ws as random dynamical systems in the sense of [18] and [25].
F urther, w e define en trop y for translation in v arian t (in distribution with resp ect
to translations in R d ) random dynamical systems restricting the definition to the
unit cub e. It turns out that this definition mak es sense b ecause of the transla-
tion in v ariance of the systems. After that, w e sho w that for translation in v arian t
random dynamical systems the defined en trop y is less then or equal to the sum
of their p ositiv e Ly apuno v exp onen ts. Moreo v er, w e establish P esin’s form ula in
the case when the system preserv es the v olume. This also implies the resp ectiv e
results for translation in v arian t Bro wnian flo ws.
W e also discuss an alternativ e approac h to the definition of en trop y . W e
define en trop y for random dynamical systems with the fixed origin using ideas
of Brin and Katok, see [9]. After that w e pro v e Ruelle’s inequalit y with resp ect
to this definition, i.e. w e b ound from ab o v e the defined en trop y b y the sum of
p ositiv e Ly apuno v exp onen ts of the systems. This implies the resp ectiv e result
for translation in v arian t random dynamical systems and translation in v arian t
Bro wnian flo ws.
5

Con ten ts
Abstract 5
1 In tro duction 8
2 Preliminaries 13
2 . 1 S t o c h a s t i c F l o w s ........................... 1 3
2.1.1 Driving Fields and Lo cal Characteristics . . . . . . . . . . 13
2.1.2 Kunita-T yp e In tegrals . . . . . . . . . . . . . . . . . . . . 16
2.1.3 Sto c hastic Flo ws and T ranslation In v arian t Bro wnian Flo ws 17
2.1.4 Represen tation of Sto c hastic Flows . . . . . . . . . . . . . 18
2.1.5 Regularit y Prop erties of T ranslation In v ariant Bro wnian
F l o w s ............................. 2 0
2.2 Random Dynamical Systems . . . . . . . . . . . . . . . . . . . . . 21
2.3 Ly apuno v Sp ectrum of T ranslation In v arian t Bro wnian Flo ws and
Random Dynamical Systems . . . . . . . . . . . . . . . . . . . . . 24
3 Definition of En trop y 30
3.1 Definition of En trop y of P artitions . . . . . . . . . . . . . . . . . 31
3.2 Class of 1-p erio dic in Distribution Sets . . . . . . . . . . . . . . . 32
3.3 Metric En trop y of T ranslation In v arian t Random Dynamical Systems 33
3.4 En trop y for t w o-sided systems . . . . . . . . . . . . . . . . . . . . 39
4 Ruelle’s Inequalit y for T ranslation In v arian t Random Dynamical
Systems 41
4 . 1 M a i n R e s u l t .............................. 4 1
4.2 Pro of of Ruelle’s inequalit y . . . . . . . . . . . . . . . . . . . . . . 43
5 P esin’s F orm ula for T ranslation In v arian t Random Dynamical
Systems 47
5 . 1 M a i n R e s u l t .............................. 4 8
5.2 Pro of of Theorem 5.1.3 . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3 Pro of of P esin’s F orm ula using Theorem 5.1.3 . . . . . . . . . . . 61
6 Lo cal Ruelle’s Inequalit y for Random Dynamical Systems 67
6 . 1 M a i n R e s u l t .............................. 6 7
6.2 Main Idea of the Pro of . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Preliminaries Before the Pro of of Lo cal Ruelle’s Inequalit y . . . . 70
6.3.1 Ly apuno v Metric . . . . . . . . . . . . . . . . . . . . . . . 70
6

6.3.2 Some T ec hnical Lemmas . . . . . . . . . . . . . . . . . . . 72
6.4 Pro of of Lo cal Ruelle’s Inequalit y . . . . . . . . . . . . . . . . . . 76
7 Op en Problems 79
A T ransformations of graphs 81
Bibliograph y 82
Index of Notation and Abbreviations 87
7

Chapter 1
In tro duction
In the thesis w e deal with certain random dynamical systems (RDSs) on R d , whic h
are translation in v arian t in distribution. W e pro vide a w a y to define en trop y for
suc h systems without assuming the existence of an in v arian t probabilit y measure.
F urther, w e estimate the defined en trop y from b elo w and ab o v e in terms of certain
lo cal c haracteristics of the systems that are called Ly apuno v exp onen ts. Later
in the c hapter w e pro vide an in tro duction to the notions of en trop y , Ly apuno v
exp onen ts, what is kno wn ab out the estimates (from b elo w and ab o v e) in the
literature and our results. W e start from a motiv ating example.
One of the essen tial topics of sto c hastic analysis is the analysis of sto c hastic
differen tial equations (SDEs) of the t yp e
φ s,t ( x ) = x +
t
Z s
b ( φ s,u ( x )) du +
t
Z s
σ ( φ s,u ( x )) dW u , 0 ≤ s ≤ t, x ∈ R d , (1.1)
where W = ( W 1 , . . . , W k ) denotes a k -dimensional Bro wnian motion and b :
R d → R d and σ : R d → R d × k denote appropriate drift and diffusion functions.
The existence and uniqueness of solutions of differen t t yp es of this equation w as
already studied, see e.g. [15], Chapter IV. Moreo v er, under some smo othness
assumptions on the functions b and σ (see for example [15], Chapter V.2), the
solution of the SDE (1.1) generates a sto c hastic flo w of homeomorphisms, that
is a family { φ s,t : s, t ∈ [0 , ∞ ) } of random diffeomorphisms on R d that satisfies
almost surely
i) φ u,t ◦ φ s,u = φ s,t for all s, t, u ∈ [0 , ∞ );
ii) φ s,s = id | R d for all s ∈ [0 , ∞ );
iii) ( s, t, x ) 7→ φ s,t is con tin uous.
Ho w ev er, it turns out that not ev ery sto chastic flo w is generated b y an SDE
of the t yp e (1.1). Roughly sp eaking, some of them in v olv e to o m uc h randomness
for only finitely man y Bro wnian motions. An example is translation in v arian t
Bro wnian flo ws (TIBFs) that will b e in tro duced in the next c hapter. Ho w ev er,
one can observ e a one-to-one corresp ondence b et w een the solution of SDEs and
sto c hastic flo ws, considering the definition of SDEs in the sense of Kunita [20].
He in tro duced a more general class of SDEs (see Section 2.1):
8

φ s,t ( x ) = x +
t
Z s
F ( φ s,u ( x ) , du ) 0 ≤ s ≤ t, x ∈ R d ,
where F : R d × [0 , ∞ ) → R d is a con tin uous semimartingale field (see Section
2.1). Kunita [20] pro v ed that there is a one-to-one corresp ondence b et w een the
solutions of SDEs of Kunita-t yp e and sto c hastic flo ws of homeomorphisms. W e
will state some of these results in Section 2.1.
An imp ortan t class of sto c hastic flo ws, whic h will b e the fo cus of in terest, are
translation in v arian t Bro wnian flo ws (in tro duced in Section 2.1). These sto c has-
tic flo ws ha v e the additional prop ert y that the homeomorphisms on disjoin t time
in terv als are indep enden t, their distributions are homogeneous in time and in-
v arian t under translations in space. A particular sub class of TIBFs whic h is
called isotropic Bro wnian flo ws (that are additionally in v arian t in distribution
with resp ect to rotations) w as extensiv ely studied in the 1980s b y Le Jan [23]
and Baxandale and Harris [5]. In particular, they ha v e calculated the Ly apuno v
exp onen ts of these flo ws in terms of the isotropic co v ariance function. Ly apuno v
exp onen ts describ e the exp onen tial rate of separation in a certain (usually ran-
dom) direction of infinitesimally close tra jectories. These exp onen ts crucially
affect the global b eha viour of the flo w. Existence of a finite Ly apuno v sp ectrum
(whic h means that all the exp onen ts are finite) in tuitiv ely tells us that the flo ws
are not to o c haotic. Indeed, there are a lot of results in the literature whic h
b ound c haoticit y of certain systems from ab o v e and ev en measure it in terms of
Ly apuno v exp onen ts. In fact, it often happ ens that a smo oth dynamical system
(DS) has en trop y , whic h is equal to the sum of p ositiv e Ly apuno v exp onen ts of
the system. Usually , these results are called Ruelle’s inequalit y if w e measure
en trop y from ab o v e b y this sum, and P esin’s form ula if we pro v e the equalit y .
Suc h results will b e discussed later in the in tro duction. Ho w ev er, it turns out
that w e can not define en trop y for TIBFs b ecause they ha v e no in v arian t proba-
bilit y measure, whic h is a crucial restriction in the classical definition of en trop y .
As w e will see later, it turns out that one can define en trop y for the flo ws using
their translation in v ariance and then pro v e the analogues of Ruelle’s inequalit y
and P esin’s form ula for TIBFs. Now let us finish the discussion of the motiv ating
example and pro vide more details of the main ob jects and results of the thesis.
The standard quan tit y to measure c haoticit y or uncertain t y is the notion of
en trop y . F or dynamical systems, one can consider the notion of the so-called met-
ric en trop y (or sometimes called Kolmogoro v-Sina ˘ ı en trop y). It w as in tro duced
b y Kolmogoro v [19] and Sina ˘ ı [39], and later w as studied b y man y authors, for
example [6], [34], [30], [44], First of all, let us explain the meaning of en trop y
is for a deterministic dynamical system with an in v arian t probabilit y measure.
The en trop y of suc h a system, giv en a partition of the space, is the asymptotic
exp onen tial rate of y es-no questions (with resp ect to the in v arian t probabilit y
measure) necessary to encrypt the tra jectory of a particle ev olving with this sys-
tem with resp ect to the partition. T aking the suprem um o v er all appropriate
partitions then pro vides the en trop y of the system.
No w let us in tro duce the notion of random dynamical systems. A random
9

dynamical system is the discrete ev olution pro cess generated b y the comp osition
of indep enden t, iden tically distributed (i.i.d.) random diffeomorphisms acting on
some state space. This notion follo ws [18] and [25], that studied these systems on
a compact state space. W e will see that sto c hastic flo ws with indep enden t and
stationary incremen ts after discretization in time can b e seen as suc h random
dynamical systems, see Section 2.2. Let us remark that Arnold in tro duced in
[1] (see Section 1.1.1) a more general class of random dynamical systems. It
has b een sho wn b y Arnold and Sc heutzo w [2] that under some mild assumptions
there exists ev en a one-to-one corresp ondence b et w een RDSs in the sense of [1]
and sto c hastic flo ws. Ho w ev er, the indep endence of incremen ts of sto c hastic flo ws
is essen tial for us, so w e stic k to a more restrictiv e notion of RDSs from [18] and
[25].
In the thesis, all the main results are represen ted for RDSs. In Chapter 2
w e pro vide a w a y to represen t TIBFs as translation in v arian t random dynamical
systems (TIRDSs) (i.e. RDSs whic h are in v arian t in distribution with resp ect to
translations). F urther, it turns out that all the main results for RDSs can b e
translated to TIBFs, see Corollary 4.1.1, Corollary 5.1.1 and Remark 6.1.1.
Kifer [18] extended the notion of en trop y to random dynamical systems: a
probabilit y measure is said to b e in v arian t for RDS if the a v erage o v er all p os-
sible diffeomorphisms preserv es the measure. Hence, entrop y of a RDS giv en a
partition of the state space is defined as for deterministic DSs, but the n um b er
of y es-no questions is additionally a v eraged with resp ect to randomness. Again
taking suprem um o v er all appropriate partitions yields the en trop y of the RDS
(see [18], Section 2.1). Th us, en trop y can b e seen as a description of the c haotic
b eha viour of t ypical random tra jectories generated b y the system. Ho w ev er,
TIRDSs ha v e no in v arian t probabilit y measure, whic h is essen tial for the defi-
nition of en trop y . T o resolv e this problem, w e rep eat the argumen ts of Kifer,
but consider only p erio dic partitions and observ e only the dynamics in the unit
cub e [0 , 1) d . It turns out that b ecause of translation in v ariance of the systems w e
can in a similar w a y define the notion of en trop y and enjo y its prop erties suc h
as scalabilit y in time or stabilit y with resp ect to the sequence of appro ximating
partitions (see Lemma 3.3.5 and Lemma 3.3.6), whic h w e need to pro v e further
results. W e mainly follo w here [18], Section 2.1. The details can b e found in
Chapter 3.
Alternativ ely to the notion of en trop y , one can measure c haoticit y of a DS on
the lo cal lev el defining the notion of Ly apuno v exp onen ts. These v alues in tuitiv ely
pro vide the rate of separation of infinitesimally close tra jectories. More precisely ,
Ly apuno v exp onen ts pro vide the exp onen tial rate of gro wth of the deriv ativ e of
the comp osed maps of the DS. There are t w o famous form ulas relating en trop y
with the Ly apuno v exp onen ts of DSs. They are called Ruelle’s inequalit y and
P esin’s form ula. W e pro vide a brief in tro duction to these t w o form ulas in the
next t w o paragraphs.
Ruelle’s inequalit y (or sometimes called Margulis-Ruelle inequalit y) states
that metric en trop y of a (random) dynamical system is b ounded from ab o v e b y
the sum of its p ositiv e Ly apuno v exp onen ts. The first result of this sort w as ob-
tained for C 1 maps b y Ruelle [35]. The first formulation for RDSs appears in [18]
(Theorem V.1.4). This pro of con tained a mistak e, and later Liu and Qian ([25])
10

and Bahnm ¨ uller and Bogensc h ¨ utz ([4]) indep enden tly pro vided the corrections to
the pro of. Later v an Bargen [42] and Bisk amp [8] pro v ed Ruelle’s inequalit y for
certain RDSs on R d . Ho w ev er, they still imp osed the existence of an in v arian t
probabilit y measure for the RDSs. It turns out that our definition of en trop y
lets us essen tially rep eat the pro of of Ruelle’s inequalit y from [42] to resp ectiv ely
obtain Ruelle’s inequalit y for TIRDSs. The details can b e found in Chapter 4.
P esin’s form ula asserts that the en trop y of a dynamical system equals the sum
of its p ositiv e Ly apuno v exp onen ts. Hence, P esin’s form ula is an impro v emen t of
Ruelle’s inequalit y . This remark able form ula w as first established for determin-
istic DSs on a compact Riemannian manifold preserving a smo oth measure (see
[31], [32] and [33]). F or some cases, it w as generalized to deterministic DSs that
preserv e only a Borel measure (see [36], [13]) and to DSs with singularities, see
[17]. The first result for RDSs w as obtained b y Ledrappier and Y oung [22]. Let us
note that P esin’s form ula t ypically requires more regularit y then Ruelle’s inequal-
it y . F or example, let us compare the first results in this direction, obtained b y
Ruelle [35] and P esin [32]. Both results concern deterministic dynamical systems
on a compact Riemannian manifold. Ho w ev er, Ruelle’s inequalit y and P esin’s
form ula require C 1 and C 2 smo othness resp ectiv ely . That is a t ypical situation,
i.e. P esin’s form ula is a stronger result, whic h ho w ev er holds for a smaller class
of systems. It turns out that our definition of en trop y lets us apply Ma ˜ n ´ e’s ideas,
see [26] to pro v e P esin’s form ula for v olume preserving TIRDSs. The details can
b e found in Chapter 5.
Another approac h to define en trop y for TIRDSs is to use another notion of
en trop y , whic h app ears in the literature, and to connect it with P esin’s form ula.
P erhaps the most natural idea in this direction is to use Brin and Katok’s def-
inition of lo cal en trop y , that defined a w a y to lo cally measure c haoticit y of a
deterministic DS on a compact metric space. They define lo cal en trop y in the
follo wing w a y . F or a giv en p oin t x they consider the Bo w en ball with radius r
around the p oin t, i.e. the set of p oin ts that sta y with the tra jectory of x during
first n iterations of the DS. Then they measure the exp onen tial rate of deca y of
measures of suc h sets in terms of lim inf and lim sup. Finally , it turns out that
adding additional limit in space, i.e. when r → 0+, they obtain the same limit,
whic h coincides a.e. with the Kolmogoro v-Sina ˘ ı en trop y , see [9]. That sho ws that
lo cal en trop y and Kolmogoro v-Sina ˘ ı en trop y are (at least in some cases) simi-
lar ob jects, so one can try to use lo cal en trop y for the definition of en trop y for
TIRDSs.
F ormally w e can in the same w a y define ”lo w er” lo cal en trop y , whic h corre-
sp onds to lim inf and ”upp er” lo cal en trop y , whic h corresp onds to lim sup. Ho w-
ev er, the lac k of compactness and the absence of an in v arian t probabilit y measure
do not giv e us a c hance to apply Brin and Katok’s ideas. It is ev en unclear if
the ”lo w er” and ”upp er” lo cal en tropies coincide. Ho w ev er, it turns out that for
a RDS with the fixed origin w e can estimate the defined lo cal en trop y , whic h
corresp onds to lim sup, from ab ov e b y the sum of p ositiv e Ly apuno v exp onen ts,
obtaining some analogue of Ruelle’s inequalit y for the RDSs, see Theorem 6.1.1.
F urther, this theorem implies the resp ectiv e result for TIRDSs, see Corollary
6.1.1, and also for TIBFs, see Remark 6.1.1. Surprisingly enough, w e can apply
some ideas from Ma ˜ n ´ e’s pap er, see [26], even though in the pap er he estimates
11

en trop y from b elo w b y the sum of the p ositiv e Ly apuno v exp onents. The details
can b e found in Chapter 6.
T o the kno wledge of the author, this is the first case of a direct connection
b et w een Brin-Katok en tropy and Ly apuno v exp onen ts, without using metric en-
trop y . Note that Duc and Siegm und (see [12]) defined lo cal metric en trop y for
certain dynamical systems and directly connected it with their Ly apuno v exp o-
nen ts. Ho w ev er, they defined it only for systems with finite time horizon, and it
turns out that their approac h can not b e applied to our case.
12

Chapter 2
Preliminaries
In this c hapter w e will pro vide an in tro duction to sto c hastic flo ws in the sense of
Kunita [20]. In particular, w e will state the main definitions and some previous
results w e will use in the thesis. W e also pro vide a brief in tro duction to random
dynamical systems and Ly apuno v exp onen ts.
The c hapter is organized as follo ws. In Section 2.1.1 w e define the notions of
driving fields and lo cal c haracteristics. In Section 2.1.2 w e in tro duce Kunita-t yp e
in tegrals. In Section 2.1.3 w e define sto c hastic flo ws and TIBFs. In Section 2.1.4
w e state the represen tation theorems for sto c hastic flo ws via sto c hastic differen tial
equations of a Kunita-t yp e. In Section 2.1.5 w e obtain certain in tegrabilit y and
regularit y prop erties of TIBFs. In Section 2.2 w e giv e a short in tro duction to
random dynamical systems and describ e ho w TIBFs can b e seen as suc h an
ev olution pro cess. In Section 2.3 w e establish Ly apuno v sp ectrum for RDSs and
TIBFs.
2.1 Sto c hastic Flo ws
In this section w e giv e an in tro duction to sto c hastic flo ws, follo wing mainly [20],
Chapters 3 and 4, and [7], Chapter 2.
2.1.1 Driving Fields and Lo cal Characteristics
W e pro vide a brief in tro duction to driving fields and lo cal c haracteristics follo wing
mainly [20], Section 3.1, and [7], Section 2.2.1.
F or m ∈ N 0 w e denote b y C m the set of m -times con tin uously differen tiable
functions f : R d → R d . In the case m = 0 w e will often denote C 0 b y C . F or
f ∈ C m define
k f k m := sup
x ∈ R d
| f ( x ) |
1 + | x | + X
1 ≤| α |≤ m
sup
x ∈ R d | D α f ( x ) | ,
and denote C m
b := { f ∈ C m : k f k m < ∞} . Then C m
b with the norm k·k m is a
Banac h space. F or δ ∈ (0 , 1] w e denote b y C m,δ the set of functions f ∈ C m suc h
that D α f for | α | = m are δ -H¨ older con tin uous. In tro ducing for f ∈ C m
13

k f k m + δ := k f k m + X
| α | = m
sup
x 6 = y
| D α f ( x ) − D α f ( y ) |
| x − y | δ
the space C m,δ
b := { f ∈ C m : k f k m + δ < ∞} with the norm k·k m + δ is again a
Banac h space.
W e sa y that a con tin uous function f : R d × [0 , ∞ ) → R d ; ( x, t ) 7→ f ( x, t )
b elongs to C m,δ
b if f ( t ) ≡ f ( · , t ) is an elemen t of C m + δ
b for an y t ∈ [0 , ∞ ) and for
an y T < ∞
T
Z
0
k f ( t ) k m + δ dt < ∞ .
If k f ( t ) k m + δ is uniformly b ounded in t then w e sa y that f b elongs to the class
C m,δ
ub .
No w let us denote for m ∈ N 0 the space ˜
C m whic h consists of functions
g : R d × R d → R d that are m -times con tin uously differen tiable with resp ect to
eac h spatial v ariable. F or g ∈ ˜
C m define
k g k ∼
m := sup
x,y ∈ R d
| g ( x, y ) |
(1 + | x | )(1 + | y | ) + X
1 ≤| α |≤ m
sup
x ∈ R d | D α
1 D α
2 g ( x, y ) |
and for δ ∈ (0 , 1]
k g k ∼
m + δ := k g k ∼
m + X
1 ≤| α |≤ m
sup
x ∈ R d | D α
1 D α
2 g k ∼
δ ,
where
k g k ∼
δ := sup
x 6 = x 0 ,y 6 = y 0
| g ( x, y ) − g ( x 0 , y ) − g ( x, y 0 ) + g ( x 0 , y 0 ) |
| x − x 0 | δ | y − y 0 | δ .
Then w e can define ˜
C m
b := n g ∈ ˜
C m : k g k ∼
m < ∞ o
and ˜
C m,δ
b := n g ∈ ˜
C m : k g k ∼
m + δ < ∞ o .
W e sa y that a con tin uous function g : R d × R d × [0 , ∞ ) → R d ; ( x, y , t ) 7→
g ( x, y , t ) b elongs to ˜
C m,δ
b , if g ( t ) ≡ g ( · , · , t ) is an elemen t of ˜
C m,δ
b for an y t ∈ [0 , ∞ )
and for an y T < ∞
T
Z
0
k g ( t ) k ∼
m + δ dt < ∞ .
If additionally k g ( t ) k ∼
m + δ is uniformly b ounded in t then we sa y that g b elongs to
the class C m,δ
ub .
Let us no w consider a family { F ( x, t ) } t ≥ 0 of R d -v alued con tin uous semimartin-
gales, where x ∈ R d , on a filtered probabilit y space (Ω , F , ( F t ) t ≥ 0 , P ). F urther,
14

consider the canonical decomp osition of the semimartingale
F ( x, t ) = M ( x, t ) + V ( x, t )
in to a lo cal martingale M ( x, t ) and a pro cess V ( x, t ) of lo cally b ounded v ariation.
The pro cess F ( x, t ) is called a con tin uous semimartingale with v alues in C m,δ (or
simply a con tin uous C m,δ -semimartingale) if t 7→ M ( x, t ) is a con tin uous lo cal
martingale with v alues in C m,δ (or simply a con tin uous C m,δ -lo cal martingale)
and V ( x, t ) is a con tin uous C m,δ pro cess, suc h that D α
x V ( x, t ), | α | ≤ m are all
pro cesses of lo cally b ounded v ariation.
W e assume that there exists a co v ariance function a : R d × R d × [0 , + ∞ ) × Ω →
R d × d and a drift function b : R d × [0 , + ∞ ) × Ω → R d suc h that
h M i ( x, · ) , M j ( y , · ) i t =
t
Z
0
a i,j ( x, y , u ) du, V i ( x, t ) =
t
Z
0
b i ( x, u ) du,
where h· , ·i denotes the quadratic v ariation pro cess at time t . W e call the pair
( a, b ) the lo cal c haracteristics of the family of semimartingales F ( x, t ), x ∈ R d .
Also a and b are called infinitesimal co v ariance and infinitesimal mean of the
family of semimartingales F ( x, t ) resp ectiv ely .
The infinitesimal co v ariance a ( x, y , t ) is said to b elong to the class B m,δ
b if
a ( x, y , t ) has a mo dification that is a predictable pro cess with v alues in ˜
C m,δ
b and
for all T < ∞
T
Z
0
k a ( t ) k ∼
m,δ dt < ∞ P -almost surely . (2.1)
Analogously , the infinitesimal mean b ( x, t ) b elongs to B m 0 ,δ 0
b if b ( x, t ) has a
mo dification that is a predictable pro cess with v alues in C m 0 ,δ 0
b and for all T < ∞
T
Z
0
k b ( t ) k m 0 ,δ 0 dt < ∞ P -almost surely . (2.2)
In this case w e sa y the pair ( a, b ) b elongs to the class ( B m,δ
b , B m 0 ,δ 0
b ). The pair
( a, b ) b elongs to the class ( B m,δ
ub , B m 0 ,δ 0
ub ) if (2.1) is replaced b y
ess sup
ω ∈ Ω
sup
0 ≤ t ≤ T k a ( t ) k ∼
m + δ < ∞
and (2.2) b y
ess sup
ω ∈ Ω
sup
0 ≤ t ≤ T k b ( t ) k ∼
m 0 + δ 0 < ∞ .
If m = m 0 and δ = δ 0 the pair ( a, b ) is said to b elong to the class B m,δ
b (or B m,δ
ub ).
W e simply write F ∈ B m,δ
b (or F ∈ B m,δ
ub ) to indicate that the lo cal c haracteristics
of the semimartingales F ( x, t ) , x ∈ R d b elong to the class B m,δ
b (or B m,δ
ub ).
15

2.1.2 Kunita-T yp e In tegrals
W e mainly follo w here [20], Section 3.2, and [7], Section 2.2.1.
Let F ( x, t ), x ∈ R d b e a family of con tin uous C -martingales suc h that lo cal
c haracteristics ( a, b ) b elongs to the class B 0 ,δ
b for some δ > 0. F urther, let { f t } t ≥ 0
b e a predictable R d -v alued pro cess suc h that for all T < ∞ P -almost surely
T
Z
0
a ( f s , f s , s ) ds < + ∞ ,
T
Z
0
b ( f s , s ) ds < + ∞ . (2.3)
If f is a simple pro cess, i.e. there exists n ∈ N , 0 = t 0 < . . . < t n < + ∞ and
functions f t i ∈ C , 0 ≤ i ≤ n satisfying
f t =
n − 1
X
i =0
f t i 1 [ t i ,t i +1 ) ( t ) + f t n 1 [ t n , + ∞ ) ( t ) ,
then the Itˆ o-Kunita sto c hastic in tegral of f with resp ect to the lo cal martingale
field M ( x, t ) is defined in the follo wing w a y
t
Z
0
M ( f s , ds ) :=
n
X
i =0 { M ( f t i ∧ t , t i +1 ∧ t ) } − M ( f t i ∧ t , t i ∧ t ) } .
Let no w f t b e a general predictable pro cess that satisfies (2.3). Then there exists
a Cauc h y-sequence { f n } of simple predictable pro cesses suc h that for an y m, n →
∞ and T < ∞ w e ha v e P -almost surely
T
Z
0
a ( f n
s , f n
s , s ) − 2 a ( f n
s , f m
s , s ) + a ( f m
s , f m
s , s ) ds → 0 .
F urther, one can sho w (see [20], Section 3.2) that the sequence  t
R 0
M ( f n
s , ds )  n
con v erges uniformly in t on compact subsets of [0 , ∞ ) in probabilit y . The limit
is called the Itˆ o-Kunita sto c hastic in tegral of f with resp ect to the lo cal martin-
gale field M ( x, t ) and is denoted b y
t
R 0
M ( f s , ds ). Th us the Itˆ o-Kunita sto c hastic
in tegral of f with resp ect to the semimartingale field F ( x, t ) is defined b y its
canonical decomp osition, i.e. for an y T < ∞
T
Z
0
F ( f s , ds ) :=
T
Z
0
M ( f s , ds ) +
T
Z
0
b ( f s , s ) ds.
Note that analogously one can define a Stratono vic h-Kunita in tegral (see [20],
Section 2.3).
16

2.1.3 Sto c hastic Flo ws and T ranslation In v arian t Bro wn-
ian Flo ws
W e pro vide a brief in tro duction to sto c hastic flo ws and TIBFs follo wing mainly
[20], Section 4.1, and [7], Section 2.2.1.
First of all w e define the notion of a sto c hastic flo w.
Definition 2.1.1. A family of r andom home omorphisms { φ s,t : s, t ∈ [0 , ∞ ) } on
R d on some pr ob ability sp ac e (Ω , F , P ) is c al le d a sto chastic flow of home omor-
phisms if almost sur ely
i) φ s,t = φ u,t ◦ φ s,u for al l s, t, u ∈ [0 , ∞ ) ;
ii) φ s,s = Id | R d for al l s ∈ [0 , ∞ ) ;
iii) ( s, t, x ) 7→ φ s,t ( x ) is c ontinuous.
It is c al le d a sto chastic flow of C k -diffe omorphisms, if additional ly almost
sur ely
iv) φ s,t ( x ) is k times differ entiable with r esp e ct to x for al l s, t ∈ [0 , ∞ ) and
the derivatives ar e c ontinuous in ( s, t, x ) .
Prop erties i ) and ii ) immediately imply that φ s,t ( ω ) − 1 is giv en b y φ t,s ( ω ).
This fact together with condition iii ) yields that φ s,t ( ω ) − 1 ( x ) is also con tin uous
in ( s, t, x ). Condition iv ) sho ws that φ s,t ( ω ) − 1 ( x ) is k times contin uously differ-
en tiable with resp ect to x . Therefore φ t,s ( ω ) is indeed a C k -diffeomorphism for
all s, t ∈ [0 , ∞ ).
Let us denote b y G the set of homeomorphisms on R d . This set forms a group
with resp ect to the comp osition of maps. F urther, it can b e equipp ed with the
metric
d 0 ( φ, φ 0 ) := ρ ( φ, φ 0 ) + ρ ( φ − 1 , ( φ 0 ) − 1 )
where
ρ ( φ, φ 0 ) := X
N ≥ 1
2 − N sup | x |≤ N | φ ( x ) − φ 0 ( x ) |
1 + sup | x |≤ N | φ ( x ) − φ 0 ( x ) | .
The metric ρ induces the so called top ology of uniform con v ergence on com-
pact sets. Then the set ( G, d 0 ) is a complete separable top ological group. A
sto c hastic flo w of homeomorphisms can b e regarded as a G -v alued con tin uous
random pro cess with index set [0 , ∞ ) × [0 , ∞ ) whic h satisfies i ) and ii ). W e call
it a sto c hastic flo w with v alues in G .
F or a m ulti index α = ( α 1 , . . . , α d ) with α i ∈ N 0 , i = 1 , . . . , d w e denote
| α | := P d
i =1 | α i | . F urther, w e denote the spatial partial differen tial op erator with
resp ect to index α b y D α . More precisely
D α := ∂ | α |
∂ x α 1
1 . . . ∂ x α d
d
.
17

Finally , denote b y D x f the differen tial of a function f ev aluated at p oin t
x ∈ R d .
Let G k ⊂ G b e the set of all C k -diffeomorphisms on R d . It is a subgroup of
G , and moreo v er, it is again a complete separable top ological group with resp ect
to the metric
d k ( φ, φ 0 ) := X
| α |≤ k
ρ ( D α φ, D α φ 0 ) + X
| α |≤ k
ρ ( D α φ − 1 , D α ( φ 0 ) − 1 ) .
A sto c hastic flo w of C k -diffeomorphisms can b e seen as a G k -v alued con tin uous
random pro cess with index set [0 , ∞ ) × [0 , ∞ ) satisfying prop erties i ) and ii ).
Analogously , w e call it a sto c hastic flo w with v alues in G k .
No w let us define the class of translation in v arian t Bro wnian flo ws.
Definition 2.1.2. A sto chastic flow φ with values in G 2 is c al le d
i) a Br ownian flow if any n ∈ N , 0 ≤ t 0 < . . . ≤ t n < ∞ the r andom variables
{ φ t i − 1 ,t i } i = 1 ,n ar e indep endent;
ii) a homo gene ous Br ownian flow, if additional ly for any h ≥ 0 the laws of
{ φ s,t : 0 ≤ s ≤ t < ∞} and { φ s + h,t + h : 0 ≤ s ≤ t < ∞} c oincide;
iii) a tr anslation invariant sto chastic flow, if the distributions of φ s,t ( · + a )
and φ s,t + a c oincide for al l s, t ∈ R + and a ∈ R d ;
iv) a tr anslatio n invariant Br ownian flow if c onditions i)-iii) ar e satisfie d and
φ is a solution of a SDE
φ s,t ( x ) = x +
t
Z s
F ( du, φ s,u ( x )) , for al l t ≥ s ≥ 0 .
wher e F ( t, x, ω ) : R + × R d × Ω → R d is a c ontinuous semimartingale with values
in C with F ∈ B 2 , 1
ub .
Note that TIBFs w ere already discussed in [37], see Remark on p. 50. In
that article condition iv ) w as not included in the definition. In fact, there w as
a similar restriction, whic h ho w ever imposed less regularity on the infinitesimal
mean and co v ariance. Here w e use a stronger condition iv ) to b e able to obtain
Ruelle’s inequalit y and P esin’s form ula for TIBFs, see Corollary 4.1.1, Corollary
5.1.1 and Remark 6.1.1.
2.1.4 Represen tation of Sto c hastic Flo ws
In Section 2.1.3 TIBFs are defined as solutions of certain Kunita-t yp e SDEs,
whic h seems to b e a serious restriction. Ho w ev er, it turns out that this is not the
case thanks to the results in [20]. No w let us pro vide more details. W e follo w
here [20], Chapter 4, and [7], Section 2.2.1.
In this section w e discuss the connection b et w een sto c hastic flo ws and SDEs
of the t yp e
dX t = F ( X t , dt ) , t ≥ s ≥ 0 , (2.4)
18

where s is a fixed p ositiv e n um b er and F is a semimartingale field.
F or fixed s ∈ [0 , ∞ ) and x ∈ R d a con tin uous R d -v alued pro cess φ s,t ( x ) ,
0 ≤ s ≤ t < ∞ adapted to {F t } is called a solution of SDE (2.4) starting at time
s in p oin t x if it satisfies
φ s,t ( x ) = x +
t
Z s
F ( φ s,u ( x ) , du ) , for all t ≥ s. (2.5)
The existence and uniqueness of a solution is sho wn in [20], Theorem 3.4.1:
Theorem 2.1.1. L et F ( x, t ) b e a c ontinuous semimartingale with values in C
with lo c al char acteristics b elonging to the class B 0 , 1
b . Then for e ach s and x the
e quation (2.5) has a unique solution.
Consider a sto c hastic flo w { φ s,t : s, t ∈ [0 , ∞ ) } with v alues in G k , k ∈ N 0 .
Let {F s,t : 0 ≤ s ≤ t < ∞} b e the filtration generated b y the flo w, which is for
s<t the least σ -algebra F s,t con taining all n ull sets and ∩ > 0 σ ( φ u,v : s −  ≤ u ≤
v ≤ t +  ) . The forw ard part { φ s,t : 0 ≤ s ≤ t < ∞} is called a forw ard C k ,δ -
semimartingale flo w, if for ev ery s the sto c hastic pro cess { φ s,t : 0 ≤ s ≤ t < ∞}
is a con tin uous C k ,δ -semimartingale adapted to {F s,t : t ∈ [ s, ∞ ) } .
Then an y sufficien tly smo oth forw ard semimartingale flo w the follo wing result
(see [20], Theorem 4.4.1) pro vides the existence and uniqueness of a con tin uous
semimartingale field satisfying (2.4):
Theorem 2.1.2. L et { φ s,t : 0 ≤ s ≤ t < ∞} b e a forwar d C k,δ -semimartingale
flow for some k ≥ 0 and δ > 0 such that for every s the lo c al char acteristics b e-
longs to the class B k ,δ
b . Then ther e exists a unique c ontinuous C k, -semimartingale
F ( x, t ) with F ( x, 0) = 0 (for al l  < δ ) with lo c al char acteristics b elonging to the
class B k ,δ
b such that for e ach s and x the pr o c ess { φ s,t , t ∈ [ s, ∞ ) } satisfies (2.5).
Pr o of. See [20], Theorem 4.4.1.
On the other hand the follo wing statemen t (see [20], Theorem 4.6.5) yields
that under certain smo othness assumptions on a s emimartingale F there exists a
solution of SDE (2.4), whic h forms a forw ard sto c hastic flo w of diffeomorphisms.
Theorem 2.1.3. L et F ( x, t ) b e a c ontinuous C -semimartingale whose lo c al char-
acteristics b elongs to the class B k ,δ
b for some k ≥ 1 and δ > 0 . Then the
solution of the sto chastic differ ential e quation (2.4) b ase d on F has a mo difi-
c ation { φ s,t : 0 ≤ s ≤ t < ∞} such that it is a forwar d sto chastic flow of
C k -diffe omorphisms. F urther, it is a forwar d C k , -semimartingale for any  < δ .
Theorem 2.1.2 and Theorem 2.1.3 pro vide the corresp ondence b et w een sto c has-
tic flo ws and semimartingale fields b y the SDE (2.4). Note that in [20] all the
ab o v e is done only on a finite time in terv al, i.e. when 0 ≤ s ≤ t ≤ T for some
fixed T < + ∞ . H o w ev er, a standard lo calizing argumen t for lo cal martingales
pro vides the results as stated ab o v e.
19

2.1.5 Regularit y Prop erties of T ranslation In v arian t Bro w-
nian Flo ws
The aim of the section is to sho w that TIBFs fulfill certain regularit y assumptions,
that w e will use later to obtain the main results.
The follo wing result pro vides in tegrabilit y of deriv ativ es of φ . Note that a
similar result with also a similar pro of w as established in [8], Section 9, pp. 140-
141.
Lemma 2.1.1. L et φ b e a TIBF. Then we have
Z log + sup
v ∈ B (0 , 1) k D v φ 0 ,n k d P < ∞ , ∀ n ∈ N , (2.6)
Z log + sup
v ∈ B (0 , 1) 
 D v ( φ − 1
0 ,n ) 
 d P < ∞ , ∀ n ∈ N , (2.7)
and Z log + sup
v ∈ B (0 , 1) 
 D 2
v φ 0 ,n 
 d P < ∞ , ∀ n ∈ N . (2.8)
Pr o of. First of all, let us quote the follo wing result b y Imk eller and Sc heutzo w
(see [16], Theorem 2.2). F or the sak e of con v enience w e form ulate it for TIBFs.
Theorem 2.1.4. L et φ b e a TIBF. Then for al l T ≥ 0 , ther e exist c, γ > 0 such
that for al l 1 ≤ | α | ≤ 2 the r andom variable
Y α := sup
y ∈ R d
sup
0 ≤ s,t ≤ T k D α
y φ s,t k exp {− γ (log + | y | ) 1 / 2 }
is Φ c -inte gr able, wher e
Φ c : [0 , + ∞ ) → [0 , + ∞ ); x 7→
∞
Z
1
exp( − ct 2 ) x t dt.
By [16], Lemma 1.1 (left inequalit y in (4)) w e ha v e for z ≥ 1 the inequalit y
exp((log z ) 2 / 4 c − (log K ) 2 / 4 c ) ≤ Φ c ( z ) . (2.9)
No w let us sho w (2 . 8). Fix n ∈ N . Let α b e a m ulti index with | α | = 2. Then
Theorem 2.1.4 for T = n and s = 0 implies
Z log + sup
v ∈ B (0 , 1) k D α
v φ 0 ,n k d P ≤ Z log + Y α d P = Z log ( Y α ∨ 1) d P
=2 √ c Z log ( Y α ∨ 1) / 2 √ cd P ;
20

Note that z ≤ exp { z 2 } , and so
2 √ c Z log ( Y α ∨ 1) / 2 √ cd P
≤ 2 √ c Z exp((log( Y α ∨ 1)) 2 / 4 c ) d P
≤ 2 √ c exp { (log K ) 2 / 4 c } Z exp((log ( Y α ∨ 1)) 2 / 4 c − (log K ) 2 / 4 c ) d P
(2 . 9)
≤ 2 √ c exp { (log K ) 2 / 4 c } Z Φ c ( Y α ∨ 1) d P
≤ 2 √ c exp { (log K ) 2 / 4 c } Z (Φ c ( Y α )+Φ c (1)) d P < ∞ ,
whic h completes the pro of of (2 . 8). In the same w a y one can pro v e (2 . 6) and
(2 . 7). Note that in the case of (2.7) w e additionally use that φ − 1
0 ,n = φ n, 0 .
2.2 Random Dynamical Systems
In this section w e in tro duce the notion of random dynamical systems in tro duced
in [18], Section 1.2 and [25], Chapter 1, § 1. W e mainly follo w here [25], Chapter
1, § 1.
It the thesis w e deal with random dynamical systems generated b y i.i.d. maps.
More precisely , a random dynamical system for us alw a ys is the discrete-time
ev olution pro cess generated b y sup erp ositions of some random diffeomorphisms
on R d . These diffeomorphisms will b e assumed to b e i.i.d. according to a certain
distribution on the set of diffeomorphisms. Note that, as it w as men tioned in the
in tro duction, this view is quite restricted and usually random dynamical systems
are defined as in [1], Section 1.1.1.
F or the sak e of con v enience w e consider the space of space of p ossible dif-
feomorphisms as the initial probabilit y space. More precisely , recall that (see
Section2.1.3) the space G 2 is the space of 2-times con tin uously differen tiable dif-
feomorphisms on R d . No w denote G 2 b y ˜
Ω. Then it is (see Section 2.1.3) a
complete separable top ological group w.r.t. the top ology of uniform con v ergence
on compact sets for all deriv ativ es up to order t w o. F urther, denote b y B ( ˜
Ω) the
Borel σ -algebra on ˜
Ω. No w fix a probabilit y measure ˜ ν on B ( ˜
Ω), according to
whic h w e will c hose the diffeomorphic maps. F urther, let
( ˜
Ω N , B ( ˜
Ω) N , ˜ ν N ) =
+ ∞
Y
i =1
( ˜
Ω , B ( ˜
Ω) , ˜ ν )
b e the infinite pro duct of copies of the probabilit y space ( ˜
Ω , B ( ˜
Ω) , ˜ ν ). Denote b y
ψ i : ˜
Ω N → ˜
Ω the i -th co ordinate function on the sequence space ˜
Ω N . Let us define
for ev ery ˜ ω = ( ψ 0 ( ˜ ω ) , ψ 1 ( ˜ ω )) , . . . ) ∈ ˜
Ω Z and n ∈ N
ψ 0 , ˜ ω = Id | R d ,
ψ n, ˜ ω = ψ n − 1 ( ˜ ω ) ◦ ψ n − 2 ( ˜ ω ) . . . ◦ ψ 0 ( ˜ ω ) .
21

The one-sided RDS generated b y these comp osed maps, that is { ψ n, ˜ ω : n ∈
N 0 , ˜ ω ∈ ( ˜
Ω N , B ( ˜
Ω) N , ˜ ν N ) } , will b e referred to as ψ .
No w w e define the notion of t w o-sided RDS. The main difference is that t w o-
sided RDSs are defined also for negativ e times. Let
( ˜
Ω Z , B ( ˜
Ω) Z , ˜ ν Z ) =
+ ∞
Y
i = −∞
( ˜
Ω , B ( ˜
Ω) , ˜ ν )
b e the infinite pro duct of copies of the probabilit y space ( ˜
Ω , B ( ˜
Ω) , ˜ ν ). Again
denote b y ψ i : ˜
Ω Z → ˜
Ω the i -th co ordinate function on the sequence space ˜
Ω Z .
Let us define for ev ery ˜ ω = ( ...,ψ − 1 ( ˜ ω ) , ψ 0 ( ˜ ω ) , ψ 1 ( ˜ ω )) , . . . ) ∈ ˜
Ω Z and n ∈ Z
ψ 0 , ˜ ω = Id | R d ,
ψ n, ˜ ω = ψ n − 1 ( ˜ ω ) ◦ ψ n − 2 ( ˜ ω ) . . . ◦ ψ 0 ( ˜ ω ) ,
ψ − n, ˜ ω = ψ − 1
− n ( ˜ ω ) ◦ ψ − 1
− n +1 ( ˜ ω ) . . . ◦ ψ − 1
− 1 ( ˜ ω ) .
The t w o-sided random dynamical system generated b y these comp osed maps,
that is { ψ n, ˜ ω : n ∈ Z , ˜ ω ∈ ( ˜
Ω Z , B ( ˜
Ω) Z , ˜ ν Z ) } , will also b e referred to as ψ .
No w w e define the notion of in v arian t measure of RDSs. In tuitiv ely a measure
is in v arian t for RDS if and only if it is preserv ed under the action of the system
on a v erage. Note that the definition is b orro w ed from [25], Chapter I, Definition
1.1; see also [18], Section 1.2 ( P ∗ -in v ariance).
Definition 2.2.1. i) L et ψ b e a one-side d RDS define d on a pr ob ability sp ac e
( ˜
Ω N , B ( ˜
Ω) N , ˜ ν N ) . Then a Bor el me asur e ˜ µ on R d is c al le d an invariant me asur e
of ψ if Z ˜ µ ( ψ − 1
1 ,ω ( A )) d ˜ ν N = ˜ µ ( A ) , ∀ A ∈ B ( R d ) .
ii) L et ψ b e a two-side d RDS on a pr ob ability sp ac e ( ˜
Ω Z , B ( ˜
Ω) Z , ˜ ν Z ) . Then a
Bor el me asur e ˜ µ on R d is c al le d an invariant me asur e of ψ if
Z ˜ µ ( ψ − 1
1 ,ω ( A )) d ˜ ν Z = ˜ µ ( A ) , ∀ A ∈ B ( R d ) .
No w define for a t w o-sided RDS ψ b y θ the left shift op erator on ˜
Ω, namely
ψ n ( θ ˜ ω ) = ψ n +1 ( ˜ ω )
for all ˜ ω = ( ...,ψ − 1 ( ˜ ω ) , ψ 0 ( ˜ ω ) , ψ 1 ( ˜ ω ) , . . . ) ∈ ˜
Ω Z . Note that θ is measurable
and with a measurable in v erse. Moreov er, θ is a measure-preserving transforma-
tion on ( ˜
Ω Z , B ( ˜
Ω) Z , ˜ ν Z ). Finally , let us note that θ is ergo dic, since
...,ψ 1 ,θ − 1 ω , ψ 1 ,ω , ψ 1 ,θ ω , . . . are indep enden t and iden tically distributed.
F urther, for a t w o-sided RDS ψ define the sk ew pro duct shift Θ : R d × ˜
Ω Z →
R d × ˜
Ω Z as
Θ( x, ˜ ω ) := ( ψ 0 ( ˜ ω ) x, θ ˜ ω ) .
Note that Θ is measurable.
22

In the same w a y for a one-sided system ψ define the left shift op erators θ + and
the sk ew pro duct shift Θ + . Then θ + is measurable measure-preserving ergo dic
transformation on ( ˜
Ω N , B ( ˜
Ω) N , ˜ ν N ), where Θ + is measurable.
No w let us define the notion of translation in v arian t random dynamical sys-
tem.
Definition 2.2.2. A (one-side d or two-side d) RDS ψ is c al le d tr anslation in-
variant r andom dynamic al system (TIRDS) if the distributions of ψ 1 ,ω ( · + a ) and
ψ 1 ,ω ( · ) + a c oincide for al l a ∈ R d .
T o the kno wledge of the author this is the first definition of TIRDSs in suc h
a sense. Ho w ev er, translation in v arince in the same sense w as already defined
for Bro wnian flo ws, see [37], Remark on p. 50; see also [10], Section 1.2, p.17,
prop ert y ( iii ).
An example of TIRDSs is discretized in time TIBFs, see the end of the section.
No w w e establish the in v ariance of the Leb esgue measure for TIRDSs.
Prop osition 2.2.1. L et ψ b e a two-side d (one-side d) TIRDS on a pr ob ability
sp ac e ( ˜
Ω Z , B ( ˜
Ω) Z , ˜ ν Z ) (( ˜
Ω N , B ( ˜
Ω) N , ˜ ν N )) . Then the L eb esgue me asur e µ is invari-
ant for ψ .
Pr o of. W e pro v e the lemma only for t w o-sided TIRDSs. The pro of for one-sided
TIRDSs is the same. W e ha v e
Z µ ( ψ − 1
1 ,ω ( A )) d ˜ ν Z = Z Z 1 ψ 1 ,ω ( x ) ∈ A ( x ) dµ ( x ) d ˜ ν Z
= Z Z 1 ψ 1 ,ω ( x ) ∈ A ( x ) d ˜ ν Z dµ ( x )
= Z ˜ ν Z ( ψ 1 ,ω ( x ) ∈ A ) dµ ( x )
= Z ˜ ν Z ( ζ x ∈ A − x ) dµ ( x ) ,
where ζ x := ψ 1 ,ω ( x ) − x. No w b ecause of translation in v ariance of ψ , the distri-
bution of ζ x do es not dep end on x , and therefore
Z ˜ ν Z ( ζ x ∈ A − x ) dµ ( x ) = Z ˜ ν Z ( ζ 0 ∈ A − x ) dµ ( x )
= Z Z 1 ζ 0 ∈ A − x ( x ) d ˜ ν Z dµ ( x )
= Z Z 1 x ∈ A − ζ 0 ( x ) dµ ( x ) d ˜ ν Z
= Z µ ( A − ζ 0 ) d ˜ ν Z = µ ( A ) ,
as required.
No w w e form ulate a consequence of Prop osition I.2.3 from [18], but for TIRDSs.
23

Prop osition 2.2.2. L et ψ b e a one-side d TIRDS on a pr ob ability sp ac e
( ˜
Ω N , B ( ˜
Ω) N , ˜ ν N ) . Then the me asur e M + := µ × ˜ ν N (define d on R d × ˜
Ω N ) is
invariant for Θ + .
Pr o of. See [18], Prop osition I.2.3 together with Prop osition 2.2.1.
Our aim no w is to construct from a homogeneous Bro wnian flo w φ a transla-
tion in v arian t random dynamical system ϕ , suc h that { ϕ n,θ m ω , m, n ∈ N 0 } coin-
cides in distribution with { φ m,m + n , m, n ∈ N 0 } .
Define
ˆ
Ω = G 2 ; ˆ ν ( A ) = P ( φ 0 , 1 ∈ A ) ,
where A ∈ B ( G 2 ). The pro cedure from the b eginning of the section generates
the triple ( ˆ
Ω N , B ( ˆ
Ω) N , ˆ ν N ) and the corresp onding one-sided random dynamical
system, denote it b y ϕ . Then it is easy to c hec k that indeed
{ ϕ n,θ m ω , m, n ∈ N 0 } d
= { φ m,m + n , m, n ∈ N 0 } .
Alternativ ely , on the last step of the describ ed pro cedure w e generate the triple
( ˆ
Ω Z , B ( ˆ
Ω) Z , ˆ ν Z ) and the corresp onding t w o-sided random dynamical system, de-
note it also b y ϕ . In an y case if φ is a TIBF, then ϕ is a TIRDS. In the thesis w e
will use these pro cedures only in Corollary 4.1.1, whic h pro vides Ruelle’s inequal-
it y for TIBFs, and in Corollary 5.1.1 (only the pro cedure of generating t w o-sided
system), whic h pro vides P esin’s form ula for TIBFs.
2.3 Ly apuno v Sp ectrum of T ranslation In v ari-
an t Bro wnian Flo ws and Random Dynami-
cal Systems
Let ξ b e a random v ariable defined on a probabilit y space (Ω , F , P ). Then the
exp ected v alue of ξ , denoted b y E P ξ , is defined as the Leb esgue in tegral
E P ξ = Z
Ω
ξ ( ω ) d P ( ω ) .
F rom no w on w e will abbreviate E instead of E P if there is no risk of am biguit y .
No w w e state assumptions for a RDS ψ , whic h corresp ond to in tegrabilit y
conditions (2.6), (2.7), (2.8).
Assumption 1: ψ is a one-sided (t w o-sided) RDS on a probabilit y space
( ˜
Ω N , B ( ˜
Ω N ) , ˜ ν N ) (( ˜
Ω Z , B ( ˜
Ω Z ) , ˜ ν Z )) and satisfies
E log + sup
v ∈ B (0 , 1) k D v ψ n,ω k < ∞ , ∀ n ∈ N .
Assumption 2: ψ is a one-sided (t w o-sided) RDS on a probabilit y space
( ˜
Ω N , B ( ˜
Ω N ) , ˜ ν N ) (( ˜
Ω Z , B ( ˜
Ω Z ) , ˜ ν Z )) and satisfies
24

E log + sup
v ∈ B (0 , 1) 
 D v ( ψ − 1
n,ω ) 
 < ∞ , ∀ n ∈ N .
Assumption 3: ψ is a one-sided (t w o-sided) RDS on a probabilit y space
( ˜
Ω N , B ( ˜
Ω N ) , ˜ ν N ) (( ˜
Ω Z , B ( ˜
Ω Z ) , ˜ ν Z )) and satisfies
E log + sup
v ∈ B (0 , 1) 
 D 2
v ψ n,ω 
 < ∞ , ∀ n ∈ N .
No w w e connect TIBFs with Assumptions 1-3.
Theorem 2.3.1. L et φ b e a TIBF and ϕ b e the r esp e ctive one-side d (two-side d)
TIRDS on a pr ob ability sp ac e ( ˆ
Ω N , B ( ˆ
Ω N ) , ˆ ν N ) (r esp e ctively ( ˆ
Ω Z , B ( ˆ
Ω Z ) , ˆ ν Z )) ,i.e.
is c onstructe d as in Se ction 2.2. Then ϕ satisfies Assumptions 1-3.
Pr o of. Assumption 1, 2 and 3 holds b ecause of (2.6), (2.7), and (2.8) resp ectiv ely .
Note that one can construct also other RDSs, that are translation in v arian t
and satisfy Assumptions 1-3. The following example is prop osed b y M. Sc heutzo w.
Example. Let a one-sided TIRDS ϕ on a probabilit y space ( ˆ
Ω N , B ( ˆ
Ω) N , ˆ ν N )
corresp onds to a TIBF φ , i.e. is constructed as in Section 2.2. Recall that
ˆ ν ( A ) = P ( φ 0 , 1 ∈ A ), where A ∈ B ( ˆ
Ω). No w define another measure ˇ ν on ˆ
Ω N in
the follo wing w a y
ˇ ν ( A ) = ( 1
2 ˆ ν ( A ) , A ∈ B ( ˆ
Ω) and id R d / ∈ A,
1
2 , { id R d } = A.
This pro cedure generates the triple ( ˆ
Ω N , B ( ˆ
Ω) N , ˇ ν N ) and the corresp onding ran-
dom dynamical system, denote it b y ˇ ϕ . In tuitiv ely , one can explain ˇ ϕ in the
follo wing w a y: w e consider ϕ , but b efore eac h iteration w e toss a symmetric coin.
Then w e apply the dynamics of ϕ in the case of heads and force the flo w sta y
the same in the case of tails. Then ˇ ϕ do es not corresp ond to a TIBF, b ecause
ˇ ϕ 1 ,ω (0) is not a normal random v ariable. Ho wev er, it is easy to c hec k that ˇ ϕ is
translation in v arian t and satisfies Assumptions 1-3.
No w w e state t w o results, whic h sho w the existence of a finite Ly apuno v
sp ectrum of one-sided RDSs with the fixed origin and of one-sided TIRDSs re-
sp ectiv ely . Note that the result b elo w is an analogue of Theorem 1.6 from [35].
Theorem 2.3.2. L et ψ b e a one-side d r andom dynamic al system on a pr ob ability
sp ac e ( Ω N , B (Ω) N , ν N ) , which satisfies Assumption 1, and also has the fixe d origin,
i.e. ψ 1 ,ω (0) = 0 , ∀ ω . Then ther e exist numb ers λ 1 > λ 2 > . . . > λ p and a forwar d
θ + -invariant me asur able set Ω N
1 with ν N (Ω N
1 )=1 , such that for al l ω ∈ Ω N
1 ther e
exists a me asur able splitting
R d = E ψ
1 ( ω ) ⊕ . . . ⊕ E ψ
p ( ω )
25

of R d over Ω N
1 into r andom subsp ac es E ψ
i ( ω ) (so-c al le d Osele dets sp ac es) with di-
mension dim E ψ
i ( ω ) = d i , i = 1 , p (so-c al le d Osele dets splitting) with the fol lowing
pr op erties
i) we have
lim
n →∞
1
n log | ( D 0 ψ 0 ,t ) v | = λ i ⇐ ⇒ v ∈ V ψ
i ( ω ) \ V ψ
i +1 ( ω ) ,
wher e V p +1 := { 0 } and for i = 1 , p
V ψ
i ( ω ) := E ψ
p ( ω ) ⊕ . . . ⊕ E ψ
i ( ω ) .
ii) the subsp ac es V ψ
i ar e θ + -invariant, i.e. ( D 0 ψ n,ω ) V ψ
i ( ω ) = V ψ
i ( θ n
+ ω ) ;
The numb ers λ i and d i ar e c al le d Lyapunov exp onents of the RDS ψ and their
multiplicities r esp e ctively.
Pr o of. Equalit y i) for ψ holds b ecause of [35], Theorem 1.6, where τ and T
should b e substituted b y θ + and D 0 ψ 1 ,ω resp ectiv ely . Note that the in tegrabilit y
conditions of the theorem hold b ecause of Assumption 1 and ergo dicit y of θ .
Finally , ii) is a trivial consequence of i).
Note that sev eral statemen ts b elo w in this section also pro vide n um b ers λ i and
d i . In ev ery case these n um b ers are also called (as in Theorem 2.3.2) Ly apunov
exp onen ts and their m ultiplicities resp ectiv ely .
Theorem 2.3.3. L et ψ b e a one-side d tr anslation invariant r andom dynamic al
system on a pr ob ability sp ac e ( ˆ
Ω N , B ( ˆ
Ω) N , ˆ ν N ) , which satisfies Assumption 1. Then
ther e exist numb ers λ 1 > λ 2 > . . . > λ p and a θ + -invariant me asur able set ˆ
Ω N
1
with ˆ ν N ( ˆ
Ω N
1 )=1 , such that for al l ω ∈ ˆ
Ω N
1 ther e exists a me asur able splitting
R d = E ψ
1 ( ω ) ⊕ . . . ⊕ E ψ
p ( ω )
of R d over ˆ
Ω N
1 into r andom subsp ac es E ψ
i ( ω ) (so-c al le d Osele dets sp ac es) with di-
mension dim E ψ
i ( ω ) = d i , i = 1 , p (so-c al le d Osele dets splitting) with the fol lowing
pr op erties
i) we have
lim
n →∞
1
n log | ( D 0 ψ 0 ,t ) v | = λ i ⇐ ⇒ v ∈ V ψ
i ( ω ) \ V ψ
i +1 ( ω ) ,
wher e V p +1 := { 0 } and for i = 1 , p
V ψ
i ( ω ) := E ψ
p ( ω ) ⊕ . . . ⊕ E ψ
i ( ω ) .
ii) the subsp ac es V ψ
i ar e θ + -invariant, i.e. ( D 0 ψ n,ω ) V ψ
i ( ω ) = V ψ
i ( θ n
+ ω ) ;
Pr o of. It suffices to pro v e the theorem for another one-sided RDS ψ on a proba-
bilit y space ( Ω N , B (Ω) N , ν N ) generated b y i.i.d. mappings
ψ 1 ,ω − ψ 1 ,ω (0) , ψ 1 ,θ ω − ψ 1 ,θ ω (0) . . . ,
26

b ecause random matrices D 0 ψ 1 ,ω and D 0 ψ 1 ,ω ha v e the same distribution. F or ψ
the theorem holds b ecause of Theorem 2.3.2.
In fact, TIBFs ha v e a finite Ly apuno v sp ectrum ev en in con tin uous time. The
follo wing prop osition pro vides the precise statemen t.
Prop osition 2.3.1. L et φ b e a TIBF. Then ther e exist numb ers λ 1 > λ 2 > . . . >
λ p and me asur able set Ω 1 with P (Ω 1 )=1 , such that for al l ω ∈ Ω 1 ther e exists a
me asur able splitting
R d = E φ
1 ( ω ) ⊕ . . . ⊕ E φ
p ( ω )
of R d over Ω 1 into r andom subsp ac es E φ
i ( ω ) (so-c al le d Osele dets sp ac es) with di-
mension dim E φ
i ( ω ) = d i , i = 1 , p (so-c al le d Osele dets splitting) with the fol lowing
pr op erty
lim
t →∞
1
t log | ( D 0 φ 0 ,t ) v | = λ i ⇐ ⇒ v ∈ V φ
i ( ω ) \ V φ
i +1 ( ω )
(her e t ∈ R + ), wher e V p +1 := { 0 } and for i = 1 , p
V φ
i ( ω ) := E φ
p ( ω ) ⊕ . . . ⊕ E φ
i ( ω ) .
Pr o of. W e can consider φ as a one-sided RDS in the sense of Arnold, see [1], Sec-
tion 1.1.1. Then w e can apply [1], Theorem 3.4.1 (C). The in tegrabilit y condition
of Theorem 3.4.1 (C) holds b ecause as in the pro of of Lemma 2.1.1 w e can sho w
that
log + sup
t ∈ [0 , 1] k D 0 φ 0 ,t k ∈ L 1 ( P ) .
Remark 2.3.1. F r om now on we stop discussing TIBFs. The only exc eptions
ar e Cor ol lary 4.1.1, Cor ol lary 5.1.1 and R emark 6.1.1, which formulate the main
r esults of the thesis in terms of TIBFs.
No w w e state t w o results, whic h sho w the existence of a finite Ly apuno v
sp ectrum of t w o-sided RDSs with the fixed origin and of t w o-sided TIRDSs re-
sp ectiv ely . Note that the result b elo w is an analogue of Theorem 3.1 from [35].
Theorem 2.3.4. L et ψ b e a two-side d r andom dynamic al system on a pr ob ability
sp ac e ( Ω Z , B (Ω) Z , ν Z ) , which satisfies Assumptions 1 and 2, and also has the fixe d
origin, i.e. ψ 1 ,ω (0) = 0 , ∀ ω . Then ther e exist numb ers λ 1 > λ 2 > . . . > λ p and
a θ -invariant me asur able set Ω Z
1 with ν Z (Ω Z
1 ) = 1 , such that for al l ω ∈ Ω Z
1 ther e
exists a me asur able splitting
R d = E ψ
1 ( ω ) ⊕ . . . ⊕ E ψ
p ( ω )
of R d over Ω Z
1 into r andom subsp ac es E ψ
i ( ω ) (so-c al le d Osele dets sp ac es) with di-
mension dim E ψ
i ( ω ) = d i , i = 1 , p (so-c al le d Osele dets splitting) with the fol lowing
pr op erties
27

i) the subsp ac es E ψ
i ar e θ -invariant, i.e. ( D 0 ψ n,ω ) E ψ
i ( ω ) = E ψ
i ( θ n ω ) ;
ii) lim
n →±∞
1
n log | ( D 0 ψ n,ω ) v | = λ i ⇐ ⇒ v ∈ E ψ
i ( ω ) \{ 0 } .
Pr o of. Equalit y ii) for ψ holds b ecause of [35], Theorem 3.1, where τ and T
should b e substituted b y θ and D 0 ψ 1 ,ω resp ectiv ely . Note that the in tegrabilit y
conditions of the theorem hold b ecause of Assumptions 1 and 2, and ergo dicit y
of θ . Finally , i) is a trivial consequence of ii).
Theorem 2.3.5. L et ψ b e a two-side d tr anslation invariant r andom dynamic al
system on a pr ob ability sp ac e ( ˆ
Ω Z , B ( ˆ
Ω) Z , ˆ ν Z ) , which satisfies Assumptions 1 and
2. Then ther e exist numb ers λ 1 > λ 2 > . . . > λ p and a θ -invariant me asur able set
ˆ
Ω Z
1 with ˆ ν Z ( ˆ
Ω Z
1 )=1 , such that for al l ω ∈ ˆ
Ω Z
1 ther e exists a me asur able splitting
R d = E ψ
1 ( ω ) ⊕ . . . ⊕ E ψ
p ( ω )
of R d over ˆ
Ω Z
1 into r andom subsp ac es E ψ
i ( ω ) (so-c al le d Osele dets sp ac es) with di-
mension dim E ψ
i ( ω ) = d i , i = 1 , p (so-c al le d Osele dets splitting) with the fol lowing
pr op erties
i) the subsp ac es E ψ
i ar e θ -invariant, i.e. ( D 0 ψ n,ω ) E ψ
i ( ω ) = E ψ
i ( θ n ω ) ;
ii) lim
n →±∞
1
n log | ( D 0 ψ n,ω ) v | = λ i ⇐ ⇒ v ∈ E ψ
i ( ω ) \{ 0 } .
Pr o of. It suffices to pro v e the theorem for another t w o-sided RDS ψ on a proba-
bilit y space ( Ω Z , B (Ω) Z , ν Z ) generated b y i.i.d. mappings
. . . ψ 1 ,θ − 1 ω − ψ 1 ,θ − 1 ω (0) , ψ 1 ,ω − ψ 1 ,ω (0) , ψ 1 ,θ ω − ψ 1 ,θ ω (0) . . . ,
b ecause random matrices D 0 ψ 1 ,ω and D 0 ψ 1 ,ω ha v e the same distribution. F or ψ
the theorem holds b ecause of Theorem 2.3.4.
Remark 2.3.2. One c an ask if two-side d and one-side d systems b ase d on the same
pr ob ability sp ac e (say ( ˜
Ω , B ( ˜
Ω) , ˜ ν ) ) have the same Lyapunov sp e ctrum. In fact it
is true and is a trivial c or ol lary of the asymptotic b ehaviour of 1
n log | ( D 0 ψ n,ω ) v |
when n → + ∞ .
F or a t w o-sided RDS ψ as in Theorem 2.3.4 or Theorem 2.3.5 define i 0 :=
max { i ∈ N : λ i > 0 } . F or ω ∈ ˆ
Ω Z
1 define the follo wing linear subspaces b y
S ψ ( ω ) := E ψ
p ( ω ) ⊕ . . . ⊕ E ψ
i 0 +1 ( ω ) , (2.10)
U ψ ( ω ) := E ψ
1 ( ω ) ⊕ . . . ⊕ E ψ
i 0 ( ω ) . (2.11)
F rom no w on w e will abbreviate E ψ
i , V ψ
i , S ψ , and U ψ b y E i , V i , S , and U
resp ectiv ely if there is no risk of am biguit y .
Remark 2.3.3. Note that the subsp ac es S and U ar e define d with r esp e ct to the
origin. In the same way we c an define subsp ac es S x and U x with r esp e ct to x , i.e.
c onsidering sp atial derivatives at x .
28

Lemma 2.3.1. L et ψ b e a two-side d RDS on a pr ob ability sp ac e (Ω Z , B (Ω) Z , ν Z ) ,
which satisfies Assumption 1 and 2, has Lyapunov exp onents λ 1 , . . . , λ p with mul-
tiplicities d 1 , . . . , d p , and also has the fixe d origin, i.e. ψ 1 ,ω (0) = 0 , ∀ ω . Then
ther e exists an invariant set of a ful l me asur e Ω Z
1 such that for every ω ∈ Ω Z
1 we
have
lim
n →∞
1
n log | det[ D 0 ψ n,ω | U ( ω ) ] | =
p
X
i =1
d i λ +
i .
Pr o of. Theorem 2.3.5 implies
lim sup
n →∞
1
n log | det[ D 0 ψ n,ω | S ( ω ) ] | ≤
p
X
i =1
d i λ i −
p
X
i =1
d i λ +
i , (2.12)
and
lim sup
n →∞
1
n log | det[ D 0 ψ n,ω | U ( ω ) ] | ≤
p
X
i =1
d i λ +
i . (2.13)
Moreo v er,
lim inf
n →∞
1
n log | det[ D 0 ψ n,ω | U ( ω ) ] |
≥ lim inf
n →∞
1
n log | det[ D 0 ψ n,ω ] / det[ D 0 ψ n,ω | S ( ω ) ] |
≥ lim inf
n →∞
1
n log | det[ D 0 ψ n,ω ] |
− lim sup
n →∞
1
n log | det[ D 0 ψ n,ω | S ( ω ) ] |
(2 . 12)
≥ lim inf
n →∞
1
n log | det[ D 0 ψ n,ω ] | −
p
X
i =1
d i λ i +
p
X
i =1
d i λ +
i =
p
X
i =1
d i λ +
i ,
where the last equalit y holds b y F ursten b erg-Kesten Theorem, see [1], Theo-
rem 3.3.3. This together with (2.13) completes the pro of of the lemma.
Remark 2.3.4. L et G b e a subset of R d . F r om now on we wil l sometimes omit
br ackets and abbr eviate ψ n,ω G inste ad of ψ n,ω ( G ) if ther e is no risk of ambiguity.
29

Chapter 3
Definition of En trop y
Kifer in [18] successfully defined the notion of en trop y for RDS (in the case of
in v arian t probabilit y measure). Ho w ev er, it turns out that some basic prop erties
of en trop y in Kifer’s setting, suc h as stabilit y with resp ect to the sequence of
appro ximating partitions (see [18], Corollary I I.2.1), can not b e pro v ed in the
same w a y as in deterministic dynamics. T o resolve the problem, he defines en trop y
of the resp ectiv e sk ew pro duct and then connects the en trop y of the RDS with the
en trop y of the sk ew pro duct. The whole pro cedure is describ ed in [18], Section
2.1.
The definition of en trop y in our case is ev en a more c hallenging task b ecause
TIRDSs ha v e no in v arian t probabilit y measure, but the Leb esgue measure, whic h
is an infinite in v arian t measure. T o define en trop y in our case, w e consider only
p erio dic partitions and lo ok at the dynamics of the system only in the fixed cub e
[0 , 1) d . A t the same time w e also, follo wing Kifer, define the notion of en trop y of
the sk ew pro duct and, as in [18], connect the defined en trop y with the en tropy of
one-sided TIRDSs.
Let us note that w e imp ose on the partitions of the state space of the sk ew
pro duct certain sp ecific assumptions, see Definition 3.2.2. These assumptions, in
particular, imply translation in v ariance of the partitions in some sense, whic h lets
us sho w in v ariance of the en trop y with resp ect to the sk ew pro duct and establish
the desired prop erties of the en trop y , see Section 3.3.
Finally , in Section 3.4 w e define en trop y for v olume preserving t w o-sided
TIRDSs, whic h is basically the adaptation of the argumen ts from Section 3.3.
Note that here w e restrict ourselv es to the v olume preserving case b ecause later
(in the pro of of P esin’s form ula) it will b e imp ortan t to ha v e the in v ariance of
conditional measures (see Theorem 3.4.1) with resp ect to the whole randomness,
i.e. with resp ect to R d × B ( ˆ
Ω) Z , and in this case the preserv ation of the v olume
is essen tial.
30

3.1 Definition of En trop y of P artitions
W e pro vide a short in tro duction to en trop y and conditional en trop y of partitions,
mainly follo wing [25], Chapter 0, § 3, and also [43], Section 6.2.
W e put 0 log 0 := 0. No w let us start the section with the definition of the
en trop y of a partition.
Definition 3.1.1. L et P b e a c ountable me asur able p artition of the pr ob ability
sp ac e (Ω , F , P ) . L et G b e a sub- σ -algebr a of F . The c onditional entr opy of P
given G is the numb er
H P ( P |G ) := − Z
Ω X
A ∈P
P ( A |G ) log P ( A |G ) d P ∈ [0 , ∞ ] .
The numb er
H P ( P ) := − X
A ∈P
P ( A ) log P ( A ) ∈ [0 , ∞ ]
is c al le d the entr opy of P .
Note that since 0 log 0 = 0, the sums in the latter definition alw a ys mak e
sense.
F or t w o partitions P 1 and P 2 w e denote their common refinemen t b y P 1 ∨
P 2 . Note also that σ ( P ) denotes the σ -algebra generated b y the elemen ts of P .
Finally , P 1 ≺ P 2 means that σ ( P 1 ) ⊂ σ ( P 2 ).
No w w e pro vide some basic prop erties of the defined en trop y .
Lemma 3.1.1. L et P 1 and P 2 b e c ountable me asur able p artitions of the pr ob a-
bility sp ac e (Ω , F , P ) . L et further G , G 0 ⊂ F b e σ -algebr as, and f : (Ω , F , P ) →
(Ω , F , P ) b e a me asur e pr eserving me asur able map. Then the fol lowing holds true
1. H P ( P 1 |G ) ≥ 0 .
2. H P ( P 1 ∨ P 2 |G ) = H P ( P 1 |G ) + H P ( P 2 | σ ( P 1 ) ∨ G ) .
3. H P ( P 1 ∨ P 2 ) = H P ( P 1 ) + H P ( P 2 | σ ( P 1 )) .
4. P 1 ≺ P 2 implies H P ( P 1 |G ) ≤ H P ( P 2 |G ) .
6. P 1 ≺ P 2 implies H P ( P 1 ) ≤ H P ( P 2 ) .
7. H P ( P 1 ) ≥ H P ( P 1 |G ) .
8. G ⊂ G 0 implies H P ( P 1 |G ) ≥ H P ( P 1 |G 0 ) .
9. H P ( P 1 ∨ P 2 |G ) ≤ H P ( P 1 |G ) + H P ( P 2 |G ) .
10. H P ( P 1 ∨ P 2 ) ≤ H P ( P 1 ) + H P ( P 2 ) .
11. H P ( f − 1 P 1 | f − 1 G ) = H P ( P 1 |G ) .
12. H P ( f − 1 P 1 ) = H P ( P 1 ) .
Pr o of. The same as in [18], Remark I I.1.1 and [18], Lemma I I.1.2. Note that in
[18] partitions are finite, but it do es not alter the pro of.
No w w e form ulate a trivial upp er b ound on the en trop y of a partition.
31

Lemma 3.1.2. The (c onditional) entr opy of a p artition is at most the lo garithm
of its c ar dinality.
Pr o of. The same as in [18], Corollary I I.1.1.
F rom no w on in this c hapter consider a one-sided TIRDS ψ on a probabilit y
space ( ˆ
Ω N , B ( ˆ
Ω) N , ˆ ν N ) (the only exception is Section 3.4, where w e discuss en trop y
for t w o-sided systems). Recall that µ is the Leb esgue measure on R d . F urther, for
m, n ∈ R , m<n denote µ m,n := µ | [ m,n ) d the restriction of µ to the cub e [ m, n ) d .
Recall that M + := µ × ˆ ν N is in v arian t for Θ + , see Prop osition 2.2.2. Finally ,
denote
M +
0 , 1 := M + | [0 , 1) d × ˆ
Ω ,
where the measure M +
S is the restriction of the measure M + to the subset S .
Note that µ 0 , 1 is a probabilit y measure, so en tropies H µ 0 , 1 (with resp ect to
( R d , B ( R d ) , µ 0 , 1 )) and H M +
0 , 1 (with resp ect to ( R d × ˆ
Ω N , B ( R d ) × B ( ˆ
Ω) N , M +
0 , 1 ))
p erfectly mak e sense.
3.2 Class of 1 -p erio dic in Distribution Sets
In this section w e presen t a certain class of subsets of R d , whic h enlarges the
class of 1-p erio dic sets (the definition see b elow) b y certain random sets (i.e. b y
certain subsets of R d × ˆ
Ω N ), whic h w e need for the pro of of P esin’s form ula. No w
let us define the notion of 1-p erio dic set and 1-p erio dic partition.
Definition 3.2.1. A set A ⊂ R d is c al le d 1 -p erio dic, if for every v ∈ Z d we have
A + v = A.
Definition 3.2.2. A c ountable me asur able p artition P of R d is c al le d 1 -p erio dic
if every element of P is a 1 -p erio dic set.
Denote b y A tr the class of elemen ts of B ( R d ) × B ( ˆ
Ω) N of the form A × ˆ
Ω N ,
where A is a 1-p erio dic set.
F urther, denote
B tr := ( m
\
i =1
Θ − n i
+ A i     
A i ∈ A tr , n i ∈ N 0 , m ∈ N ) .
Finally , let us define the notion of 1-p erio dic in distribution set and 1-p erio dic
in distribution partition.
Definition 3.2.3. A set B ⊂ R d × ˆ
Ω N is c al le d 1 -p erio dic in distribution, if
B ∈ B tr .
Definition 3.2.4. A c ountable me asur able p artition P of R d × ˆ
Ω N is c al le d 1 -
p erio dic in distribution if every element of P is a 1-p erio dic in distribution set.
32

3.3 Metric En trop y of T ranslation In v arian t Ran-
dom Dynamical Systems
W e start the section with a crucial result: w e pro v e in v ariance of the (conditional)
measure M +
0 , 1 of a 1-p erio dic in distribution partition with resp ect to the sk ew
pro duct Θ + . This result lets us define en trop y of ψ and then dev elop en trop y
theory (with resp ect to 1-p erio dic in distribution partitions) in a similar wa y to
the case of standard settings.
Recall that in this c hapter one-sided TIRDS ψ is defined on a probabilit y
space ( ˆ
Ω N , B ( ˆ
Ω) N , ˆ ν N ).
Theorem 3.3.1. L et A b e a me asur able 1 -p erio dic in distribution set. Then we
have
M +
0 , 1 (Θ − 1
+ A | Θ − 1
+ ( R d × B ( ˆ
Ω) N )) = M +
0 , 1 ( A | R d × B ( ˆ
Ω) N ) ◦ Θ + , (3.1)
and
M +
0 , 1 (Θ − 1
+ A ) = M +
0 , 1 ( A ) . (3.2)
Pr o of. Let us sho w (3.1). F or a measurable subset A of R d × ˆ
Ω N denote b y A ω
the restriction of A to R d × { ω } .
Then b y translation in v ariance of ψ , ev aluation of the left hand side for giv en
ω pro vides
LH S ( ω ) = E [ µ 0 , 1 ( ψ − 1
1 ,ω ( A θ + ω )) | Θ − 1
+ ( R d × B ( ˆ
Ω) N )]
= E " X
v ∈ Z d
µ 0 , 1 ( ψ − 1
1 ,ω ( A θ + ω ∩ ( v + [0 , 1) d ))) | Θ − 1
+ ( R d × B ( ˆ
Ω) N ) #
= E " X
v ∈ Z d
µ [0 , 1) d − v ( ψ − 1
1 ,ω ( A θ + ω ∩ [0 , 1) d )) | Θ − 1
+ ( R d × B ( ˆ
Ω) N ) #
= E h µ ( ψ − 1
1 ,ω ( A θ + ω ∩ [0 , 1) d )) | Θ − 1
+ ( R d × B ( ˆ
Ω) N ) i =: I .
No w b y in v ariance of M + , see Prop osition 2.2.2, w e ha v e
I = µ ( A θ + ω ∩ [0 , 1) d ) = µ 0 , 1 ( A θ + ω ) = RH S ( ω ) ,
as required. The pro of of (3.2) is the same (in fact is ev en easier).
No w w e pro v e in v ariance of en trop y of a 1-p erio dic in distribution partition
with resp ect to the sk ew pro duct Θ + .
Theorem 3.3.2. L et P b e a c ountable me asur able 1 -p erio dic in distribution p ar-
tition of R d × ˆ
Ω N with finite entr opy, i.e. H M +
0 , 1 ( P ) < ∞ . Then we have
33

H M +
0 , 1 (Θ − 1
+ P | (Θ − 1
+ ( R d × B ( ˆ
Ω) N )) = H M +
0 , 1 ( P | R d × B ( ˆ
Ω) N ) , (3.3)
and
H M +
0 , 1 (Θ − 1
+ P ) = H M +
0 , 1 ( P ) . (3.4)
Pr o of. It suffices to sho w the theorem for finite partitions, b ecause en trop y of
an infinite partition { C 1 , C 2 , . . . } with finite en trop y can b e appro ximated b y
en tropies of finite partitions { C 1 ,..., C n } , when n go es to infinit y . F or finite
partitions the theorem is a direct corollary of Theorem 3.3.1.
The follo wing lemma defines metric en trop y of the sk ew pro duct.
Lemma 3.3.1. L et ξ b e a c ountable me asur able 1 -p erio dic in distribution p artition
with finite entr opy. Then ther e exist
h M + (Θ + , ξ | R d × B ( ˆ
Ω) N ) := lim
n →∞
1
n H M +
0 , 1 n − 1
_
i =0
Θ − i
+ ξ | R d × B ( ˆ
Ω) N ! (3.5)
and
h M + (Θ + , ξ ) := lim
n →∞
1
n H M +
0 , 1 n − 1
_
i =0
Θ − i
+ ξ ! . (3.6)
The numb ers
h M + (Θ + | R d × B ( ˆ
Ω) N ) := sup
ξ
h M +
0 , 1 (Θ + , ξ | R d × B ( ˆ
Ω) N )
and
h M + (Θ + ) := sup
ξ
h M + (Θ + , ξ )
ar e c al le d metric entr opy of Θ + given r andomness and metric entr opy of Θ +
r esp e ctively. The supr emum is taken over al l finite 1 -p erio dic in distribution
p artitions.
Pr o of. Note that the pro of of the theorem is similar to the pro of of Theorem
I I.1.1 from [18].
Denote b y
a n := H M +
0 , 1 n − 1
_
i =0
Θ − i
+ ξ     
R d × B ( ˆ
Ω) N !
Then b y Statemen t 9 from Lemma 3.1.1 w e ha v e
34

a n + m = H M +
0 , 1 n + m − 1
_
i =0
Θ − i
+ ξ     
R d × B ( ˆ
Ω) N !
= H M +
0 , 1 n − 1
_
i =0
Θ − i
+ ξ     
R d × B ( ˆ
Ω) N !
+ H M +
0 , 1 Θ − n
+
m − 1
_
i =0
Θ − i
+ ξ     
R d × B ( ˆ
Ω) N !
= H M +
0 , 1 Θ − n
+
m − 1
_
i =0
Θ − i
+ ξ     
R d × B ( ˆ
Ω) N ! + a n := I ;
no w Statemen t 8 from Lemma 3.1.1 implies
I ≤ H M +
0 , 1 Θ − n
+
m − 1
_
i =0
Θ − i
+ ξ     
Θ − n
+ ( R d × B ( ˆ
Ω) N ) ! + a n = a m + a n ,
where the last equalit y holds b y Statemen t 11 from Lemma 3.1.1. Th us, w e obtain
a m + n ≤ a m + a n , m, n ∈ N . (3.7)
Then (3.7) together with subadditivit y argumen ts pro vides the existence of the
left hand side of (3.5). Moreo v er, (3.7) implies for all p ositiv e in tegers n inequalit y
a n ≤ na 1 , whic h means that the left hand side of (3.5) is finite. Analogously , one
can sho w that the left hand side of (3.6) exists and finite.
Lemma 3.3.2. If ξ = { A 1 , . . . , A k } and η = { B 1 , . . . , B m } ar e finite me asur able
p artitions of R d and ˆ
Ω N r esp e ctively, then
E H µ 0 , 1 n − 1
_
i =0
ψ − 1
i,ω ξ ! = H M +
0 , 1 n − 1
_
i =0
Θ − i
+ ( ξ × η ) | R d × B ( ˆ
Ω) N ! ,
wher e ξ × η := { A i × B j : 1 ≤ i ≤ k , 1 ≤ j ≤ m } .
Pr o of. The pro of is similar to a part of the pro of of Theorem I I.1.4 (i) from [18].
Fix n ∈ N . F urther, fix sets i 0 , . . . , i n − 1 , j 0 ,...j n − 1 ∈ N . Then w e ha v e
M +
0 , 1 n − 1
\
k =0
Θ − k
+ ( A i k × B j k ) | R d × B ( ˆ
Ω) N ! = M +
0 , 1 ( C n | R d × B ( ˆ
Ω) N ) ,
where C n :=
n − 1
T
k =0 { ( x, ω ) : ψ k ,ω ( x ) ∈ A i k , and θ k ∈ B j k } . F urther
M +
0 , 1 ( C n | R d × B ( ˆ
Ω) N ) = µ 0 , 1 n − 1
\
k =0
ψ − 1
k ,ω ( A i k ) ! 1 B i 0 ( ω ) . . . 1 B i n − 1 ( θ n − 1 ω ) .
35

Th us
M +
0 , 1 n − 1
\
k =0
Θ − k
+ ( A i k × B j k ) | R d × B ( ˆ
Ω) N ! = µ 0 , 1 n − 1
\
k =0
ψ − 1
k ,ω ( A i k ) !
× 1 B i 0 ( ω ) . . . 1 B i n − 1 ( θ n − 1 ω ) ,
and the latter equalit y directly implies the lemma.
Remark 3.3.1. L emma 3.3.2 implies that for a finite 1 -p erio dic me asur able p ar-
tition ξ = { A 1 , . . . , A k } of R d and a finite me asur able p artition η = { B 1 , . . . , B m }
of ˆ
Ω N , the numb er
H M +
0 , 1 n − 1
_
i =0
Θ − i
+ ( ξ × η ) | R d × B ( ˆ
Ω) N !
do es not dep end on η . Ther efor e, we also c an define h M + (Θ + , ξ × η | R d × B ( ˆ
Ω) N )
(which is e qual to h M + (Θ + , ξ × ˆ
Ω N | R d × B ( ˆ
Ω) N ) ).
No w w e are ready for the main definition of the thesis. The follo wing theorem
pro vides the definition of en trop y for TIRDSs.
Theorem 3.3.3. L et P b e a finite 1 -p erio dic p artition of R d . Then ther e exists
h µ ( ψ , P ) := lim
n →∞
E 1
n H µ 0 , 1 n − 1
_
i =0
ψ − 1
i,ω P ! .
The numb er
h µ ( ψ ) := sup
P
h µ ( ψ , P )
is c al le d metric entr opy of ψ . The supr emum is taken over al l finite 1 -p erio dic
p artitions.
Pr o of. Because of Lemma 3.3.1 and Lemma 3.3.2 (with η = ˆ
Ω N ) w e kno w that
h µ ( ψ , P ) = h M + (Θ + , P × ˆ
Ω N | R d × B ( ˆ
Ω) N ) , (3.8)
and the righ t hand side exists b ecause of Lemma 3.3.1. The lemma is pro v en.
No w w e dev elop en trop y theory as in [18], Section 2.1. W e start from the
follo wing result.
Lemma 3.3.3. If ξ = { A 1 , . . . , A k } and η = { B 1 , . . . , B m } ar e finite me asur able
p artitions of R d and ˆ
Ω N r esp e ctively. F urther, let ξ b e 1 -p erio dic. Then we have
h µ ( ψ , ξ ) = h M + (Θ + , ξ × η | R d × B ( ˆ
Ω) N ) ,
wher e ξ × η := { A i × B j : 1 ≤ i ≤ k , 1 ≤ j ≤ m } .
36

Pr o of. This is a straigh t consequence of Lemma 3.3.1, Lemma 3.3.2 and Theorem
3.3.3.
Lemma 3.3.4. We have
h µ ( ψ ) = h M + (Θ + | R d × B ( ˆ
Ω) N ) .
Pr o of. Because of Remark 3.3.1 the pro of can b e done in the same w a y as in
Kifer’s b o ok, see [18], Theorem I I.1.4 (ii).
F or a p ositiv e in teger n denote b y ψ n the random dynamical system with
rescaled time, i.e. ψ n
m,ω ( x ) := ψ nm,ω ( x ). Indeed, then it is easy to c hec k that ψ n
is a one-sided TIRDS.
Lemma 3.3.5. F or any n ∈ N we have
h µ ( ψ n ) = nh µ ( ψ ) .
Pr o of. The same as in [18], Lemma I I.1.4.
Lemma 3.3.6. If P 1 , P 2 , . . . is a se quenc e of finite 1 -p erio dic p artitions such
that lim n →∞ diam ( P n ∩ [0 , 1) d )=0 wher ein diam ( P ) = sup C ∈P diam ( C ) is the
diameter of the p artition P . Then
h µ ( ψ ) = lim
n →∞ h µ ( ψ , P n ) .
Pr o of. Because of Lemma 3.3.4 and (3.8) it suffices to sho w that
lim
n →∞ h M + (Θ + , P n × ˆ
Ω N | R d × B ( ˆ
Ω) N ) = h M + (Θ + | R d × B ( ˆ
Ω) N ) ,
and this equalit y can b e obtained in the same w a y as Lemma 3.3.4.
Remark 3.3.2. It is natur al to exp e ct that one c an alternatively define entr opies
of Θ + using the dynamics of the whole sp ac e. Now let us b e mor e pr e cise. L et ξ
b e a finite 1 -p erio dic in distribution p artition. Denote
M +
c := 1
(2 c ) d M + | [ − c,c ) d × ˆ
Ω , and M +
0 , 1 ,v := 1
(2 c ) d M + | ([0 , 1) d + v ) × ˆ
Ω ,
wher e v ∈ R d . Then M +
c and M +
0 , 1 ,v ar e pr ob ability me asur es, so entr opies H M +
c
(with r esp e ct to ( R d × ˆ
Ω N , B ( R d ) × B ( ˆ
Ω) N , M +
c and ( R d × ˆ
Ω N , B ( R d ) × B ( ˆ
Ω) N , M +
0 , 1 ,v )
p erfe ctly make sense.
Then one c ould try to define entr opy of Θ + with r esp e ct to ξ in one of the
fol lowing ways:
h 0
M + (Θ + , ξ | R d × B ( ˆ
Ω) N ) := lim
n →∞ lim
N →∞
1
n H M +
N n − 1
_
i =0
Θ − i
+ ξ | R d × B ( ˆ
Ω) N ! ,
37

or
h 00
M + (Θ + , ξ | R d × B ( ˆ
Ω) N ) := lim
N →∞ lim
n →∞
1
n H M +
N n − 1
_
i =0
Θ − i
+ ξ | R d × B ( ˆ
Ω) N ! ;
mor e over, it is natur al to exp e ct that the right hand sides of the latter two defini-
tions c oincide with h M + (Θ + , ξ | R d × B ( ˆ
Ω) N ) . However, it is not cle ar if the limits
in these definitions exist and if the right hand sides (in the c ase of existenc e)
c oincide with the define d entr opy. However, one c an show that
h M + (Θ + , ξ | R d × B ( ˆ
Ω) N ) ≤ lim inf
n →∞ lim inf
N →∞
1
n H M +
N n − 1
_
i =0
Θ − i
+ ξ | R d × B ( ˆ
Ω) N ! , (3.9)
and
h M + (Θ + , ξ | R d × B ( ˆ
Ω) N ) ≤ lim inf
N →∞ lim inf
n →∞
1
n H M +
N n − 1
_
i =0
Θ − i
+ ξ | R d × B ( ˆ
Ω) N ! .
(3.10)
Inde e d, by Jensen ’s ine quality, applie d to g ( x ) := − x log x , we have ∀ A ∈ P
X
v ∈ Z d ∩ [ − N ,N ) d Z
R d × ˆ
Ω Z
g ( M +
0 , 1 ,v ( A |G )) d M +
0 , 1 ,v ≤ (2 N ) d Z
R d × ˆ
Ω Z
g ( M +
N ( A |G )) d M +
N ,
wher e G := R d × B ( ˆ
Ω) N . This yields
X
v ∈ Z d ∩ [ − N ,N ) d
H M +
0 , 1 ,v  P | R d × B ( ˆ
Ω) N  ≤ (2 N ) d H M +
N  P | R d × B ( ˆ
Ω) N  ,
Final ly, sinc e for every finite 1 -p erio dic in distribution p artition P and ∀ v , w ∈ Z d
we have
H M +
0 , 1 ,v  P | R d × B ( ˆ
Ω) N  = H M +
0 , 1 ,w  P | R d × B ( ˆ
Ω) N  ,
we obtain
H M +
0 , 1  P | R d × B ( ˆ
Ω) N  ≤ H M +
N  P | R d × B ( ˆ
Ω) N  .
The latter ine quality implies (3.9) and (3.10) as r e quir e d.
38

3.4 En trop y for t w o-sided systems
Consider a t w o-sided TIRDS ψ defined on a probabilit y space ( ˆ
Ω Z , B ( ˆ
Ω) Z , ˆ ν Z ).
F urther, let ψ b e v olume preserving, i.e. the measure M := µ × ˆ ν Z (defined on
R d × ˜
Ω Z ) is in v arian t for the sk ew pro duct Θ. No w w e briefly provide the same
pro cedure as in Section 3.3, but for t w o-sided systems.
Denote b y A Z
tr the class of elemen ts of B ( R d ) × B ( ˆ
Ω) Z of the form A × ˆ
Ω Z ,
where A is a 1-p erio dic set. F urther, denote
B Z
tr := ( m
\
i =1
Θ − n i A i     
A i ∈ A tr , n i ∈ N 0 , m ∈ N ) ;
w e sa y that a partition P is called 1-p erio dic in distribution if P ∈ B Z
tr . Define
M 0 , 1 := M | [0 , 1) d × ˆ
Ω . No w let us form ulate an analogue of Theorem 3.3.1 for t w o-
sided systems.
Theorem 3.4.1. L et A ∈ B Z
tr . Then we have
M 0 , 1 (Θ − 1 A | Θ − 1 ( R d × B ( ˆ
Ω) Z )) = M 0 , 1 ( A | R d × B ( ˆ
Ω) Z ) ◦ Θ , (3.11)
and
M 0 , 1 (Θ − 1 A ) = M 0 , 1 ( A ) . (3.12)
Pr o of. Let us sho w (3.11). Indeed, b y translation in v ariance of ψ , ev aluation of
the left hand side for giv en ω pro vides
LH S ( ω ) = E [ µ 0 , 1 ( ψ − 1
1 ,ω ( A θ ω )) | R d × B ( ˆ
Ω) Z ]
= E " X
v ∈ Z d
µ 0 , 1 ( ψ − 1
1 ,ω ( A θ ω ∩ ( v + [0 , 1) d ))) | R d × B ( ˆ
Ω) Z #
= E " X
v ∈ Z d
µ [0 , 1) d − v ( ψ − 1
1 ,ω ( A θ ω ∩ [0 , 1) d )) | R d × B ( ˆ
Ω) Z #
= E h µ ( ψ − 1
1 ,ω ( A θ ω ∩ [0 , 1) d )) | R d × B ( ˆ
Ω) Z i =: I .
No w b y in v ariance of M w e ha v e
I = µ ( A θ ω ∩ [0 , 1) d )) = µ 0 , 1 ( A θ ω ) = RH S ( ω ) ,
as required. The pro of of (3.12) is the same (in fact is ev en easier).
Remark 3.4.1. A l l the r esults fr om Se ction 3.3 also hold for two-side d systems
with r esp e ct to B Z
tr (and R d × B ( ˆ
Ω) Z ). T o pr ove them, it suffic es to r ep e at the
ar guments fr om Se ction 3.3. In p articular, entr opy h µ ( ψ ) is wel l define d. F or
the sake of c ompleteness we formulate the definition of h µ ( ψ ) . Note that it is the
same as in The or em 3.3.3.
39

Definition 3.4.1. L et P b e a finite 1 -p erio dic p artition of R d . Then ther e exists
h µ ( ψ , P ) := lim
n →∞
E 1
n H µ 0 , 1 n − 1
_
i =0
ψ − 1
1 ,ω P ! .
The numb er
h µ ( ψ ) := sup
P
h µ ( ψ , P )
is c al le d metric entr opy of ψ . The supr emum is taken over al l finite 1 -p erio dic
p artitions.
40

Chapter 4
Ruelle’s Inequalit y for
T ranslation In v arian t Random
Dynamical Systems
Ruelle’s inequalit y is a relation b et w een en trop y of a DS and its Ly apuno v
exp onen ts. Namely , it states that the en trop y is less than or equal to the sum of
the p ositiv e Ly apuno v exp onen ts of the system. The form ula w as first established
b y Ruelle for deterministic DSs acting on a compact Riemannian manifold, see
[35]. Later differen t authors pro v ed Ruelle’s inequalit y in differen t settings, see
e.g. the discussion of the inequalit y in the in tro duction. Nev ertheless, to the
b est of our kno wledge, this form ula has nev er b een established b efore for systems
without in v arian t probabilit y measure. The problem is that en trop y in this case
is ill-p osed. It is ill-p osed for TIRDSs as w ell, b ecause these systems do not ha v e
an in v arian t probabilit y measure, but the Leb esgue measure, whic h is an infinite
in v arian t measure.
In this c hapter w e use the definition of en trop y as in Theorem 3.3.3. This
definition lets us pro v e Ruelle’s inequalit y for one-sided TIRDSs rep eating the
standard argumen ts. In this c hapter w e follo w closely v an Bargen, see [42] (and
also [43], Section 6.3), that pro v ed Ruelle’s inequalit y for certain sto c hastic flo ws
on R d .
4.1 Main Result
In this section w e pro v e the follo wing theorem
Theorem 4.1.1. L et ψ b e a one-side d TIRDS define d on a pr ob ability sp ac e
( ˆ
Ω N , B ( ˆ
Ω) N , ˆ ν N ) , which satisfies Assumptions 1 and 2, and has Lyapunov exp o-
nents λ 1 , . . . , λ p with multiplicities d 1 , . . . , d p . Then we have
h µ ( ψ ) ≤
p
X
i =1
d i λ +
i .
41

Note that in the same w a y , as w e pro v e Theorem 4.1.1, w e can pro v e Ru-
elle’s inequalit y for t w o-sided v olume preserving TIRDSs. Note that this will be
imp ortan t only for Remark 5.1.1. No w let us form ulate the resp ectiv e result.
Prop osition 4.1.1. L et ψ b e a two-side d TIRDS on a pr ob ability sp ac e
( ˆ
Ω Z , B ( ˆ
Ω) Z , ˆ ν Z ) , which satisfies Assumptions 1 and 2, and has Lyapunov exp o-
nents λ 1 , . . . , λ p with multiplicities d 1 , . . . , d p . F urther, let the me asur e M =
µ × ˆ ν Z b e invariant for the skew pr o duct Θ . Then we have
h µ ( ψ ) ≤
p
X
i =1
d i λ +
i .
Theorem 4.1.1 immediately imply Ruelle’s inequalit y for TIBFs. Let us for-
m ulate the result precisely .
Corollary 4.1.1. L et φ b e a TIBF and ϕ b e the r esp e ctive one-side d TIRDS on
a pr ob ability sp ac e ( ˆ
Ω N , B ( ˆ
Ω) N , ˆ ν N ) (i.e. is c onstructe d as in Se ction 2.2), which
has Lyapunov exp onents λ 1 , . . . , λ p with multiplicities d 1 , . . . , d p . Then we have
h µ ( ϕ ) ≤
p
X
i =1
d i λ +
i .
No w let us come bac k to Theorem 4.1.1. Note that the pro of of the theorem
is similar to the pro of of Theorem 4.1 in [42].
F or a r > 0 and S ⊂ R d denote b y B r ( S ) the r -neigh b ourho o d of S , i.e.
B r ( S ) := [
x ∈ S
B ( x, r );
no w w e form ulate the follo wing purely deterministic result from geometry .
Lemma 4.1.1. L et A : R d → R d b e a line ar mapping and let R d b e e quipp e d
with the usual Euclide an norm |·| . L et further δ 1 ( A ) ≥ . . . ≥ δ d ( A ) denote the
singular values of A . Then ther e exists a c onstant C ( d ) which only dep ends on
d such that for any  > 0 the numb er of disjoint b al ls with r adius 
2 , which c an
interse ct B 2  ( AB  (0)) do es not exc e e d
C ( d )
d
Y
u =1
( δ u ( A ) ∨ 1) .
Pr o of. See [21], Lemma I I.2.3.
Denote
P k := { v + [0 , 2 − k ) d + Z d | v ∈ 2 − k Z d ∩ [0 , 1) d } ;
finally , w e state the follo wing lemma
42

Lemma 4.1.2. F or every p ositive inte gers k and n we have
h µ ( ψ n , P k ) ≤ E H µ 0 , 1 ( ψ − 1
n,ω P k | σ ( P k )) .
Pr o of. By Prop osition 2.2.2, the pro of is the same as in [4], Corollary 1.
4.2 Pro of of Ruelle’s inequalit y
In this section w e pro v e Theorem 4.1.1.
Fix n ∈ N . F or k ∈ N define the set ˆ
Ω N
k of ω for whic h w e ha v e the follo wing
statemen t: for an y  ≤ √ d 2 − k and x, y ∈ [0 , 1) d the inequalit y | x − y | ≤  implies
| ψ n,ω ( x ) − ψ n,ω ( y ) − ( D x ψ n,ω )( x − y ) | ≤ . (4.1)
Note that for k ∈ N w e ha v e trivial inclusions
ˆ
Ω N
k ⊂ ˆ
Ω N
k +1 , k ∈ N , and ˆ
Ω N = ∞
[
k =1
ˆ
Ω N
k .
W e ha v e
nh µ ( ψ ) = h µ ( ψ n ) = lim
k →∞ h µ ( ψ n , P k ) ≤ lim inf
k →∞
E H µ 0 , 1 ( ψ − 1
n,ω P k | σ ( P k )) ,
where the last inequalit y holds b y Lemma 4.1.2. No w let us en umerate the ele-
men ts of P k as P k, 1 ,..., P k , 2 k d . Then
E H µ 0 , 1 ( ψ − 1
n,ω P k | σ ( P k )) = − E
2 kd
X
i =1
µ 0 , 1 ( P k ,i )
×
2 kd
X
j =1
µ 0 , 1 ( ψ − 1
n,ω P k ,j | σ ( P k ,i )) log µ 0 , 1 ( ψ − 1
n,ω P k ,j | σ ( P k ,i ))
= − E 2 − k d
2 kd
X
i =1
2 kd
X
j =1
µ 0 , 1 ( ψ − 1
n,ω P k ,j | σ ( P k ,i )) log µ 0 , 1 ( ψ − 1
n,ω P k ,j | σ ( P k ,i )) .
No w w e can estimate
43

nh µ ( ψ ) ≤ lim inf
k →∞
E H µ 0 , 1 ( ψ − 1
n,ω P k | σ ( P k ))
≤ lim sup
k →∞ − E 1 ˆ
Ω N
k 2 − k d
×
2 kd
X
i =1
2 kd
X
j =1
µ 0 , 1 ( ψ − 1
n,ω P k ,j | σ ( P k ,i )) log µ 0 , 1 ( ψ − 1
n,ω P k ,j | σ ( P k ,i ))
+ lim sup
k →∞ − E 1 ˆ
Ω N \ ˆ
Ω N
k 2 − k d
×
2 kd
X
i =1
2 kd
X
j =1
µ 0 , 1 ( ψ − 1
n,ω P k ,j | σ ( P k ,i )) log µ 0 , 1 ( ψ − 1
n,ω P k ,j | σ ( P k ,i )) := I 1 + I 2 .
No w let us en umerate all the v ectors v ∈ 2 − k Z d as v (1) := 0 , v (2) , v (3) , . . . Fix
a p ositiv e in teger i with v ( i ) ⊂ [0 , 1) d . W e will estimate the n um b er of sets
v ( j ) + [0 , 2 − k ) d that in tersect ψ n,ω ( v ( i ) + [0 , 2 − k ) d ) to estimate I 1 via Lemma
3.1.2. Note that diam ( P k ) = √ d 2 − k , and hence for ev ery ω ∈ ˆ
Ω N
k b y (4.1) w e
ha v e
ψ n,ω ( v ( i ) + [0 , 2 − k ) d ) ⊂ ψ n,ω ( v ( i )) + B √ d 2 − k   D v ( i ) ψ n,ω  B (0 , √ d 2 − k )  .
Therefore for ev ery p ositiv e in tegers i and j and ω ∈ ˆ
Ω N
k , prop ert y
 v ( j ) + [0 , 2 − k ) d  ∩ ψ n,ω ( v ( i ) + [0 , 2 − k ) d ) 6 = ∅
implies
B √ d 2 − k ( v ( j )) ∩  ψ n,ω ( v ( i )) + B √ d 2 − k   D v ( i ) ψ n,ω  B (0 , √ d 2 − k )  6 = ∅ ,
and the latter inequalit y yields
B √ d 2 − k − 1 ( v ( j )) ∩  ψ n,ω ( v ( i )) + B √ d 2 − k +1   D v ( i ) ψ n,ω  B (0 , √ d 2 − k )  6 = ∅ .
Therefore w e ha v e for ω ∈ ˆ
Ω N
k b y Lemma 4.1.1 applied to A := D v ( i ) ψ n,ω
#  j : ( v ( j ) + [0 , 2 − k ) d ) ∩ ψ n,ω ( v ( i ) + [0 , 2 − k ) d ) 6 = ∅  ≤ K ( n, ω , i ) ,
where
K ( n, ω , i ) := C ( d )
d
Y
u =1
( δ u ( D v ( i ) ψ n,ω ) ∨ 1) .
the latter inequalit y implies b y Lemma 3.1.2
44

I 1 = lim sup
k →∞ − E 1 ˆ
Ω N
k 2 − k d
2 kd
X
i =1
2 kd
X
j =1
µ 0 , 1 ( ψ − 1
n,ω P k ,j | σ ( P k ,i )) log µ 0 , 1 ( ψ − 1
n,ω P k ,j | σ ( P k ,i ))
≤ lim sup
k →∞
E 1 ˆ
Ω N
k 2 − k d
2 kd
X
i =1
log K ( n, ω , i )
≤ lim sup
k →∞
E 2 − k d
2 kd
X
i =1
log K ( n, ω , i )
= lim sup
k →∞
E log K ( n, ω , 1) ,
where the last inequalit y holds b ecause translation in v ariance of ψ implies the
indep endence of the distribution of K ( n, ω , i ) with resp ect to i . No w b y the
definition of K ( n, ω , 1) w e ha v e
lim sup
k →∞
E log K ( n, ω , 1) = log C ( d ) + E " d
X
u =1
log + ( δ u ( D 0 ψ n,ω )) # .
Th us w e ha v e
I 1 ≤ log C ( d ) + E " d
X
u =1
log + ( δ u ( D 0 ψ n,ω )) # . (4.2)
T o handle the term I 2 we will again estimate the n um b er of elemen ts P k ,j that
can in tersect ψ n,ω ( v ( i ) + [0 , 2 − k ) d ) . T o do this w e put
L ( n, ω ) := sup
z ∈ B ( v ( i ) , 2 √ d ) k D z ψ n,ω k
and observ e that w e ha v e b y the mean v alue theorem for x, y ∈ v ( i ) + [0 , 2 − k ) d
and arbitrary ω ∈ ˆ
Ω N
| ψ n ( x, ω ) − ψ n ( y , ω ) | ≤ L ( n, ω ) | x − y | ,
whic h implies
ψ n,ω ( v ( i ) + [0 , 2 − k ) d ) ⊂ ψ n,ω B ( v ( i ) , √ d 2 − k ) ⊂ B ( ψ n,ω ( v ( i )) , L ( n, ω ) √ d 2 − k ) .
By Lemma 4.1.1 applied to A := L ( n, ω )1 R d w e conclude
#  j : v ( j ) + [0 , 2 − k ) d ∩ ψ n,ω ( v ( i ) + [0 , 2 − k ) d ) 6 = ∅  ≤ C ( d )( L ( n, ω ) ∨ 1) d
whic h yields
45

I 2 = lim sup
k →∞
E 1 ˆ
Ω N \ ˆ
Ω N
k 2 − k d
2 kd
X
i =1
2 kd
X
j =1
µ 0 , 1 ( ψ − 1
n,ω P k ,j | σ ( P k ,i )) log µ 0 , 1 ( ψ − 1
n,ω P k ,j | σ ( P k ,i ))
≤ lim sup
k →∞
E 
 1 ˆ
Ω N \ ˆ
Ω N
k 2 − k d
2 kd
X
i =1
log C ( d ) + d log + L ( n, ω ) 

≤ lim sup
k →∞
E h 1 ˆ
Ω N \ ˆ
Ω N
k log C ( d ) + d log + L ( n, ω ) i
≤ lim sup
k →∞
E h 1 ˆ
Ω N \ ˆ
Ω N
k log C ( d ) + d log + L ( n, ω ) i .
The latter estimation together with (4.2) pro vides
nh µ ( ψ ) ≤ 2 log C ( d )+ E " d
X
u =1
log + ( δ u ( D 0 ψ n,ω )) #
+ lim sup
k →∞
E h 1 ˆ
Ω N \ ˆ
Ω N
k d log + L ( n, ω ) i .
Since E log + L ( n, ω ) is finite whic h follo ws from Assumption 1, the last term
v anishes. Hence w e ha v e
nh µ ( ψ ) ≤ 2 log C ( d ) + E " d
X
u =1
log + ( δ u ( D 0 ψ n,ω )) # .
No w w e divide b y n to obtain
h µ ( ψ ) ≤ lim sup
n →∞
1
n E " d
X
u =1
log + ( δ u ( D 0 ψ n,ω )) # ≤
p
X
u =1
d u λ u ,
where the last inequalit y holds b ecause of [25], p. 54, inequalit y (2.6) and [25],
Prop osition I.3.2. The theorem is pro v en.
46

Chapter 5
P esin’s F orm ula for T ranslation
In v arian t Random Dynamical
Systems
P esin’s form ula is a relation b et w een Kolmogoro v-Sina ˘ ı en trop y of a smo oth dy-
namical system and its p ositiv e Ly apuno v exp onen ts. The form ula w as first
established b y P esin for deterministic DSs on a compact Riemannian manifold,
whic h preserv e a smo oth in v arian t probabilit y measure, see [32], [31], and [33].
Later differen t authors pro v ed P esin’s form ula in differen t settings, see e.g. the
discussion of the form ula in the in tro duction. Nev ertheless, to the b est of our
kno wledge, this form ula has nev er b een established b efore for systems without
in v arian t probabilit y measure. The problem is that en trop y in this case is ill-
p osed. It is ill-p osed for TIRDSs as w ell, b ecause these systems do not ha v e
an in v arian t probabilit y measure, but the Leb esgue measure, whic h is an infinite
in v arian t measure.
In this c hapter w e pro v e P esin’s form ula for v olume preserving t w o-sided
TIRDSs with en trop y defined as in Definition 3.4.1. T o estimate en trop y from
b elo w in our case, w e follo w closely Ma ˜ n ´ e’s approac h, see [26] and [3]. The idea of
Ma ˜ n ´ e is to estimate en trop y from b elo w b y the exp onen tial rate of deca y of certain
n um b ers. More precisely , these n um b ers are denoted as measures of Bo w en balls
with certain state-dep enden t radii, where Bo w en ball (with cen ter x and radius
r ) is the set of p oin ts that sta y on the distance at most r from the tra jectory
of x during the first n iterations of the system. These radii are c hosen so that
they are not to o small, but the Bo w en balls with these radii are thin enough if
w e measure the thic kness with resp ect to the unstable direction. This lets Ma ˜ n ´ e
estimate measures of Bo w en balls from ab o v e as required.
Our pro of is divided in to t w o parts. In the first part (Section 5.2) w e consider
a t w o-sided RDS with the fixed origin and b ound from ab o v e the measures of
Bo w en ball of the fixed p oin t with certain random radii. This pro cedure is similar
to Ma ˜ n ´ e’s approac h, but in this case dynamics on the state space is substituted
b y dynamics of the shift. It turns out that the trivialit y of the dynamics on the
state space (w e alw a ys sta y at the origin) lets us rep eat the ideas of Ma ˜ n ´ e, ev en
though the probabilit y space, where the shift is defined, is non-compact. In the
second part (Section 5.3) w e also follo w [26] and [3]. Note that in our case w e
47

measure the exp onen tial rate of deca y in the sense of lim inf and not in the sense
of lim sup as Ma ˜ n ´ e did. This giv es us the p ossibilit y to simplify the pro of for our
case.
W e form ulate the main result for t w o-sided systems b ecause w e use the ideas
of Ma ˜ n ´ e [26] and Bahnm ¨ uller [3], whic h rely on negativ e times as w ell. F urther,
there are t w o reasons for us to restrict ourselv es to the v olume preserving case.
First of all, the definition of en trop y in the t w o-sided case (see Section 3.4) relies
on the preserv ation of the v olume. Another reason is that Ma ˜ n ´ e in [26] (and
then Bahnm ¨ uller in [3]) pro v ed P esin’s form ula for t w o-sided systems with an
absolutely con tin uous in v arian t measure. It turns out that to rep eat the ideas of
[26] and [3] w e ha v e to stic k to the v olume preserving case.
5.1 Main Result
In this c hapter w e pro v e the follo wing theorem
Theorem 5.1.1. L et ψ b e a two-side d TIRDS define d on a pr ob ability sp ac e
( ˆ
Ω Z , B ( ˆ
Ω) Z , ˆ ν Z ) , which satisfies Assumptions 1-3, and has Lyapunov exp onents
λ 1 , . . . , λ p with multiplicities d 1 , . . . , d p . F urther, let the me asur e M = µ × ˆ ν Z b e
invariant for the skew pr o duct Θ . Then we have
h µ ( ψ ) ≥
p
X
i =1
d i λ +
i .
Remark 5.1.1. Be c ause of R uel le’s ine quality, se e Pr op osition 4.1.1, the ine qual-
ity, which app e ars in The or em 5.1.1, is e quivalent to
h µ ( ψ ) =
p
X
i =1
d i λ +
i ,
which is a usual form of Pesin ’s formula in the liter atur e.
Remark 5.1.1 together with Theorem 2.3.1 immediately implies P esin’s for-
m ula for TIBFs. Let us form ulate the result precisely .
Corollary 5.1.1. L et φ b e a TIBF and ϕ b e the r esp e ctive two-side d TIRDS on a
pr ob ability sp ac e ( ˆ
Ω Z , B ( ˆ
Ω) Z , ˆ ν Z ) (i.e. is c onstructe d as in Se ction 2.2), which has
Lyapunov exp onents λ 1 , . . . , λ p with multiplicities d 1 , . . . , d p . F urther, let µ × ˆ ν Z
(define d on R d × ˜
Ω Z ) b e invariant for the skew pr o duct Θ . Then we have
h µ ( ϕ ) =
p
X
i =1
d i λ +
i .
Remark 5.1.2. The class of TIBFs, for which Cor ol lary 5.1.1 holds, is not
empty. F or example, it holds for volume pr eserving isotr opic Br ownian flows
(these flows ar e discusse d for example in [11], [10] Chapter 2 and Chapter 5, and
[41]).
48

No w let us come bac k to the main result of the c hapter. The pro of of the
theorem is similar to the pro of of P esin’s form ula b y Ma ˜ n ´ e, see [26]; see also [3].
Before w e start explaining the pro of of the theorem, let us define lo cal en trop y
of ψ with random radius δ , whic h is the k ey ob ject in the pro of of the theorem.
F or a t w o-sided RDS ψ and a function δ : ˆ
Ω Z → (0 , ∞ ) define
R ψ ,δ,x
n :=
n
\
j =0
ψ − 1
j,ω B ( ψ j,ω ( x ) , δ ( θ j ω )) .
Note that R ψ ,δ,x
n in the case of deterministic δ are usually called Bo w en balls with
cen ter x and radius δ .
No w w e define ob jects, whic h are similar to lo cal en trop y and sho w that these
ob jects are constan ts.
Theorem 5.1.2. L et ψ b e a two-side d TIRDS define d on a pr ob ability sp ac e
( ˆ
Ω Z , B ( ˆ
Ω) Z , ˆ ν Z ) and δ : ˆ
Ω Z → (0 , 1] b e a me asur able function. Then ther e exist
deterministic numb ers h loc ( ψ , δ , x ) and h loc ( ψ , δ , x ) (mayb e e qual to + ∞ ) such
that for ˆ ν Z -a.a. ω
h loc ( ψ , δ, x ) = lim sup
n →∞ − 1
n log µ ( R ψ ,δ,x
n ) (5.1)
and
h loc ( ψ , δ, x ) = lim inf
n →∞ − 1
n log µ ( R ψ ,δ,x
n ) . (5.2)
Remark 5.1.3. Note that h loc and h l oc ar e similar to lo c al entr opy. However, in
the definition of lo c al entr opy we additional ly p ass to the limit in δ , and δ is a
(deterministic) numb er.
Pr o of of The or em 5.1.2. Without loss of generalit y assume that x = 0. Let us
form ulate a prop osition that implies the theorem
Prop osition 5.1.1. L et ψ b e a two-side d RDS define d on a pr ob ability sp ac e
( Ω Z , B (Ω) Z , ν Z ) with the fixe d origin, i.e. ψ 1 ,ω (0) = 0 , ∀ ω ∈ Ω Z , and δ :
Ω Z → (0 , 1] b e a me asur able function. Then ther e exist deterministic numb ers
h loc ( ψ , δ, 0) and h loc ( ψ , δ , 0) (mayb e e qual to + ∞ ) such that for ν Z -a.a. ω
h loc ( ψ , δ, 0) = lim sup
n →∞ − 1
n log µ ( R ψ ,δ, 0
n ) (5.3)
and
h loc ( ψ , δ, 0) = lim inf
n →∞ − 1
n log µ ( R ψ ,δ, 0
n ) . (5.4)
Indeed, w e can apply Prop osition 5.1.1 to t w o-sided RDS ψ generated b y i.i.d.
mappings
. . . ψ 1 ,θ − 1 ω − ψ 1 ,θ − 1 ω (0) , ψ 1 ,ω − ψ 1 ,ω (0) , ψ 1 ,θ ω − ψ 1 ,θ ω (0) . . . ,
49

and so translation in v ariance of ψ implies the theorem. No w let us pro ve Propo-
sition 5.1.1. During the pro of w e abbreviate ψ instead of ψ . W e pro v e (5.3). The
pro of of (5.4) is the same.
It suffices to sho w that
ξ := lim sup
n →∞ − 1
n log µ n
\
i =0
ψ − 1
i,ω ( B (0 , δ ( θ i ω ))) ! (5.5)
is a constan t ν Z -almost ev erywhere. Recall that θ is ergo dic, and so in order to
pro v e (5.5) it suffices to c hec k that
ξ ( ω ) = ξ ( θ ω ) , ν Z − a.e. (5.6)
No w let us pro v e (5.6). Note that
ξ = lim sup
n →∞ − 1
n log µ  R ψ ,δ, 0
n  .
F rom no w on during the pro of of the prop osition w e abbreviate R n instead of
R ψ ,δ, 0
n . W e start from the follo wing c hain of computations
ψ − 1
1 ,ω ( R n ( θ ω )) = ψ − 1
1 ,ω n
\
i =0
ψ − 1
i,θ ω ( B (0 , δ ( θ i +1 ω ))) !
= n
\
i =0
ψ − 1
i +1 ,ω ( B (0 , δ ( θ i +1 ω ))) !
⊃ n +1
\
i =0
ψ − 1
i,ω ( B (0 , δ ( θ i ω ))) ! = R n +1 ( ω ) .
Th us,
R n +1 ( ω ) ⊂ ψ − 1
1 ,ω ( ω )( R n ( θ ω )) .
No w w e ha v e
µ ( R n +1 ( ω )) ≤ µ  ψ − 1
1 ,ω ( ω )( R n ( θ ω )) 
= Z
ψ − 1
1 ,ω ( ω )( R n ( θ ω ))
1 µ ( dx )
= Z
R n ( θ ω )
det[ D y ψ − 1
1 ,ω ] µ ( dy )
≤ sup
y ∈ B (0 , 1) 
 D y ψ − 1
1 ,ω 
 ! µ ( R n ( θ ω )) ,
where the last inequalit y holds b ecause R n ( θ ω ) ⊂ B (0 , 1) . Th us
50

µ ( R n ( ω )) ≤ sup
y ∈ B (0 , 1) 
 D y ψ − 1
1 ,ω 
 ! µ ( R n ( θ ω )) .
Then sup y ∈ B (0 , 1) 
 D y ψ − 1
1 ,ω 
 is a random v ariable, whic h do es not dep end on n .
Therefore,
lim sup
n →∞ − 1
n log µ ( R n ( ω )) ≥ lim sup
n →∞ − 1
n log µ ( R n ( θ ω )) . (5.7)
No w let us finalize the pro of of (5.6). W e kno w that
i) ξ ( ω ) ≥ ξ ( θ ω ), ν Z -a.e. (due to (5.7));
ii) ξ ( ω ) and ξ ( θ ω ) ha v e the same distribution;
The latter t w o condidions directly imply (5.6). Th us, w e ha v e obtained (5.3).
The pro of of (5.4) is the same. The prop osition is pro v en.
The rest of the c hapter consists of t w o sections. In Section 5.2 w e pro v e
the follo wing crucial theorem, which estimates h loc in terms of the Ly apuno v
exp onen ts. Note that this is an analogue of a claim in [26], see p. 101; see also
[3], inequalit y (11).
Theorem 5.1.3. L et ψ b e a two-side d TIRDS define d on a pr ob ability sp ac e
( ˆ
Ω Z , B ( ˆ
Ω) Z , ˆ ν Z ) which satisfies Assumptions 1-3 and also has Lyapunov exp onents
λ 1 , . . . , λ p with multiplicities d 1 , . . . , d p . Then for al l  > 0 ther e exists ρ : ˆ
Ω Z →
(0 , 1] with log ρ ∈ L 1 ( ˆ ν Z ) such that
h loc ( ψ , ρ, 0) ≥ (
p
X
i =1
d i λ +
i ) − . (5.8)
Finally , in Section 5.3 w e pro v e Theorem 5.1.1 using Theorem 5.1.3, whic h is
basically estimation of en trop y from b elo w b y h loc .
5.2 Pro of of Theorem 5.1.3
W e mainly follo w here [26] and [3], Section 6.
It suffices to pro v e Theorem 5.1.3 for a t w o-sided RDS ψ on a probabilit y
space ( Ω Z , B (Ω) Z , ν Z ) (instead of ψ ) generated b y i.i.d. mappings
. . . ψ 1 ,θ − 1 ω − ψ 1 ,θ − 1 ω (0) , ψ 1 ,ω − ψ 1 ,ω (0) , ψ 1 ,θ ω − ψ 1 ,θ ω (0) . . .
Fix  ∈ (0 , 1) . Note that b oth sides of (5.8) are deterministic, and therefore
it suffices to sho w that that there exist N ∈ N and a measurable set K  with
ν Z ( K  ) ≥ 1 −  suc h that for all ω ∈ K  w e ha v e
lim inf
n →∞ − 1
nN log µ ( R ψ ,ρ, 0
nN ) ≥ (
p
X
i =1
d i λ +
i ) − . (5.9)
51

Define
C ( ω ) := log + sup
v ∈ B (0 , 1) ,E ≤ R d    det[ D v ( ψ 1 ,ω ) − 1 | E ]   ∨   det[ D v ψ 1 ,ω | E ]    ; (5.10)
further, define
G (  ) := 1
2 sup { κ : 2 κ + 2 sup
L : ν Z ( L ) ≤ κ
E C + 1 L ≤  } (5.11)
No w let us define N and K  . Denote ψ N
n,ω := ψ N n,ω Recall that the subspaces S
and U are defined b y (2.10) and (2.11).
Fix large enough N so that w e can define sets K ,i ⊂ Ω Z , i = 1 , 5 with
ˆ ν Z ( K ,i ) > 1 − 
5 and n um b ers α 1 > α 2 > 1 in the follo wing w a y
1. W e ha v e ω ∈ K , 1 if and only if
   D 0 ψ N
n,ω v    ≥ α n
1 | v | , n ≥ 1 , v ∈ U ( ω ) . (5.12)
2. W e ha v e ω ∈ K , 2 if and only if
   D 0 ψ N
n,ω v    ≤ α n
2 | v | , n ≥ 1 , v ∈ S ( ω ) . (5.13)
3. W e ha v e ω ∈ K , 3 if and only if
log    det[ D 0 ψ N
n,ω | U ( ω ) ]    ≥ nN (
p
X
i =1
d i λ +
i − G (  )) , n ≥ 1 . (5.14)
4. Fix M so large that ω ∈ K , 4 if and only if
sup { γ ( U ( ω ) , S ( ω )) } ≤ M , (5.15)
where γ is defined in the App endix.
5. Fix c > 0 and a ∈ (0 , 1] suc h that ω ∈ K , 5 if and only if for ev ery
v ∈ B (0 , a ), and for ev ery subspace E ⊂ R d whic h is an ( S ( ω ) , U ( ω ))-graph with
disp ersion ≤ c (see Definition A.0.1) w e ha v e
   log    det[ D v ψ N
1 ,ω | E ]    − log    det[ D 0 ψ N
1 ,ω | U ( ω ) ]       ≤ G (  ) (5.16)
Note that (5.12) and (5.13) and (5.15) hold b ecause of Theorem 2.3.4; (5.14)
holds b ecause of Lemma 2.3.1, Theorem 3.3.3, and (5.16) holds b ecause of the
spatial smo othness of ψ .
Finally , put
K  :=
5
\
i =1
K ,i .
Then w e ha v e ν Z ( K  ) > 1 −  .
52

Lemma 5.2.1. F or al l c > 0 ther e exists a r andom variable ζ ∈ (0 , 1) with
log ζ ∈ L 1 ( ν Z ) such that if ω ∈ K  and θ N n ω ∈ K  for some p ositive inte ger n ,
then every ( S ( ω ) , U ( ω )) -gr aph G with disp ersion ≤ c and G ⊂ B (0 ,
n − 1
Q
i =0
ζ ( θ N i ω ))
is taken by ψ N
n,ω to an ( S ( θ N n ω ) , U ( θ N n ω )) -gr aph with disp ersion ≤ c .
Pr o of. Apply Lemma A.0.1 with β 1 := α n
1 , β 2 := α n
2 , α = M and with r :=
Q n − 1
i =0 ζ ( θ N i ω ). T ak e in (A.1) β 1 and β 2 whic h mak es the restriction on δ 0 stronger
but indep enden t of n . Now fix δ 0 as in Lemma A.0.1 so hat it do es not dep end
on n . T o pro v e Lemma 5.2.1, it suffices to sho w Assumption ( b ) in Lemma A.0.1.
F or this purp ose w e define
C 1
sup ( ω ) := max 1 , sup
v ∈ B (0 , 1) k D v ψ N
1 ,ω k ! ,
C 2
sup ( ω ) := max 1 , sup
v ∈ B (0 , 1) k D 2
v ψ N
1 ,ω k ! ,
and
C sup ( ω ) := 2(( C 1
sup ( ω )) 2 + C 2
sup ( ω )) .
Claim 5.2.1. F or al l
v , w ∈ B 0 , 1 /
n − 1
Y
i =0
C 1
sup ( θ N i ω ) !
we have
k D v ψ N
n,ω − D w ψ N
n,ω k ≤
n − 1
Y
i =0
C sup ( θ N i ω ) | v − w | .
Pr o of. By the mean v alue theorem w e ha v e
k D v ψ N
1 ,ω − D w ψ N
1 ,ω k ≤ C 2
sup ( ω ) | v − w | .
W e pro v e the claim b y induction with resp ect to n . The case n = 1 is clear, since
the last inequalit y holds true and C sup ( ω ) > C 2
sup ( ω ). Supp ose the claim holds
for n . Then b y the c hain rule
k D v ψ N
n +1 ,ω − D w ψ N
n +1 ,ω k
= k D ψ N
n,ω v ψ N
1 ,θ N n ω D v ψ N
n,ω − D ψ N
n,ω w ψ N
1 ,θ N n ω D w ψ N
n,ω
+ D ψ N
n,ω v ψ N
1 ,θ N n ω D w ψ N
n,ω − D ψ N
n,ω v ψ N
1 ,θ N n ω D w ψ N
n,ω k
≤k D ψ N
n,ω v ψ N
1 ,θ N n ω − D ψ N
n,ω w ψ N
1 ,θ N n ω kk D w ψ N
n,ω k
+ k D ψ N
n,ω v ψ N
1 ,θ N n ω kk D v ψ N
n,ω − D w ψ N
n,ω k := I .
53

Note that the mean v alue theorem implies for 0 ≤ m ≤ n + 1:
| ψ N
m,ω v | = | ψ N
m,ω v − ψ N
m,ω 0 | ≤ n
Y
i = m
C 1
sup ( θ N i ω ) ! − 1
≤ 1 . (5.17)
The latter inequalit y and the mean v alue theorem implies
k D ψ N
n,ω v ψ N
1 ,θ N n ω − D ψ N
n,ω w ψ N
1 ,θ N n ω k ≤ C 2
sup ( θ N n ω ) | ψ N
n,ω v − ψ N
n,ω w | .
Moreo v er, b ecause of the definition of C 1
sup and the c hain rule and (5.17) w e hav e
k D w ψ N
n,ω k ≤
n − 1
Y
i =0
C 1
sup ( θ N i ω ) .
F urther, b ecause of (5.17) w e ha v e
k D ψ N
n,ω v ψ N
1 ,θ N n ω k ≤ C 1
sup ( θ N n ω ) .
The latter three inequalities together with the induction assumption imply
I ≤ C 2
sup ( θ N n ω ) | ψ N
n,ω v − ψ N
n,ω w |
n − 1
Y
i =0
C 1
sup ( θ N i ω )
+ C 1
sup ( θ N n ω )
n − 1
Y
i =0
C sup ( θ N i ω ) | v − w |
≤ C 2
sup ( θ N n ω )(
n − 1
Y
i =0
C 1
sup ( θ N i ω )) 2
+ ≤
n
Y
i =0
C sup ( θ N i ω ) | v − w | ,
where the latter inequalit y holds b y the definition of C sup . Th us, w e ha v e
| ψ N
n,ω v − ψ N
n,ω w |
n − 1
Y
i =0
C 1
sup ( θ N i ω ) ≤
n − 1
Y
i =0
L 2 ( θ N i ω ) | v − w |
≤
n − 1
Y
i =0
C sup ( θ N i ω ) | v − w | .
The claim is pro v en.
Denote ζ ( ω ) := δ 0 /C sup ( ω ) (w e can assume δ 0 < 1). Then, according to Claim
5.2.1, for v ∈ B  0 ,
n − 1
Q
i =0
ζ ( θ N i ω )  w e obtain
54

k D 0 ψ N
n,ω − D v ψ N
n,ω k ≤
n − 1
Y
i =0
C sup ( θ N i ω ) | v |
≤
n − 1
Y
i =0
C sup ( θ N i ω ) ζ ( θ N i ω ) = δ n
0 ≤ δ 0 .
Finally , log ζ is in tegrable b ecause of Assumptions 1 and 3. The lemma is pro v en.
W e write
D r ( ω ) := { y 1 + y 2 : y 1 ∈ S ( ω ) , y 2 ∈ U ( ω ) , | y 1 | < r , | y 2 | < r } .
F urther, let k 1 and k 2 b e constan ts suc h that for all ω ∈ K  and for all p ositiv e r
B (0 , k 1 r ) ⊂ D r ( ω ) ⊂ B (0 , k 2 r ); (5.18)
this is p ossible b ecause of (5.15). Note that D r ( ω ) is an op en subset of R d .
No w let us define ρ = ρ  ( ψ , ω ) . If ω ∈ K  define N K ( ω ) as the minim um
p ositiv e in teger, suc h that θ N N K ( ω ) ∈ K  . By P oincar ´ e recurrence theorem N K
is w ell defined for a.a. ω ∈ K  . W e extend N K to Ω Z b y putting N K ( ω ) := 0 for
ω ∈ Ω Z \ K  . Recall that the n um b er a is defined in the b eginning of the section,
see (5.16).
Definition 5.2.1. Define the r andom variable ρ = ρ  ( ψ , ω ) by
ρ ( ω ) := k 1
k 2
min( a,
N K ( ω ) − 1
Y
i =0
ζ ( θ N i ω )) ,
wher e ζ is define d in L emma 5.2.1.
W e pro v e Theorem 5.1.3 with ρ defined as in Definition 5.2.1.
No w recall that log ζ is in tegrable. Then log ρ is in tegrable as w ell b ecause of
the follo wing lemma
Lemma 5.2.2. L et A ∈ B ( Ω) Z and N A the r eturn function of A If f is a non-
ne gative and inte gr able r andom variable, then
E
N A ( ω ) − 1
X
i =0
f ( θ N i ω )1 A ( ω ) ≤ E f .
Pr o of. F or j ≥ 1 define W j := { ω ∈ A : N A ( ω ) = j } . Then up to a set of
probabilit y zero w e ha v e
∞
[
n =0
θ N n ( A ) = ∞
[
j =1
j − 1
[
i =0
θ N i ( W j ) , (5.19)
where the latter equalit y holds b ecause b oth sides represen t the ”w andering” of
A with resp ect to θ N ; the sets in the RHS are disjoin t, b ecause θ N i ( W j ) is the
55

set of p oin ts, that started from A exactly i steps b efore and will come bac k to A
in j − i steps. No w w e ha v e
E
N A ( ω ) − 1
X
i =0
f ( θ N i ω )1 A ( ω ) = E
N A ( ω ) − 1
X
i =0
∞
X
j =1
f ( θ N i ω )1 W j ( ω )
= ∞
X
j =1
E
N A ( ω ) − 1
X
i =0
f ( θ N i ω )1 W j ( ω )
= ∞
X
j =1
E
j − 1
X
i =0
f ( θ N i ω )1 W j ( ω )
= ∞
X
j =1
j − 1
X
i =0
E f ( θ N i ω )1 W j ( ω )
= ∞
X
j =1
j − 1
X
i =0 Z
W j
f ◦ θ N i d ν Z .
No w let us mak e a c hange of the measure
ν Z
i = θ N i ν Z ;
Then measures ν Z
i and ν Z ha v e the same distributions, and hence
∞
X
j =1
j − 1
X
i =0 Z
W j
f ◦ θ N i d ν Z = ∞
X
j =1
j − 1
X
i =0 Z
θ N i ( W j )
f d ν Z .
Finally , since the sets θ N i ( W j ) are disjoin t and equalit y (5.19) holds, w e ha v e
∞
X
j =1
j − 1
X
i =0 Z
θ N i ( W j )
f d ν Z = Z
∞
S
j =1
j − 1
S
j =1
θ N i ( W j )
f dν Z
(5 . 19)
= Z
∞
S
n =0
θ N n ( A )
f d ν Z ≤ E f ,
whic h completes the pro of of the lemma.
F rom no w on w e will abbreviate R ψ ,ρ
n instead of R ψ ,ρ, 0
n if there is no risk
of am biguit y . Fix an y ω ∈ K  . There exists B ( ω ) > 0, dep ending only on
( S ( ω ) , U ( ω )), suc h that for all n ≥ 0
µ ( R ψ ,ρ
nN ) = B ( ω ) Z
S ( ω )
µ y ( y + U ( ω )) ∩ R ψ ,ρ
nN dµ s ( y ) ,
where µ s denotes the Leb esgue measure on S , and µ y the Leb esgue measure on
56

y + U . Define
A := { y ∈ S : y + U ∩ R ψ ,ρ
nN 6 = ∅} .
This is a b ounded subset of S . Th us
Z
S
µ y ( y + U ) ∩ R ψ ,ρ
nN dµ s ( y ) ≤ sup
y ∈ S
µ y (( y + U ) ∩ R ψ ,ρ
nN ) µ s ( A ) .
Therefore, to sho w (5.9) it suffices to pro v e that
lim inf
n →∞ inf
y ∈ S − 1
nN log µ y (( y + U ) ∩ R ψ ,ρ
nN ) ≥ (
p
X
i =1
d i λ +
i ) − . (5.20)
No w define for y ∈ S ( ω )
Λ y
n ( ω ) := { v ∈ y + U ( ω ) : ψ N
j ( ω ) v ∈ D ρ ( θ N j ω ) /k 1 ( θ N j ω ) , 0 ≤ j ≤ n } .
By the definition of R ψ ,ρ
nN and (5.18) w e ha v e
Λ y
n ( ω ) ⊃ ( y + U ( ω )) ∩ R ψ ,ρ
nN .
Therefore, to sho w (5.20), it suffices to pro v e that
lim inf
n →∞ inf
y ∈ S − 1
nN log µ y (Λ y
n ∩ R ψ ,ρ
nN ) ≥ (
p
X
i =1
d i λ +
i ) − . (5.21)
Lemma 5.2.3. If ω ∈ K  , θ N n ω ∈ K  , y ∈ S ( ω ) and Λ y
n ( ω ) 6 = ∅ , then ψ N
n,ω Λ y
n ( ω )
is an ( S ( θ N n ω ) , U ( θ N n ω )) -gr aph with disp ersion ≤ c .
Pr o of. Fix ω ∈ K  . W e pro v e the lemma b y induction with resp ect to n . F or
n = 0, b y the definition of Λ, w e ha v e
Λ y
0 ( ω ) = ( y + U ( ω )) ∩ D ρ ( ω ) /k 1 ( ω ) .
Therefore Λ y
0 ( ω ) is an op en subset of the ( S ( θ N n ω ) , U ( θ N n ω ))-graph y + U ( ω )
with disp ersion 0. W e assume that the claim is v alid for n . If θ N ( n + m ) ω ∈ K  ,
θ N ( n + j ) ω ∈ Ω Z /K  , 1 ≤ j ≤ m − 1 and Λ y
n + m ( ω ) 6 = ∅ , then the definition of Λ
implies
ψ N
n + m,ω Λ y
n + m ( ω ) = ψ N
m,θ N n ω ψ N
n,ω Λ y
n + m ( ω )
⊂ ψ N
m,θ N n ω ψ N
n,ω Λ y
n ( ω )
∩ D ρ ( θ N ( n + m ) ω ) /k 1 ( θ N ( n + m ) ω ) .
By our assumption ψ N
n,ω Λ y
n ( ω ) is an ( S ( θ N n ω ) , U ( θ N n ω ))-graph with disp ersion
≤ c and w e ha v e
57

ψ N
n,ω Λ y
n ( ω ) ⊂ D ρ ( θ N n ω ) /k 1 ( θ N n ω ) (5 . 18)
⊂
⊂ B (0 , ρ ( θ N n ω ) k 2 /k 1 )
⊂ B 
 0 ,
N K ( θ N n ω ) − 1
Y
i =0
ζ ( θ N ( n + i ) ω ) 

= B (0 ,
m − 1
Y
i =0
ζ ( θ N ( n + i ) ω )) .
By Lemma 5.2.1 w e ha v e ψ N
m,θ N n ω ψ N
n,ω Λ y
n ( ω ) is an ( S ( θ N n ω ) , U ( θ N n ω ))-graph with
disp ersion ≤ c , and so is its op en subset ψ N
n + m,ω Λ y
n + m ( ω ).
No w let us sho w (5.21). Cho ose D > 0 suc h that D > v ol dim U ( G ) for ev ery
C 1 ( S ( ω ) , U ( ω ))-graph with disp ersion ≤ c con tained in D ρ ( ω ) /k 1 ( ω ) , ω ∈ K  ,
where v ol m ( S ) denotes m -dimensional v olume of the set S ⊂ R d . If θ N n ω ∈ K 
and y ∈ S ( ω ), w e obtain b y Lemma 5.2.3 and the transformation form ula for
differen tiable maps
D > v ol dim U ( ψ N
n,ω Λ y
n ( ω )) = Z
Λ y
n ( ω )    det[ D v ψ N
n,ω | U ( ω ) ]    dµ y ( v )
≥ Z
Λ y
n ( ω ) ∩ R ψ ,ρ
nN    det[ D v ψ N
n,ω | U ( ω ) ]    dµ y ( v ) .
(5.22)
W e put
J n,N : = { 0 ≤ j ≤ n : θ N j ω ∈ K  } ,
and
J c
n,N := { 0 ≤ j ≤ n : θ N j ω ∈ Ω Z \ K  } .
Recall that C ( ω ) is defined via (5.10). No w define
C N ( ω ) := log + sup
v ∈ R ψ ,ρ
N ,E ≤ R d     det[ D v ( ψ N
1 ,ω ) − 1 | E ]    ∨    det[ D v ψ N
1 ,ω | E ]     ,
where E ≤ R d means that E is a subspace of R d . W e ha v e
  det[ D v ( ψ 1 ,ω ) − 1 | E ]   ≤ 
 D v ( ψ 1 ,ω ) − 1 

dim E ,
and
  det[ D v ψ 1 ,ω | E ]   ≤ 
 D v ψ 1 ,ω 

dim E ;
hence
C + ( ω ) ≤ d log + sup
v ∈ B (0 , 1)  
 D v ( ψ 1 ,ω ) − 1 
 ∨ 
 D v ψ 1 ,ω 
  .
58

Therefore, C + ∈ L 1 ( ν Z ) b y Assumptions 1 and 2. Moreo v er C +
N ∈ L 1 ( ν Z ) b ecause
C +
N ( ω ) ≤
N − 1
X
i =0
C + ( θ i ω ) . (5.23)
No w fix v ∈ Λ y
n ( ω ) ∩ R ψ ,ρ
N n . By the c hain rule w e ha v e
log    det[ D v ψ N
n,ω | U ( ω ) ]   
=
n − 1
X
j =0
log    
det[ D ψ N
j,ω v ψ N
1 ,θ N j ω |  D v ψ N
j,ω  U ( ω ) ]    
= X
j ∈ J n − 1 ,N
log    
det[ D ψ N
j,ω v ψ N
1 ,θ N j ω |  D v ψ N
j,ω  U ( ω ) ]    
+ X
j ∈ J c
n − 1 ,N
log    
det[ D ψ N
j,ω v ψ N
1 ,θ N j ω |  D v ψ N
j,ω  U ( ω ) ]    
= X
j ∈ J n − 1 ,N
log    
det[ D ψ N
j,ω v ψ N
1 ,θ N j ω |  D v ψ N
j,ω  U ( ω ) ]    
− X
j ∈ J c
n − 1 ,N
log    
det[ D ψ N
j +1 ,ω v ( ψ N
1 ,θ N j ω ) − 1 |  D v ψ N
j +1 ,ω  U ( ω ) ]    
≥ X
j ∈ J n − 1 ,N
log    
det[ D ψ N
j,ω v ψ N
1 ,θ N j ω |  D v ψ N
j,ω  U ( ω ) ]     − X
j ∈ J c
n − 1 ,N
C +
N ( θ N j ω ) ,
where the last inequalit y holds b ecause v ∈ R ψ ,ρ
N n . No w Lemma 5.2.3 implies that
 D v ψ N
j,ω  U ( ω )i sa( U ( θ j ω ) , S ( θ j ω ))-graph with Ly apuno v norm ≤ c . F urther,
b y the definition of Λ w e ha v e
ψ N
j,ω v ∈ D ρ ( θ nj ω ) /k 1
(5 . 18)
⊂ B  0 , k 2 ρ ( θ N j ω ) /k 1  ⊂ B (0 , a ) .
59

Therefore, (5.16) implies
X
j ∈ J n − 1 ,N
log    
det[ D ψ N
j,ω v ψ N
1 ,θ N j ω |  D v ψ N
j,ω  U ( ω ) ]     − X
j ∈ J c
n − 1 ,N
C +
N ( θ N j ω )
≥ X
j ∈ J n − 1 ,N
log    det[ D 0 ψ N
1 ,θ N j ω | U ( θ N j ω ) ]    − G (  ) n − X
j ∈ J c
n − 1 ,N
C +
N ( θ N j ω )
≥
n − 1
X
j =0
log    det[ D 0 ψ N
1 ,θ N j ω | U ( θ N j ω ) ]    − G (  ) n − 2 X
j ∈ J c
n − 1 ,N
C +
N ( θ N j ω )
= log    det[ D 0 ψ N
n,ω | U ( ω ) ]    − G (  ) n − 2 X
j ∈ J c
n − 1 ,N
C +
N ( θ N j ω )
(5 . 14)
≥ N n (
p
X
i =1
d i λ +
i − G (  )) − G (  ) n − 2 X
j ∈ J c
n − 1 ,N
C +
N ( θ N j ω ) .
Th us, for all v ∈ Λ y
n ( ω ) w e ha v e
log    det[ D v ψ N
n,ω | U ( ω ) ]    ≥ N n (
p
X
i =1
d i λ +
i − G (  )) − G (  ) n
− 2 X
j ∈ J c
n − 1 ,N
C +
N ( θ N j ω ) .
(5.24)
Then (5.22) and (5.24) together imply for all y ∈ S ( ω )
D >µ y (Λ y
n ∩ R ψ ,ρ
N n )
× exp 
 nN (
p
X
i =1
d i λ +
i − G (  )) − G (  ) n − 2 X
j ∈ J c
n − 1 ,N
C +
N ( θ N j ω ) 

≥ µ y (Λ y
n ∩ R ψ ,ρ
N n )
× exp 
 nN (
p
X
i =1
d i λ +
i − G (  )) − G (  ) n − 2 X
j ∈ J c
n − 1 ,N
N − 1
X
i =0
C + ( θ N j + i ω ) 
 ,
where the last inequalit y holds b ecause of (5.23). By taking logarithms and
dividing b y n w e obtain
− 1
n log µ y (Λ y
n ( ω )) > − 1
n log D + N (
p
X
i =1
d i λ +
i − G (  )) − G (  )
− 2
n X
j ∈ J c
n − 1 ,N
N − 1
X
i =0
C + ( θ N j + i ω ) .
(5.25)
60

By Birkhoff ’s ergo dic theorem w e ha v e
lim
n →∞
1
n X
j ∈ J c
n − 1 ,N
N − 1
X
i =0
C + ( θ N j + i ω ) = lim
n →∞
1
n X
j ∈ J c
n − 1 ,N
N − 1
X
i =0
C + ( θ N j + i ω )1 Ω Z \ K  ( θ N j ω )
= E
N − 1
X
i =0
C + ( θ i ω )1 Ω Z \ K  ( ω ) .
No w b ecause of the definition of G (  ) w e ha v e
E
N − 1
X
i =0
C + ( θ i ω )1 Ω Z \ K  ( ω ) ≤ N (  − G (  )) ,
and therefore
N G (  ) + lim
n →∞
2
n X
j ∈ J c
n − 1 ,N
C +
N ( θ N j ω ) ≤ N  − N G (  ) .
Com bining the last inequalit y with (5.25) and dividing b y N , w e obtain for n
large enough ( n do es not dep end on y )
− 1
nN log µ y (Λ y
n ( ω ) ∩ R ψ ,ρ
N n ) > − 1
nN log D + (
p
X
i =1
d i λ +
i −  ) + ( N − 1) G (  )
N .
Th us, taking inf o v er y in the latter inequalit y first, and then taking lim inf in
b oth sides completes the pro of of (5.21). The theorem is pro v en.
5.3 Pro of of P esin’s F orm ula using Theorem 5.1.3
In this section w e pro v e Theorem 5.1.1. Let us start with the follo wing lemma
Lemma 5.3.1. If x n ∈ [0 , 1) for n ≥ 1 and
∞
X
n =1
nx n < ∞
then (with 0 log 0 := 0 )
− ∞
X
n =1
x n log x n < ∞ .
Pr o of. See [26], Lemma 1.
Recall that M = µ × ˆ ν Z is in v arian t for Θ. Recall that M 0 , 1 := M | [0 , 1) d × ˆ
Ω Z .
F urther, for a coun table measurable partition Z of R d × ˆ
Ω Z denote b y Z ω the
restriction of Z to R d × { ω } . No w we form ulate an analogue of Lemma 2 from
[26]; see also Lemma 3.3 from [3].
61

Lemma 5.3.2. L et δ : R d × ˆ
Ω Z → (0 , 1] b e a me asur able function with
− Z
[0 , 1) d
E log δ ( x, ω ) dx < ∞ . (5.26)
Then ther e is a c ountable me asur able p artition Z of R d × ˆ
Ω Z with entr opy
H M 0 , 1 ( Z ) < ∞ such that for al l x ∈ R d and for al l ω ∈ ˆ
Ω Z we have
diam Z ω ( x ) ≤ δ ( x, ω ) ,
wher e diam ( · ) denotes the diameter of a set in R d , and for a p artition P of R d
the set P ( x ) denotes the element of P which c ontains x .
Pr o of. W e mainly follo w here Lemma 2 from [26]; see also Lemma 3.3 from [3].
Put U n := { ( x, ω ):4 − ( n +1) < δ ( x, ω ) ≤ 4 − n } , n ≥ 0. This generates a
coun table measurable partition U = { U n } n ≥ 0 of R d × ˆ
Ω Z . The in tegrabilit y of
log δ implies for ev ery m ≥ 1
m
X
n =0
n M 0 , 1 ( U n ) ≤
m
X
n =0 Z
U n
− log δ ( x, ω ) d M 0 , 1 ≤ Z
R d × ˆ
Ω Z
− log δ ( x, ω ) d M 0 , 1 < ∞ .
Hence,
∞
X
n =0
n M 0 , 1 ( U n ) < ∞ . (5.27)
Recall that P k := { v + [0 , 2 − k ) d + Z d | v ∈ 2 − k Z d ∩ [0 , 1) d } . Define a partition Z in
the follo wing w a y
Z := { U n ∩ P × ˆ
Ω Z : P ∈ P 2 n + d , n ≥ 0 } .
Then
H µ 0 , 1 ( Z ) = ∞
X
n =0
( − X
z ∈Z ,Z ⊂ U n
M 0 , 1 ( Z ) log M 0 , 1 ( Z )) .
Therefore
− X
z ∈Z ,Z ⊂ U n
M 0 , 1 ( Z ) log M 0 , 1 ( Z ) ≤ M 0 , 1 ( U n )
× − X
z ∈Z
M 0 , 1 ( Z ∩ U n )
M 0 , 1 ( U n ) log M 0 , 1 ( Z ∩ U n )
M 0 , 1 ( U n ) − log M 0 , 1 ( U n ) !
≤ M 0 , 1 ( U n )( d ( n + d/ 2) log 4 − log M 0 , 1 ( U n )) ,
where the last inequalit y holds b ecause of Lemma 3.1.2. No w summing o v er n
w e get
62

H µ 0 , 1 ( Z ) ≤ 2 log 4 ∞
X
n =0
d ( n + d/ 2) M 0 , 1 ( U n ) − ∞
X
n =0
M 0 , 1 ( U n ) log M 0 , 1 ( U n ) .
By (5.27) and Lemma 5.3.1 w e obtain
H M 0 , 1 ( Z ) < ∞ .
By the definition of Z it is clear that
diam Z ω ( x ) ≤ δ ( x, ω )
whic h finishes the pro of. The lemma is pro v en.
No w fix  > 0. Let us define δ = δ  ( x, ω ). W e define it in the same w a y as
ρ , but no w with resp ect to the mo ving p oin t φ n,ω ( x ) and not with resp ect to the
origin, as in the case of ρ . T o a v oid am biguit y w e pro vide the strict definition of
δ .
Definition 5.3.1. R e c al l that C ( ω ) and G (  ) ar e define d via (5.10) and (5.11)
r esp e ctively. Fix lar ge enough N so that we c an define sets K x
,i ⊂ ˆ
Ω Z , i = 1 , 5
with ˆ ν Z  K x
,i  > 1 − 
5 , i = 1 , 5 and numb ers α 1 > α 2 > 1 in the fol lowing way
(r e c al l that the subsp ac es S x and U x ar e define d as the subsp ac es S and U , but
with r esp e ct to x and not with r esp e ct to zer o)
1. We have ω ∈ K x
, 1 if and only if
  D x ψ N
n,ω v   ≥ α n
1 | v | , n ≥ 1 , v ∈ U x ( ω ) .
2. We have ω ∈ K , 2 if and only if
  D x ψ N
n,ω v   ≤ α n
2 | v | , n ≥ 1 , v ∈ S x ( ω ) .
3. We have ω ∈ K , 3 if and only if
log   det[ D x ψ N
n,ω | U x ( ω ) ]   ≥ nN (
p
X
i =1
d i λ +
i − G (  )) .
4. Fix M so lar ge that ω ∈ K x
, 4 if and only if
sup { γ ( U x ( ω ) , S x ( ω )) } < M ,
wher e γ is define d in the App endix.
5. Fix c > 0 and a ∈ (0 , 1] such that ω ∈ K x
, 5 if and only if for every
v ∈ B ( x, a ) , and for every subsp ac e E ⊂ R d which is an ( S x ( ω ) , U x ( ω )) -gr aph
with disp ersion ≤ c (se e Definition A.0.1) we have
  log   det[ D v ψ N
1 ,ω | E ]   − log   det[ D x ψ N
1 ,ω | U x ( ω ) ]     ≤ G (  ) .
63

F or ω ∈ K  define N x
K ( ω ) as the minimum p ositive inte ger, such that θ N N x
K ( ω ) ∈
K x
 . We extend N x
K to ˆ
Ω Z by putting N x
K ( ω ) := 0 for ω ∈ ˆ
Ω Z \ K x
 . Now define
C 1
sup ( x, ω ) := max 1 , sup
v ∈ B ( x, 1) k D v ψ N
1 ,ω k ! ,
C 2
sup ( x, ω ) := max 1 , sup
v ∈ B ( x, 1) k D 2
v ψ N
1 ,ω k ! ,
and
ζ ( x, ω ) := δ 0
2((( C 1
sup ( x, ω )) 2 + C 2
sup ( x, ω )) ,
wher e δ 0 is as in L emma A.0.1, when β 1 := α n
1 , β 2 := α n
2 , α = M and with
r := Q n − 1
i =0 ζ ( ψ N i,ω ( x ) θ N i ω ) . A dditional ly take in (A.1) β 1 and β 2 which makes
the r estriction on δ 0 str onger but indep endent of n . Now fix δ 0 as in L emma A.0.1
so hat it do es not dep end on n (we c an assume δ 0 < 1 ). Then define
D x
r ( ω ) := { y 1 + y 2 : y 1 ∈ S x ( ω ) , y 2 ∈ U x ( ω ) , | y 1 | < r , | y 2 | < r } .
F urther, let k 1 and k 2 b e c onstants such that for al l ω ∈ K x
 and for al l p ositive r
B (0 , k 1 r ) ⊂ D x
r ( ω ) ⊂ B (0 , k 2 r ) .
Final ly, define
δ ( x, ω ) := k 1
k 2
min( a,
N x
K ( ω ) − 1
Y
i =0
ζ ( ψ N
i,ω ( x ) , θ N i ω )) .
No w tak e the partition Z giv en b y Lemma 5.3.2, whic h corresp onds to δ = δ  .
Note that for suc h a δ holds (5.26), b ecause
E log ρ ( ω ) < ∞ ,
and ψ is translation in v arian t. Define
Z ω
n :=
n − 1
_
i =0
ψ − 1
i,ω Z θ i ω .
No w let us sho w that
h µ ( ψ ) ≥ Z
R d × ˆ
Ω Z
lim inf
n →∞ − 1
n log µ 0 , 1  ( Z ω
n ∩ [0 , 1) d )( x )  d M 0 , 1 . (5.28)
Enlarge the class B Z
tr to B 0
tr in the follo wing w a y
B 0
tr := ( m
\
i =1
Θ − n i A i     
A i ∈ Z ∪ B tr , n i ∈ N 0 , m ∈ N ) .
64

As in the pro of of Theorem 3.4.1 w e obtain for ev ery elemen t A ∈ B 0
tr
M 0 , 1 (Θ − 1 A | Θ − 1 ( R d × B ( ˆ
Ω) Z )) = M 0 , 1 ( A | R d × B ( ˆ
Ω) Z ) ◦ Θ; (5.29)
the latter equalit y directly implies for all coun table B 0
tr -measurable partitions of
R d × ˆ
Ω Z
H M 0 , 1 (Θ − 1 Z | (Θ − 1 ( R d × B ( ˆ
Ω) Z )) = H M 0 , 1 ( Z | R d × B ( ˆ
Ω) Z ) , (5.30)
Therefore, as in Lemma 3.3.1, there exists the follo wing limit
h M (Θ , Z | R d × B ( ˆ
Ω) Z ) := lim
n →∞
1
n H M 0 , 1 n − 1
_
i =0
Θ − i Z | R d × B ( ˆ
Ω) Z ! (5.31)
Let no w Z := { Z 1 , Z 2 , . . . } and Z ( k ) := { Z 1 , Z 2 ,..., Z k , ∪ i ≥ k +1 Z i } . Then the
n um b er H M 0 , 1 ( Z ( k ) ) can b e appro ximated from b elo w b y H M 0 , 1 ( Z ( k ) ) ∧ H M 0 , 1 ( ξ ×
η ), where ξ = { A 1 , . . . , A k } and η = { B 1 , . . . , B m } are finite measurable partitions
of R d and ˆ
Ω N resp ectiv ely . F urther, the n um b er H M 0 , 1 ( Z ) can b e appro ximated
from b elo w b y H M 0 , 1 ( Z ( k ) ). Therefore, as in Theorem I I.1.4 (ii) from [18] w e
obtain
h M (Θ , Z | R d × B ( ˆ
Ω) Z ) ≤ h M (Θ | R d × B ( ˆ
Ω) Z ) ≤ h µ ( ψ ) . (5.32)
No w w e ha v e
h M (Θ , Z | R d × B ( ˆ
Ω) Z ) = lim
n →∞
1
n Z
ˆ
Ω Z
H M 0 , 1 ( Z ω
n ) d ˆ ν Z
= lim inf
n →∞
1
n Z
ˆ
Ω Z
H M 0 , 1 ( Z ω
n ) d ˆ ν Z
≥ Z
ˆ
Ω Z
lim inf
n →∞
1
n H M 0 , 1 ( Z ω
n ) d ˆ ν Z
≥ Z
ˆ
Ω Z
lim inf
n →∞ Z
R d
− 1
n log µ 0 , 1  ( Z ω
n ∩ [0 , 1) d )( x )  dµ 0 , 1 d ˆ ν Z
≥ Z
R d × ˆ
Ω Z
lim inf
n →∞ − 1
n log µ 0 , 1  ( Z ω
n ∩ [0 , 1) d )( x )  d M 0 , 1 ,
whic h together with (5.32) completes the pro of of (5.28).
Note that ∀ x ∈ [0 , 1) d
lim inf
n →∞ − 1
n log µ 0 , 1  ( Z ω
n ∩ [0 , 1) d )( x )  ≥ h loc ( ψ , δ  ( x, ω ) , x ) . (5.33)
Indeed, b y the construction of Z w e ha v e
65

Z ω
n ( x ) ⊂ R ψ ,δ  ( x,ω ) ,x
n ,
whic h yields (5.33). Therefore, b y (5.28) w e ha v e
h µ ( ψ ) ≥ Z
R d × ˆ
Ω Z
h loc ( ψ , δ  ( x, ω ) , x ) d M 0 , 1 . (5.34)
No w let us apply Theorem 5.1.3. This theorem implies the follo wing corollary
Corollary 5.3.1. L et ψ b e a two-side d TIRDS define d on a pr ob ability sp ac e
( ˆ
Ω Z , B ( ˆ
Ω) Z , ˆ ν Z ) which satisfies Assumptions 1-3 and also has Lyapunov exp onents
λ 1 , . . . , λ p with multiplicities d 1 , . . . , d p . Then for al l x ∈ R d and for al l  > 0 we
have
h loc ( ψ , δ  ( x, ω ) , x ) ≥ (
p
X
i =1
d i λ +
i ) − ,
wher e δ  ( x, ω ) is define d in Definition 5.3.1.
Pr o of. In the pro of of Theorem 5.1.3 w e consider ρ ( ω ) = ρ  ( ω ), whic h is defined
via Definition 5.2.1. It turns out that b y the definition of δ  ( x, ω ), see Defini-
tion 5.3.1, and the definition of ρ ( ω ), these t w o random v ariables ha v e the same
distribution. Therefore translation in v ariance of ψ implies the corollary .
Th us, Corollary 5.3.1 together with (5.34) implies
h µ ( ψ ) ≥ (
p
X
i =1
d i λ +
i ) − .
Finally , put  → 0. The theorem is pro v en.
66

Chapter 6
Lo cal Ruelle’s Inequalit y for
Random Dynamical Systems
In this c hapter w e define en trop y for t w o-sided RDSs using the idea of Brin and
Katok. They define the notion of local en trop y , whic h is similar to the measure-
theoretic one, but measures disorder of a system only around the tra jectory of a
particular p oin t, see [9]. W e define en tropy similarly but adapt the definition to
random dynamical systems case. F urther, w e pro v e a lo cal analogue of Ruelle’s
inequalit y with the defined en trop y . Namely , for the systems with the fixed
origin, w e pro v e that the defined en trop y is less than or equal to the sum of
p ositiv e Ly apuno v exp onen ts of the system. As a corollary , w e also obtain the
resp ectiv e result for t w o-sided TIRDSs, see Corollary 6.1.1. Note that this result
also implies Ruelle’s inequalit y , where en trop y for TIBFs is defined as in Brin
and Katok’s pap er, see the remark in the next section.
The c hapter is organized as follo ws. In Section 6.1 w e form ulate the main
result. In Section 6.2 w e pro vide the main idea of the pro of. In Section 6.3 w e
establish the Ly apuno v metric for the t w o-sided RDSs and pro v e some tec hnical
lemmas. In Section 6.4 w e pro v e the main result.
6.1 Main Result
Let ψ b e a random dynamical system on a probabilit y space ( Ω Z , B (Ω) Z , ν Z )
with the fixed origin, i.e. ψ 1 ,ω (0) = 0, ∀ ω ∈ Ω Z . First of all, let us define en trop y
of ψ . Recall that
R ψ ,, 0
n :=
n
\
j =0
ψ − 1
j,ω ( B (0 ,  )) .
F urther, recall that Prop osition 5.1.1 pro vides the existence of a deterministic
v alue (ma yb e equal to + ∞ ) h loc ( ψ , , 0) such that
h loc ( ψ , , 0) = lim sup
n →∞ − 1
n log µ ( R ψ ,, 0
n ) , a.s.
Note that in this c hapter  is alw a ys a deterministic num b er; finally , define en trop y
67

of ψ in the follo wing w a y
h loc ( ψ ) := lim
 → 0+ h loc ( ψ , , 0) .
Note that suc h a limit exists b ecause of the monotonicit y argumen ts. No w w e
are ready to state the main result of the c hapter.
Theorem 6.1.1. L et ψ b e a r andom dynamic al system on a pr ob ability sp ac e
( Ω Z , B (Ω) Z , ν Z ) , which satisfies Assumptions 1-3 and has the fixe d origin, i.e.
ψ 1 ,ω (0) = 0 , ∀ ω . L et further ψ have Lyapunov exp onents λ 1 , . . . , λ p with multi-
plicities d 1 , . . . , d p . Then we have
h loc ( ψ ) ≤
p
X
i =1
d i λ +
i .
This theorem immediately pro vides the resp ectiv e result for t w o-sided TIRDSs.
Let us form ulate it.
Corollary 6.1.1. L et ψ b e a two-side d tr anslation invariant r andom dynamic al
system on a pr ob ability sp ac e ( ˆ
Ω Z , B ( ˆ
Ω) Z , ˆ ν Z ) , which satisfies Assumptions 1-3.
L et further ψ has Lyapunov exp onents λ 1 , . . . , λ p with multiplicities d 1 , . . . , d p .
Then we have
h loc ( ψ ) ≤
p
X
i =1
d i λ +
i .
Pr o of. It suffices to apply Theorem 6.1.1 to a t w o-sided RDS ψ generated b y i.i.d.
mappings
. . . ψ 1 ,θ − 1 ω − ψ 1 ,θ − 1 ω (0) , ψ 1 ,ω − ψ 1 ,ω (0) , ψ 1 ,θ ω − ψ 1 ,θ ω (0) . . . ,
and so translation in v ariance of ψ implies the corollary .
Remark 6.1.1. Cor ol lary 6.1.1 implies that for every TIBF φ we have P -almost
sur ely
lim
 → 0+ lim sup
n →∞ − 1
n log µ n
\
j =0
φ − 1
0 ,j ( B ( φ 0 ,j (0 , ω ) ,  )) ! ≤
p
X
i =1
d i λ +
i .
This c orr esp onds to the upp er b ound on lo c al entopy in Brin and Katok’s
p ap er, se e [9]. They showe d that in p articular for an er go dic DS f with a finite
invariant me asur e m (under c ertain assumptions on f and m ), for m -almost al l
x the value
lim
 → 0+ lim sup
n →∞ − 1
n log m n
\
j =0
f − n ( B ( f n ( x ) ,  )) !
do es not exc e e d metric entr opy.
68

6.2 Main Idea of the Pro of
No w w e pro vide the idea of the pro of of Theorem 6.1.1. Note that this idea w as
prop osed b y A. Blumen thal.
Essen tially , w e exploit Ma ˜ n ´ e’s and also some Thieullien’s ideas, see [26] and
[40].
F or the sak e of simplicit y supp ose that w e ha v e no zero Ly apuno v exp onen ts.
In a small region around the origin, whic h w e denote b y D n, ⊂ R ψ ,,ω
n (whic h is
almost the same as D 0 ,,n in Thieullen’s pap er, see [40], p. 239), w e obtain uniform
h yp erb olicit y in the Ly apuno v metric (as Thieullen did; see [40], Prop osition
I I.3.3). Let y ∈ D n, , D y
n, = D y
n, ( ψ , , ω ) := D n, ∩ ( U ( ω ) + x ), and the subspaces
S and U are defined as in (2.10) and (2.11). Then w e ha ve
1 . µ ( ψ n ( D y
n, )) = Z
D y
n,
| det[ D v ψ n | U ] | dµ y ( v )
. Z
D y
n,
| det[ D 0 ψ n | U ] | dµ y ( v )
= µ y ( D n, ) | det[ D 0 ψ n | U ] |
≈ µ y ( D n, ) exp ( n
p
X
i =1
d i λ +
i ) .
Here . and ≈ means that when w e tak e ” lim
n →∞ − 1
n log ” of b oth sides, w e migh t
obtain some error, but it is negligible when  go es to zero. The first ”appro xi-
mate” inequalit y holds b ecause expansion in unstable direction helps us to sho w
that µ ( ψ n ( R y
n )) is not to o small (see Lemma 6.3.2). The second ”appro ximate”
inequalit y is rather tec hnical and is formally prov en in Lemma 6.3.3 (see also
Lemma 6.3.4).
It turns out that D n, is a set suc h that its pro jection on S (with resp ect to U )
co v ers a ball in S cen tered at zero and with radius & 1. In tuitiv ely , this happ ens
b ecause stable direction S do es not let pro jections of p oin ts on S escap e from
the unit ball to o quic kly , see Lemma 6.3.2. Moreo v er, b ecause of the calculations
ab o v e, the v olume of the ”width” with resp ect to U is of the size appro ximately
at least
V n = exp ( − n
p
X
i =1
d i λ +
i ) .
Therefore, the v olume of D n, is appro ximately at least V n , and th us the same
can b e said ab out R ψ ,,ω
n , whic h completes the pro of of the theorem.
69

6.3 Preliminaries Before the Pro of of Lo cal Ru-
elle’s Inequalit y
In this section w e pro vide Ly apuno v c harts for the t w o-sided RDS ψ , and also
pro v e some tec hnical lemmas.
6.3.1 Ly apuno v Metric
W e mainly rep eat here the construction of Ly apuno v metric, whic h is done in
[40]. Note that the original idea can b e found in [33]. Ho w ev er, w e adapt it to
our case, where w e ha v e dynamics generated b y θ , whereas in [33] and [40] they
define it for the deterministic dynamics on state space.
Denote b y Ω Z
1 ⊂ Ω Z the set of ω for whic h Theorem 2.3.4 holds. Recall that
ν Z (Ω Z
1 ) = 1. W e start from the follo wing lemma
Lemma 6.3.1. F or every  > 0 ther e exists a Bor el set (which do es not dep end
on  ) Ω Z
2 ⊂ Ω Z and a me asur able function l  : Ω Z
2 → (0 , ∞ ) such that ν Z (Ω Z
2 )=1
and for al l ω ∈ Ω Z
2
i) k D v ψ 1 ,ω − D w ψ 1 ,ω k ≤ l  ( ω ) | v − w | , v , w ∈ B (0 , 1) ;
ii) l  ( θ n ω ) ≤ l  ( ω ) e n , n ≥ 0 .
Pr o of. See [25], Lemma I I I.1.4 (see also [8], Lemma 4.4 or [7], Lemma 5.2.4).
Note that w e can define suc h an l  as there b ecause of Assumption 3.
Recall that i 0 := max { i ∈ N : λ i > 0 } . Define b := λ i 0 . Then b > 0. No w fix
 > 0 with the follo wing restrictions
Restriction 1:  < b
2 . W e will use this restriction in Prop osition 6.3.1 and
also in Restriction 4, see the end of the section.
Restriction 2: ( e b −  −  ) − 1 < 1 −  . W e will use this restriction in
Lemma 6.3.2
In the end of the section w e pro vide t w o additional restrictions, whic h use the
definition of Ly apuno v metric defined b elo w.
No w w e are ready to define Ly apuno v metric, whic h is treated for example in
[33] and [40]. W e state the ”random v ersion”.
As in [33], for ω ∈ Ω Z
1 define a norm k·k ω ,n on R d in the follo wing w a y
k v k ω ,n := √ d
+ ∞
X
m =0
e − m k D 0 ψ m,θ n ω ( v ) k , v ∈ S ( θ n ω ) ,
k w k ω ,n := √ d
0
X
m = −∞
e − bm + m k D 0 ψ m,θ n ω ( w ) k , w ∈ U ( θ n ω ) ,
and
k v + w k ω ,n := max {k v k ω ,n , k w k ω ,n } , v ∈ S ( θ n ω ) , w ∈ U ( θ n ω ) .
The sequence of norms k · k ω ,n is usually called Lyapunov metric or Lyapunov
norm at ω (and p oin t 0). Note that for an in teger n w e ha v e k v k 0 ,θ n ω = k v k n,ω .
70

The follo wing lemma rep eats Prop osition I I.2.3 in [40], but is adapted to our
settings.
Prop osition 6.3.1. Ther e exists a function ρ  : Ω Z
1 ∩ Ω Z
2 → (0 ,  ] with the fol-
lowing pr op erties
i) ρ  ( θ n ω ) ≥ e − n ρ  ( ω ) , n ≥ 0 ;
ii) k ( D 0 ψ 1 ,θ n ω ) v k ω ,n +1 ≤ e a +  k v k ω ,n , v ∈ S ( θ n ω ) , n ∈ Z ;
iii) k ( D 0 ψ 1 ,θ n ω ) w k ω ,n +1 ≥ e b −  k w k ω ,n , w ∈ U ( θ n ω ) , n ∈ Z ;
iv) | v |≤k v k ω , 0 ≤ | v | /ρ  ( ω ) , v ∈ R d ;
v) F or every v , w ∈ R d , such that k v k ω , 0 , k w k ω , 0 ≤ ρ  ( ω ) we have
k D v ψ 1 − D w ψ 1 k ω , 1 ≤  ;
vi) F or every inte ger n we have
k Pr U ( θ n ω ) ,S ( θ n ω ) k ω ,n = k Pr S ( θ n ω ) ,U ( θ n ω ) k ω ,n = 1 .
Pr o of. W e imp ose ρ  ( ω ) ≤  ∧ ( /l  ( ω )). Then Lemma 6.3.1 implies iv). The rest
is due to [33], Theorem 1.5.1.
Define
B ω ,n ( x, r ) := { v ∈ R d : k v − x k ω ,n ≤ r } ,
ρ ,n ( ω ) := ρ  ( ω ) e − n ,
and
D ,n ( ω ) := { v : ∀ k = 0 , n : k ψ k ,ω ( v ) k ω ,k < ρ ,k ( ω ) } =
n
\
k =0
ψ − 1
k ,ω B ω ,k (0 , ρ ,k ( ω )) .
Note that
D ,n ( ω ) ⊂ R ψ ,,ω
n . (6.1)
Indeed, for ev ery v ∈ D ,n ( ω ) w e ha v e
| v |≤k v k ω , 0 ≤ ρ  ( ω ) ≤ ,
where the first inequalit y holds b ecause of Prop osition 6.3.1 iv). Th us, (6.1) is
pro v en.
No w w e are ready to state t w o additional restrictions on  .
Restriction 3: Fix r ∈ (0 , 1
10 ). F urther, let c = c ( r ) ≤ r and  =  ( r ) ≤ r (in
particular,  ∨ c < 1
10 ) b e so small, that there exists a measurable set K r suc h that
ν Z ( K r ) ≥ 1 − r and if ω ∈ K r and v ∈ B (0 ,  ), then for ev ery subspace E ⊂ R d
whic h is an ( S ( ω ) , U ( ω ))-graph in Ly apuno v norm k · k 0 ,ω with disp ersion ≤ c
(see Definition A.0.1), w e ha v e
  log | det[ D v ψ 1 ,ω | E ] | − log   det[ D 0 ψ 1 ,ω | U ( ω ) ]     ≤ r . (6.2)
71

Note that w e can find suc h c and  b ecause of con tin uit y argumen ts. W e will use
this restriction in Lemma 6.3.3.
Restriction 4: Let  b e so small, that Lemma A.0.1 holds for c , where δ 0 =  ,
F = ψ 1 ,ω , β 1 = e b −  , β 2 = e  , α = 1, r = ρ  ( ω ), E 1 = S ( ω ), E 2 = U ( ω ), E = R d
with norm k·k ω , 0 , and E 0 = R d with norm k·k ω , 1 ≡ k · k θ ω , 0 . Note that w e can
mak e suc h a substitution b ecause of Restriction 1, and also b y Prop osition 6.3.1
ii), iii), v) and vi). W e will use this restriction in Lemma 6.3.2 and Lemma 6.3.3.
Note that this restriction is θ -in v ariant.
6.3.2 Some T ec hnical Lemmas
Define
ρ 0
,n ( ω ) := ρ  ( ω ) e − 4 n .
Lemma 6.3.2. F or every ω ∈ Ω Z
1 ∩ Ω Z
2 , for every n ∈ N 0 and for every x with
k x k ω , 0 < ρ 0
,n ( ω ) define
G x,n ( ω ) := ( x + U ( ω )) ∩ D ,n ( ω ) .
Then
i) G x,n is a ( U ( θ n ω ) , S ( θ n ω )) -gr aph in Lyapunov norm k·k ω ,n with disp ersion
≤ c ;
ii) We have
ψ n,ω ( x + U ( ω )) ∩ B ω ,n (0 , ρ ,n ( ω )) = ψ n,ω ( G x,n ( ω ));
iii) The set ψ n,ω ( G x,n ( ω )) ∩ S ( θ n ω ) c onsists of a single p oint, which b elongs
to B ω ,n (0 , e 3 n k x k ω , 0 ) .
Pr o of. W e use induction. F or n = 0 the statemen t of the theorem is trivial.
Supp ose that it holds for 1 , n − 1. Let us pro v e the statemen t of the theorem for
n .
First of all, let us pro v e i) for n . Note that b y Prop osition 6.3.1 v) and the
mean v alue inequalit y w e ha v e
sup
v ,w ∈ B l,ω (0 ,ρ  ( ω )) ,v 6 = w
k ψ 1 ,θ l ω ( v ) − ψ 1 ,θ l ω ( w ) − ( D 0 ψ 1 ,θ l ω )( v − w ) k ω ,l +1
k v − w k ω ,l ≤ . (6.3)
Restriction 4, Prop osition 6.3.1 ii), iii), v), vi), and consecutiv e application of
Lemma A.0.1 to ω , . . . , θ j − 1 ω together imply that for ev ery j = 0 , n
ψ j,ω (( U ( ω ) + x ) ∩ D ,n ( ω ))
is a ( U ( θ j ω ) , S ( θ j ω ))-graph in Ly apuno v norm k · k ω ,j with disp ersion ≤ c . i) for
n is pro v en.
No w let us pro v e ii) for n . By induction assumption w e ha v e for i = 1 , n − 1
( x + U ( ω )) ∩ ψ − 1
i − 1 ,ω B ω ,i − 1 (0 , ρ ,i − 1 ( ω )) ⊃ ( x + U ( ω )) ∩ ψ − 1
i,ω B ω ,i (0 , ρ ,i ( ω )) . (6.4)
72

It suffices to sho w that
ψ n,ω ( x + U ( ω )) ∩ B ω ,n (0 , ρ ,n ( ω )) ⊂ ψ n,ω ( G x,n ( ω )) . (6.5)
and so to pro v e (6.5) it suffices to show (6.4) for i = n . Supp ose that it is not
true. Then there exists x 0 ∈ ( x + U ( ω )) ∩ D ,n ( ω ), suc h that k ψ n − 1 ,ω ( x 0 ) k ω ,n − 1 >
(1 −  ) ρ ,n − 1 ( ω ). T o obtain con tradiction, w e sho w that
k ψ n − 1 ,ω ( x 0 ) k ω ,n − 1 < (1 −  ) ρ ,n − 1 ( ω ) .
Denote b y x n − 1 := ψ n − 1 ,ω ( G x,n − 1 ) ∩ S ( θ n − 1 ω ), whic h exists b y induction assump-
tion (see iii)). Then ψ − 1
n − 1 ,ω ( x n − 1 ) b elongs to D ,n ( ω ), b ecause w e ha v e
k ψ 1 ,θ n − 1 ω ( x n − 1 ) k ω ,n < ρ ,n . (6.6)
Indeed, b y Prop osition 6.3.1 ii) and b y induction assumption (see iii))
k Pr S ( θ n ω ) ,U ( θ n ω ) (( ψ 1 ,θ n − 1 ω ( x n − 1 )) k ω ,n
(6 . 3)
≤ ( e  +  ) e 3( n − 1)  k x k ω , 0
≤ ( e  +  ) e 3( n − 1)  ρ 0
,n − 1 < ρ ,n
2 < ρ ,n
(6.7)
and
k Pr U ( θ n ω ) ,S ( θ n ω ) ( ψ 1 ,θ n − 1 ω ( x n − 1 )) k ω ,n
(6 . 3)
≤ e 3( n − 1)  k x k ω , 0
≤ e 3( n − 1)  ρ 0
,n − 1 < ρ ,n
2 < ρ ,n ,
(6.8)
whic h completes the pro of of (6.6). Denote b y x ∗ := ψ n − 1 ,ω ( x 0 ). By Prop osition
6.3.1 iii) and induction assumption (see i)) w e ha v e
k Pr U ( θ n − 1 ω ) ,S ( θ n − 1 ω ) ( x ∗ ) k ω ,n − 1 ≤k Pr U ( θ n − 1 ω ) ,S ( θ n − 1 ω ) ( x ∗ − x n − 1 ) k ω ,n − 1
+ k Pr U ( θ n − 1 ω ) ,S ( θ n − 1 ω ) ( x n − 1 ) k ω ,n − 1 =: I ;
no w inequalit y (6 . 3) implies
I ≤k Pr U ( θ n ω ) ,S ( θ n ω ) ( ψ 1 ,θ n − 1 ω ( x ∗ ) − ψ 1 ,θ n − 1 ω ( x n − 1 )) k ω ,n ( e b −  −  ) − 1 + 0
(6 . 8)
≤ ( ρ ,n + e 3( n − 1)  ρ 0
,n − 1 )( e b −  −  ) − 1
= ρ ,n − 1 e −  (1 +  )( e b −  −  ) − 1 < ρ ,n − 1 ( e b −  −  ) − 1 < (1 −  ) ρ ,n − 1 ,
where the last inequalit y holds b ecause of Restriction 2. Recall that  ∨ c < 1
10 ,
see Restriction 3. Then w e ha v e
73

k Pr S ( θ n − 1 ω ) ,U ( θ n − 1 ω ) ( x ∗ ) k ω ,n − 1 ≤k Pr S ( θ n − 1 ω ) ,U ( θ n − 1 ω ) ( x ∗ − x n − 1 ) k ω ,n − 1
+ k Pr S ( θ n − 1 ω ) ,U ( θ n − 1 ω ) ( x n − 1 ) k ω ,n − 1
≤ 2 cρ ,n − 1 + e  e 3( n − 1)  k x k ω , 0 ,
where the last inequalit y holds b y induction assumption (see iii)). It is easy to
c hec k that
2 cρ ,n − 1 + e  e 3( n − 1)  k x k ω , 0 < (1 −  ) ρ ,n − 1 ,
and therefore
k x ∗ k ω ,n − 1 ≤ max {k Pr U ( θ n − 1 ω ) ,S ( θ n − 1 ω ) ( x ∗ ) k ω ,n − 1 , k Pr S ( θ n − 1 ω ) ,U ( θ n − 1 ω ) ( x ∗ ) k ω ,n − 1 }
< (1 −  ) ρ ,n − 1 .
The con tradiction is obtained. ii) for n is pro v en.
Note that (6.7) and (6.8) sho w that
k Pr S ( θ n ω ) ,U ( θ n ω ) (( ψ 1 ,θ n − 1 ω ( x n − 1 )) k ω ,n < ρ ,n
2
and
k Pr U ( θ n ω ) ,S ( θ n ω ) ( ψ 1 ,θ n − 1 ω ( x n − 1 )) k ω ,n < ρ ,n
2 ,
and therefore i) and ii) imply that the set ψ n,ω ( G x,n ) ∩ U ( θ n ω ) consists of a single
p oin t. Denote it b y x n . Finally , b y induction assumption (see iii)) and (6.8), w e
ha v e
k Pr S ( θ n ω ) ,U ( θ n ω ) ( x n ) k ω ,n ≤k Pr S ( θ n ω ) ,U ( θ n ω ) ( x n − ψ 1 ,θ n − 1 ω ( x n − 1 )) k ω ,n
+ k Pr S ( θ n ω ) ,U ( θ n ω ) ( ψ 1 ,θ n − 1 ω ( x n − 1 )) k ω ,n
≤ ce 3( n − 1)  k x k ω , 0 + ( e  +  ) e 3( n − 1)  k x k ω , 0
< ( e  + 2  ) e 3( n − 1)  k x k ω , 0 ≤ ( e  + e 2  − 1) e 3( n − 1)  k x k ω , 0
≤ e 3 n k x k ω , 0 ,
whic h pro v es iii) for n. The lemma is pro v en.
Recall that r and K r are defined in Restriction 3, see (6.2). Define
J n := { 0 ≤ j ≤ n : θ j ω ∈ K r } ,
and
J c
n := { 0 ≤ j ≤ n : θ j ω ∈ Ω Z \ K r } .
F urther, define
C ( ω ) := log + sup
v ∈ B (0 , 1) ,E ≤ R d | det[ D v ψ 1 ,ω | E ] | ∨ sup
v ∈ B (0 , 1) ,E ≤ R d   det[ D v ( ψ − 1
1 ,ω ) | E ]   ! .
74

where E ≤ R d means that E is a subspace of R d .
Lemma 6.3.3. F or every ω ∈ Ω Z
1 ∩ Ω Z
2 and for every v ∈ D ,n ( ω ) such that
k Pr S ( ω ) ,U ( ω ) ( v ) k ω , 0 ≤ ρ 0
,n we have
log   det[ D v ψ n,ω | U ( ω ) ]   ≤ log   det[ D 0 ψ n,ω | U ( ω ) ]   + F n,r ( ω ) ,
wher e F n,r ( ω ) := r n + 2 P
j ∈ J c
n − 1
C + ( θ j ω ) .
Pr o of. By the c hain rule w e ha v e
log   det[ D v ψ n,ω | U ( ω ) ]   =
n − 1
X
j =0
log   det[ D ψ j,ω ( v ) ψ 1 ,θ j ω | ( D v ψ j,ω ) U ( ω ) ]  
= X
j ∈ J n − 1
log   det[ D ψ j,ω ( v ) ψ 1 ,θ j ω | ( D v ψ j,ω ) U ( ω ) ]  
+ X
j ∈ J c
n − 1
log   det[ D ψ j,ω ( v ) ψ 1 ,θ j ω | ( D v ψ j,ω ) U ( ω ) ]   := I ;
no w let us b ound the latter expression from ab o v e
I ≤ X
j ∈ J n − 1
log   det[ D ψ j,ω ( v ) ψ 1 ,θ j ω | ( D v ψ j,ω ) U ( ω ) ]   + X
j ∈ J c
n − 1
C + ( θ j ω ) .
Let us sho w that
X
j ∈ J n − 1
log   det[ D ψ j,ω ( v ) ψ 1 ,θ j ω | ( D v ψ j,ω ) U ( ω ) ]   ≤ X
j ∈ J n − 1
log   det[ D 0 ψ 1 ,θ j ω | U ( θ j ω ) ]   + r n.
Indeed, ψ j,ω ( v ) ∈ B (0 ,  ), b ecause v ∈ D ,n ( ω ). Moreo v er, b y Lemma 6.3.2 i), for
ev ery j = 0 , n
ψ j,ω (( U ( ω ) + v ) ∩ D ,n ( ω ))
is a ( U ( θ j ω ) , S ( θ j ω ))-graph in Ly apuno v norm k · k ω ,j with disp ersion ≤ c . There-
fore, ( D v ψ j,ω ) U ( ω ) is also a ( U ( θ j ω ) , S ( θ j ω ))-graph with Ly apuno v norm ≤ c .
Th us, Restriction 3 completes the pro of of the inequalit y . Th us, w e ha v e
log   det[ D v ψ n,ω | U ( ω ) ]   ≤ X
j ∈ J n − 1
log   det[ D 0 ψ 1 ,θ j ω | U ( θ j ω ) ]   + r n + X
j ∈ J c
n − 1
C + ( θ j ω )
= log   det[ D 0 ψ n,ω | U ( ω ) ]   − X
j ∈ J c
n − 1
log   det[ D 0 ψ 1 ,θ j ω | U ( θ j ω ) ]   + r n + X
j ∈ J c
n − 1
C + ( θ j ω ) ,
where the last equalit y holds b ecause of the c hain rule. Finally , b y the definition
of C , for ev ery j ∈ J c
n w e ha v e
− log   det[ D 0 ψ 1 ,θ j ω | U ( θ j ω ) ]   = log    det[ D 0 ψ − 1
1 ,θ j ω | U ( θ j +1 ω ) ]    ≤ C + ( θ j ω ) ,
whic h completes the pro of of the lemma.
75

Lemma 6.3.4. Ther e exists a p ositive numb er G r with G r → 0 , r → 0+ , such
that for ν Z -a.a. ω we have
lim sup
n →∞
1
n F n,r ≤ G r .
Pr o of. W e ha v e
| det[ D v ψ 1 ,ω | E ] | ≤ k D v ψ 1 ,ω k dim E
and
  det[ D v ψ − 1
1 ,ω | E ]   ≤ 
 D v ψ − 1
1 ,ω 

dim E .
Hence
C + ( ω ) ≤ d log + sup
v ∈ B (0 , 1) k D v ψ 1 ,ω k ∨ d log + sup
v ∈ B (0 , 1) 
 D v ψ − 1
1 ,ω 
 ! ,
and therefore, C + ∈ L 1 ( ν Z ) b y Assumptions 1 and 2. No w define
G r := r + 2 sup
L : ν Z ( L ) ≤ r
E C + 1 L .
It is easy to see that indeed G r → 0 when r → 0+. F urther, b y Birkhoff ’s ergo dic
theorem for ν Z -a.a. ω w e ha v e
lim
n →∞
1
n X
j ∈ J n − 1
C + ( θ j ω ) = E C 1 K r + ≤ ( G r / 2) − r ,
and therefore
r + lim
n →∞
2
n
n − 1
X
j =0
C + ( θ j ω ) ≤ G r .
The lemma is pro v en.
6.4 Pro of of Lo cal Ruelle’s Inequalit y
In this Section w e pro v e Theorem 6.1.1.
Recall that r ,  and c are fixed (see Section 6.3.1). Fix ω ∈ Ω Z
1 ∩ Ω Z
2 . There
exists B = B ( ω ) > 0, whic h dep ends only on ( S ( ω ) , U ( ω )) suc h that for all
n ∈ N 0
76

µ ( D ,n ) = B Z
S
µ y (( y + U ) ∩ D ,n ) dµ s ( y )
≥ B Z
B (0 ,ρ 0
n, ρ n, ) ∩ S
µ y (( y + U ) ∩ D ,n ) dµ s ( y )
≥ B µ s ( B (0 , ρ 0
n, ρ n, ) ∩ S ) inf
y ∈ B (0 ,ρ 0
n, ρ n, ) ∩ S µ y (( y + U ) ∩ D ,n ) ,
where µ s denotes the Leb esgue measure on S , and µ y the Leb esgue measure on
y + U . Th us, w e ha v e
µ ( D ,n ) ≥ e − 5 dn B 1 ( ω ) inf
B (0 ,ρ 0
n, ρ n, ) ∩ S µ y (Λ y
n, ) , (6.9)
where B 1 ( ω ) := B µ s ( B (0 , ρ 0
, 0 ρ , 0 ) ∩ S ), and Λ y
n, := ( y + U ) ∩ D ,n . Fix y ∈
B (0 , ρ 0
n, ρ n, ) ∩ S . Then b y Prop osition 6.3.1 iii) w e ha v e
B ( y , ( ρ ,n ) 2 / 2) ⊂ B ω ,n ( y , ρ ,n / 2) ⊂ B ω ,n (0 , ρ ,n ) ,
where the last implication holds b ecause y ∈ B (0 , ρ 0
n, ρ n, ) ∩ S . Note that b y
Lemma 6.3.2 i) the set ψ n,ω Λ y
n, is a ( U ( θ n ω ) , S ( θ n ω ))-graph with disp ersion ≤ c .
Recall that for a set S ⊂ R d the v alue v ol m ( S ) denotes the m -dimensional v olume
of S . Then b y Lemma 6.3.2 w e ha v e
 1 (( ρ ,n ) 2 / 2) d < v ol dim U ( ψ n,ω Λ y
n, ) ,
where  1 > 0 is a deterministic lo w er b ound on p ossible dim U -dimensional v olume
of a ( U ( θ n ω ) , S ( θ n ω ))-graph of class C 1 in a ball of radius 1, passing through 0.
F urther, b y transformation form ula w e ha v e
v ol dim U ( ψ n,ω Λ y
n, ) = Z
Λ y
n,
| det[ D v ψ n,ω | U ] | dµ y ( v )
≤  | det[ D 0 ψ n,ω | U ] | e F n,r  µ y (Λ y
n, ) ,
where the last inequalit y holds b y Lemma 6.3.3. Therefore,
 1 (( ρ ,n ) 2 / 2) d ≤  | det[ D 0 ψ n,ω | U ] | e F n,r  inf
y ∈ B (0 ,ρ 0
n, ρ n, ) ∩ S µ y (Λ y
n, )
(6 . 9)
≤ e 5 dn B − 1
1  | det[ D 0 ψ n,ω | U ] | e F n,r  µ ( D ,n ) .
By taking logarithms and dividing b y n w e obtain
77

− 1
n log µ ( D ,n )
≤ 5 d − 1
n log   1 ( ρ 2
 / 2) d B 1  | det[ D 0 ψ n,ω | U ] | e F n,r  − 1 
=5 d − 1
n log   1 ( ρ 2
 / 2) d B 1  + 1
n log  | det[ D 0 ψ n,ω | U ] | e F n,r  .
(6.10)
No w put n → ∞ . W e ha v e
h loc ( ψ , , ω ) ν Z -a.a.
≤ lim sup
n →∞ − 1
n log µ ( R ψ ,, 0
n ) ≤ lim sup
n →∞ − 1
n log µ ( D ,n )
≤ 5 d + lim sup
n →∞
1
n log  | det[ D 0 ψ n,ω | U ] | e F n,r 
ν Z -a.a.
≤ 5 d + G r + lim sup
n →∞
1
n log ( | det[ D 0 ψ n,ω | U ] | ) ,
where the last inequalit y holds b y Lemma 6.3.4. F urther, b y Lemma 2.3.1
lim sup
n →∞
1
n log ( | det[ D 0 ψ n,ω | U ] | ) ν Z -a.a.
=
p
X
i =1
d i λ +
i .
Th us,
h loc ( ψ , , ω ) ν Z -a.a.
≤ 5 d + G r +
p
X
i =1
d i λ +
i .
Finally , put r → 0+ (recall that  dep ends on r , see Resriction 3). The theorem
is pro v en.
78

Chapter 7
Op en Problems
The first question whic h arises from the thesis is whether w e can obtain P esin’s
form ula for t w o-sided TIRDSs, that are not v olume preserving. The problem is
that the pro of relies on the existence and in v ariance of the unstable direction
of t w o-sided RDS with the fixed origin. That forced us to use negativ e times
and also to define en trop y for partitions that dep end on negativ e times. But
then it is unclear wh y Θ should ha v e a smo oth in v arian t measure. An alternativ e
approac h could b e to find a substitution for the unstable direction, whic h do es not
dep end on negativ e times. More precisely , w e can consider a randomly distributed
direction, where the randomness do es not dep end on the randomness of the RDS.
F urther, the distribution should coincide with a stationary with resp ect to θ
distribution for the Mark o v c hain on the space of (dim U )-fold exterior p ow er of
R d , whic h corresp onds to the space of p ossible directions, transv ersal to S ( ω ).
Then w e do not use negativ e times an ymore, but still enjo y stationarit y of the
unstable direction. That could lead to a pro of of an analogue of Theorem 5.1.3.
Then one migh t hop e to deduce P esin’s form ula from the analogue.
Another op en question is whether w e can obtain a lo cal P esin’s form ula, i.e. is
it true that lo cal en trop y of t w o-sided RDSs with the fixed origin (or of t w o-sided
TIRDSs) is equal to the sum of p ositiv e Ly apuno v exp onen ts? In this case w e
ha v e to estimate the v olume of the resp ectiv e Bo w en balls from ab o v e. Ho w ever,
w e can not use the approac h from Chapter 6, where w e use Ly apuno v c harts,
b ecause w e meet the follo wing problem: the Ly apuno v c harts are basically c harts
that ha v e a random v olume whic h is small for some ω , so the Bo w en balls can’t b e
co v ered b y these c harts. That means that w e ha v e to find an approac h to estimate
the v olume of the Bo w en balls b ey ond the Ly apuno v c harts, whic h seems to b e a
c hallenging problem. W e prop ose t w o approac hes whic h ma y resolv e the problem
for TIBFs (ma yb e in some partial cases only).
The first approac h is to add linear drift to w ards the origin, obtaining another
sto c hastic flo w with an in v arian t probabilit y measure. F or example, in the case
of isotropic Bro wnian flo ws, w e obtain isotropic Ornstein-Uhlen b ec k flo ws, that
are describ ed, for example, in [42]. In this case w e exp ect that the Bo w en balls
of the obtained flo ws ha v e larger v olume in distribution b ecause of the definition
of Bo w en balls and the added drift. Ho w ev er, it turns out to b e a non-trivial
statemen t and one has to c hec k it. F urthermore, ev en for the new flo w, the
question of the existence of a lo cal P esin’s form ula is not trivial b ecause of lac k
79

of compactness.
Another approac h could b e to p erio dize the initial TIBF, i.e. to obtain another
flo w so that its lo cal b eha viour around the tra jectory of a fixed p oin t (sa y zero)
is the same in distribution, but the spatial b eha viour is spatially p erio dic ω -wise.
T o p erio dize the flo w, one should c hange the co v ariance tensor of the system
so that the obtained one coincides with the co v ariance tensor of the initial flo w in
some neigh b ourho o d of zero, but also b ecomes p erio dic. The problem is that the
c hanged co v ariance tensor should b e p ositiv e-definite. Therefore, for the tensors
with high smo othness, suc h a task seems to b e more c hallenging. P erhaps one
should find a w a y to p erio dize tensors with a singularit y at zero and then to
extend the result to the flo ws with a smo oth tensor.
In the thesis w e try to use the definition of Brin and Katok, but one can also
think ab out some other ”lo cal” definitions of en trop y . A p ossible approac h is
to consider information function of the resp ectiv e RDS (sa y at zero). Shannon-
McMillan-Breiman Theorem asserts that for ergo dic dynamical systems infor-
mation function often coincides a.e. with Kolmogoro v-Sina ˘ ı en trop y of the sys-
tem. T o approac h the problem, one can try to pro v e the analogue of Shannon-
McMillan-Breiman Theorem for TIRDSs, where instead of Kolmogoro v-Sina ˘ ı en-
trop y w e consider en trop y defined in Chapter 3.
No w let us discuss a p ossibilit y to establish P esin’s formula in the case of
Kunita-t yp e SDEs (on R d ; for the sak e of simplicit y let d = 1) with delay . Let us
b e more precise. Denote b y
x t ( s ) := x ( t + s ) , s ∈ [ − 1 , 0] , t ≥ 0;
no w consider the follo wing dela y equations



dx ( t ) = F ( x t ) dt + M ( dt, x ( t )) , t ≥ 0 ,
x 0 ∈ C ([ − 1 , 0] , R ) ,
i.e. Kunita-t yp e dela y equations, where M is a translation in v arian t martin-
gale field, F is a translation in v arian t with resp ect to constan ts drift term, i.e.
F ( x t ) ≡ F ( x t − c ), c ∈ R . F rom [29] and [28] w e kno w that, under certain mild
assumptions, the equation ab o v e generates a sto c hastic flo w whic h ev en has Ly a-
puno v sp ectrum. Resp ectiv ely , a discretized flo w can b e seen as a one-sided RDS
on the state space C ([0 , 1]). Hence, one migh t think of defining en trop y for suc h
RDSs (p erhaps similar to the definition in Chapter 3) and pro v e an analogue of
P esin’s form ula for suc h systems.
80

App endix A
T ransformations of graphs
Here w e, follo wing [26], pp. 98–99 with [27], in tro duce graphs with b ounded
disp ersion and also state a lemma ab out transformation of suc h graphs.
Note that all v ector spaces men tioned in the App endix are finite-dimensional.
Definition A.0.1. L et ( E , k · k ) b e a norme d ve ctor sp ac e with splitting E =
E 1 ⊕ E 2 . We c al l a subset G of E an ( E 1 , E 2 ) -gr aph if ther e is an op en set
U ⊂ E 2 and a C 1 map f : U → E 1 such that
G = { f ( x ) + x : x ∈ U } .
The disp ersion of G is define d by
sup
x,y ∈ U,x 6 = y
k f ( x ) − f ( y ) k
k x − y k
Henc e we c an c onclude that disp ersion of a gr aph is like Lipschitz c onstant in
the c o or dinate system, gener ate d by E 1 and E 2
Let π 1 : E → E 2 b e the pro jection on to E 1 with k ernel E 2 and let π 2 : E → E 1
b e the pro jection on to E 2 with k ernel E 1 . Then w e define
γ ( E 1 , E 2 ) := max( k π 1 k , k π 2 k ) .
No w w e are ready to form ulate the main result of App endix.
Lemma A.0.1. Given β 1 > β 2 > 1 , α > 0 , and c > 0 , then for every
δ 0 ∈ (0 , min { β 1 α − 1 (1 + c ) − 1 , ( β 1 − β 2 ) cα − 1 (1 + c ) − 2 } ) (A.1)
the fol lowing pr op erty holds. If E = E 1 ⊕ E 2 with γ ( E 1 , E 2 ) ≤ α , and F is a C 1
emb e dding of a b al l B (0 , r ) ⊂ E into another Banach sp ac e E 0 satisfying
(a) D 0 F is an isomorphism and γ (( D 0 F ) E 1 , ( D 0 F ) E 2 ) ≤ α ;
(b) k D 0 F − D x F k ≤ δ 0 for al l x ∈ B (0 , r ) ;
(c) k ( D 0 F ) v k ≥ β 1 k v k for al l v ∈ E 2 ;
(d) k ( D 0 F ) v k ≤ β 2 k v k for al l v ∈ E 1 ;
then for every ( E 1 , E 2 ) -gr aph G with disp ersion ≤ c c ontaine d in the b al l B (0 , r ) ,
its image F ( G ) is a (( D 0 F ) E 1 , ( D 0 F ) E 2 ) -gr aph G with disp ersion ≤ c .
81

Pr o of. See [26], Lemma 3 with [27].
82

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86

Index of Notation and
Abbreviations
( · ) + p ositiv e part of ( · )
|·| Euclidean norm
k·k op erator norm coming from |·|
a.a. almost all
a.e. almost ev erywhere
a.s. almost surely
B ( · ) Borel σ -algebra
B r ( A ) set of p oin ts that are on the distance at most r from
set A
B ( x, r ) closed r -ball, cen tered at x ∈ R d
B tr class of 1-p erio dic in distribution sets in one-sided case
B Z
tr class of 1-p erio dic in distribution sets in t w o-sided case
B 0
tr class of sets, see p. 64
d natural n um b er, dimension of the state space
DS dynamical system
diam ( A ) diameter of set A
diam ( P ) sup C ∈P diam ( C ), diameter of partition P
D v spatial deriv ativ e
e.g. exempli gratia, for example
H P ( ξ |G ) conditional en trop y of partition ξ giv en σ -algebra G
h M (Θ) en trop y of sk ew pro duct Θ
h M (Θ | R d × B ( ˆ
Ω) Z ) en trop y of sk ew pro duct Θ giv en randomness
h M (Θ , P | R d × B ( ˆ
Ω) Z ) en trop y of sk ew pro duct Θ giv en randomness with
resp ect to partition P
h µ ( ψ ) en trop y of ψ
h µ ( ψ , P ) en trop y of ψ with resp ect to partition P
id R d iden tit y map on R d
i.i.d. indep enden t, iden tically distributed
i.e. id est, this is
N set of p ositiv e in tegers
N 0 set of non-negativ e in tegers
M := µ × ν N
M 0 , 1 M | [0 , 1) d × ˆ
Ω
M + := µ × ν Z
87

M +
0 , 1 M + | [0 , 1) d × ˆ
Ω
Pr S 1 ,S 2 ( A ) pro jection of set A on subspace S 1 , whic h is parallel to
subspace S 2
R set of real n um b ers
R + set of non-negativ e real n um b ers
RDS random dynamical system
S stable direction
SDE sto c hastic differen tial equation
TIBF translation in v arian t Bro wnian flo w
TIRDS translation in v arian t random dynamical system
v ol m ( S ) m -dimensional v olume of S
U unstable direction
Θ sk ew pro duct of t w o-sided RDS
Θ + sk ew pro duct of one-sided RDS
θ left shift op erator of t w o-sided RDS
θ + left shift op erator of one-sided RDS
λ i Ly apuno v exp onen t
µ Leb esgue measure on R d
µ m,n := µ | [ m,n ) d restriction of Leb esgue measure µ to cub e [ m, n ) d .
88

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