Deterministically Driven Random
Walks in a Random Environment
Colin Malcolm William Little
M.Sc
A thesis submitted to
The University of Surrey
for the degree of
Doctor of Philosophy
Department of Mathematics
University of Surrey
December 2012
Abstract
We introduce the concept of a deterministically driven random walk in a random environment on a state space S , focusing on the case where S is countable. Since our construction has a purely deterministic representation we
refer to it as a deterministic walk in a deterministic environment (DWDE).
For the deterministic walk in a xed environment we establish properties
analogous to those found in Markov chain theory, but for processes that do
not in general have the Markov property. In the nite state space setting,
we establish hypotheses for the recurrence or transience of a deterministic
walk, and the existence of asymptotic occupation times. In the case of a
DWDE on Z, we establish hypotheses that ensure that it is either recurrent
or transient. An immediate consequence of this result is that a symmetric
DWDE on Z is recurrent. Moreover, in the transient case, we show that the
probability that the DWDE diverges to +∞ is either 0 or 1. In certain cases
we compute the direction of divergence in the transient case.
c Colin Little 2012
This thesis and the work to which it refers are the results of my own efforts. Any ideas, data, images or text resulting from the work of others
(whether published or unpublished) are fully identied as such within
the work and attributed to their originator in the text, bibliography or
in footnotes. This thesis has not been submitted in whole or in part for
any other academic degree or professional qualication. I agree that
the University has the right to submit my work to the plagiarism detection service TurnitinUK for originality checks. Whether or not drafts
have been so-assessed, the University reserves the right to require an
electronic version of the nal document (as submitted) for assessment
as above.
Colin Little
Contents
1 Introduction
1
2 Random Walks in a Random Environment
8
2.1 Markov Chains and Simple Random Walks . . . . . . .
2.2 Random Walks on Z in a Fixed Environment . . . . . .
2.3 Random Walks on Z in a Random Environment . . . .
8
13
17
3 The Deterministic Walk in a Deterministic Environment
26
3.1 Introducing the Deterministic Walk . . . . . . . . . . .
3.2 Examples of Deterministic Walks on a Finite State Space
3.2.1 Example: a Deterministic Markov Chain . . . .
3.2.2 Counter-example to Transitivity . . . . . . . . .
3.3 Examples of Deterministic Walks on Z . . . . . . . . .
3.3.1 Deterministic Walks in a Homogeneous Environment . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Deterministic Walks in a Deterministic Environment . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 A Deterministic Version of a Random Walk on Z
3.4 Main Questions . . . . . . . . . . . . . . . . . . . . . .
27
29
29
32
33
33
35
35
37
4 Properties of a Deterministic Walk on a Finite State
Space
39
4.1 Markov maps . . . . . . . . . . . . . . . . . . . . . . .
4.2 Transitivity of a Deterministic Walk on a Finite State
Space . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i
40
42
4.3 Asymptotic Occupation Times for a Deterministic Walk
on a Finite State Space . . . . . . . . . . . . . . . . . .
5 Properties of Markov Maps with Strong Distortion
47
51
6 Asymptotic Properties of a Deterministic Walk in a
Fixed Environment
57
7 Zero-One Laws for a Deterministic Walk on Z in a Deterministic Environment
63
7.1
7.2
7.3
7.4
7.5
A Deterministic Walk on Z in a Deterministic Environment
Measurability Lemmas . . . . . . . . . . . . . . . . . .
A 4-case Zero-One Law for the DWDE . . . . . . . . .
The Linkage Property . . . . . . . . . . . . . . . . . .
A 3-case Zero-One Law for the DWDE . . . . . . . . .
8 Transient DWDEs on Z
64
65
67
69
71
80
ii
Acknowledgments
Special thanks are due to my Ph.D supervisor, Prof. Ian Melbourne,
for suggesting the topic upon which this research has been based, and
for his help and advice throughout. I would also like to thank Prof.
Jon Aaronson, Prof. Henk Bruin and Dr. Dalia Terhesiu for numerous
helpful discussions, and Dr. Claudia Wul for undertaking the job of
being the examiner of my M.Phil/Ph.D transfer report.
I would like to thank the community as a whole within of Department of Mathematics at the University of Surrey for providing such an
agreeable environment in which to work.
Finally, I would like to thank the EPSRC for funding this research.
iii
Chapter 1
Introduction
The purpose of this thesis is provide a foundational background for
the study of deterministically driven random walks in a random environment. We consider the simplest possible situations in which the
deterministic walk occurs on a nite, or a countably innite, state
space.
Much attention in the area of deterministically driven random walks
in a random environment has focused lately on applications to problems
in randomly generated dynamical billiards. Before giving an account
of such work we briey describe what a dynamical billiard is by means
of an example. Among the most well-known examples of a dynamical
billiard is that of a Lorentz gas, which models the free motion of a point
particle in Euclidean space Rd (d ≥ 2) subject to elastic collisions with
a xed, countable, array of dispersing scatterers. Each scatterer is
an open, bounded, connected, and strictly convex domain of Rd , with
smooth boundary ∂ . The scatterers are assumed to be pairwise disjoint.
When the conguration of scatterers is periodic, this setup is called a
periodic Lorentz gas. When the free ight of the particle is bounded,
the billiard is said to have a nite horizon. Given a nite horizon
, π ],
billiard ow, there is a natural global cross-section M = ∂ × [ −π
2 2
and the Poincaré map T : M → M is called the billiard map.
Sinai [27] showed that a nite horizon planar periodic Lorentz gas is
recurrent and ergodic. Bunimovich, Sinai and Chernov [8] proved the
central limit theorem and weak invariance principle (WIP) for such
systems, while Melbourne & Nicol [24] showed that almost every tra-
1
jectory is approximable by a sample path of a Brownian motion. (This
property is known as an almost sure invariance principle (ASIP).) In a
recent development of the subject of deterministic walks in a random
environment Dolgopyat et al [14] have shown that the WIP of a planar
periodic Lorentz gas is (in some sense) robust to random perturbations
of nite regions of an otherwise periodic arrangement of scatterers. One
important feature of the results that we present here is that they do
not require a periodicity condition on the environment.
Another class of random dynamical billiards considers the situation
in which a space (e.g. a strip in R2 , R2 , etc) is tessellated by countably many translated copies of the same polygon. A billiard is then
constructed by randomly assigning a conguration of scatterers to each
polygon. Once an environment of scatterers has been generated the resulting billiard dynamics are run. Such systems are known as random
Lorentz gases (or random Lorentz tubes in the strip setting). Lenci et
al [11, 12] show that (under certain hypotheses on the geometry of scatterer congurations) almost all realisations of a random 2-dimensional
Lorentz tube are recurrent, and that (under somewhat more restrictive hypotheses) this can also be shown for higher dimensional cases.
Lenci & Troubetzkoy [21] identify very special hypotheses under which
a given xed 2-dimensional Lorentz tube is recurrent and ergodic, and
for which the rst return map is K -mixing, and they consider random
Lorentz tubes whose typical realisation satises these hypotheses.
Examples of other (non-billiard) applications of deterministic walks
in random environments can be found in Dolgopyat [13], and Simula
& Stenlund [26].
We observe that a common feature in the above examples is that
once an environment (of scatterers) has been generated, it remains unchanged thereafter. A dierent class of deterministically driven random
walk in a random environment deals with situations in which the initial conguration of the environment is randomly chosen, and is then
allowed to evolve over time. For example, Stenlund [30] considers the
asymptotic behaviour of a random Sinai billiard on a 2-torus in which
the position of a single scatterer is randomly updated after each collision (for which he establishes an ASIP). We do not consider such
2
systems here, but note (as Stenlund does) that the situation in which
the environment is frozen is typically much harder to analyse than the
situation in which it updates randomly after collisions.
In this work we are concerned with understanding a simpler problem:
specically, we investigate the recurrence and transience properties of
a deterministic walk in a random environment on a nite or countable state space. Our primary motivation lies in the fact that (to our
knowledge) the existing literature does not address such questions in a
systematic way. The secondary motivation consists in the hope that the
results described here will provide a foundation upon which to tackle
harder problems such as the billiard systems described above.
Our setup bears certain similarities to the now classical subject
of random walks in a random environment (RWRE), which considers
ensembles of randomly generated Markov chains on a common state
space. But unlike in that situation we will be concerned more generally
with ensembles of dynamical systems that are deterministic in nature,
and which do not have the Markov property.
In Chapter 3, we formally introduce our setup. By way of further
motivation for the questions considered there, we here give a brief account of RWREs in the one-dimensional setting; giving in Chapter 2 a
more detailed account of the results that are of primary interest.
We start by recalling the elementary case of the simple random
walk on Z. A simple random walk (SRW) considers the situation in
which a walker, starting at some state (e.g. 0), performs a random
path on the integers Z by taking independent steps in such a way
that, for some xed p ∈ (0, 1), at any given time, the probability of
his taking a step to the left is p, while the probability of his taking a
step to the right is 1 − p. Polya showed (see p.118-119 of [4]) that this
process is recurrent for the case where p = 21 , in that the walker returns
to 0 with probability 1. (Moreover, the walker visits every integer
innitely often with probability 1.) However, in the situation where
p > 12 (p < 12 ) it follows from the Strong Law of Large Numbers that
the process is non-recurrent, and moreover that with probability 1 the
walker will drift to +∞ (−∞, respectively). In addition to this uniform
3
pointwise behaviour, in the recurrent case, the SRW also satises an
important distributional property: namely, the central limit theorem
(CLT). Specically, letting Un denote the position of the walker at time
Un
n, it can be shown that the distribution of √
converges to the standard
n
Normal distribution N (0, 1), as n → ∞.
Clearly, the setup of the SRW is very simple, and it natural to ask
what criteria govern the asymptotics of a general Markov chain on Z,
and by extension to the RWRE setting, what can we say about the
behaviour of entire ensembles of Markov chains on Z whose transition
probabilities are generated according to some stochastic or deterministic process? In Chapter 2, we will turn to these questions in depth in
the one-dimensional case1 .
In Chapter 2, we will be particularly interested in the following
type of RWRE on Z. Let (αn )n∈Z be a sequence of independent and
identically distributed (i.i.d.) random variables taking values in (0, 1).
For each realisation (αn )n of this process we dene a Markov chain
(Un )n≥0 on Z satisfying, (i) P (U0 = 0) = 1 and (ii) for all n ≥ 0 and
k ∈ Z, P (Un+1 = k + 1|Un = k) = αk and P (Un+1 = k − 1|Un =
k) = 1 − αk . Each realisation of the process (αn )n thus denes an
environment of transition probabilities on Z, and as such the process
(αn )n is called a random environment on Z. A setup of this kind is
called a simple RWRE on Z. More complex setups for the RWRE have
focused on situations where the environment is generated by dierent
types of dynamics, and by allowing jumps other than ±1.
The rst mathematical results for simple RWREs go back to the
works of Koslov [20], Kesten et al [18], and Solomon [29] in the mid1970s. Since then a great deal has been established about the qualitative and quantitative properties of RWREs in the one dimensional
setting.
The seminal results in the eld are due to Solomon [29] and Sinai
[28]. Solomon was the rst to establish a criterion for classifying the (almost sure) pointwise asymptotic behaviour of the simple RWRE on Z.
In particular he showed this behaviour was completely determined by
1
Full and relatively modern surveys of RWREs, including the higher dimensional cases,
can be found in [32] and [34].
4
0
). Solomon also proved a Law of Large Numbers for
the sign of E(ln 1−α
α0
simple RWREs, and showed the existence of transient simple RWREs
for which the growth rate of the position of the walk is sub-linear. Sinai
[28] famously showed that in the recurrent case the distributional behaviour of the simple RWRE exhibits anomalously sub-diusive growth
according to a (ln n)2 law.
Subsequent developments in the eld have extended these results to
more general settings. Key [19] obtained a more general classication
result for the case of RWRE on Z in an i.i.d. environment for which
jumps are uniformly bounded. Bolthausen & Goldsheid [5] further extended the classication of RWREs on Z with bounded jumps to the
situation in which the random environment is ergodic and stationary.
Moreover, [5] considered the more general strip model in which the
state space is {1, . . . , k} × Z, for some k ∈ N. Bremont [7], under certain additional assumptions, further extended the classication of one
dimensional RWREs with bounded jumps to the setting of a Gibbsian
environment on a subshift of nite type. As yet, relatively little is
known about the classication of higher dimensional RWREs.
Under certain additional hypotheses to those considered in [28],
Letchikov [22] established the same sub-diusive (ln n)2 behaviour in
the setting of recurrent RWREs on Z that have bounded jumps. Bolthausen
& Goldsheid [6] established distributional limit laws for the position of a
recurrent RWRE on a strip in an i.i.d. environment. In particular, they
showed that the RWRE on a strip typically exhibits Sinai type (ln n)2 ,
with the CLT being the degenerate case occurring only under special
conditions. Goldsheid [17] and Peterson [25] independently proved that
in case of a transient simple RWRE, in almost every environment the
position of the walk satises a CLT. Specically, [17] and [25] show
that for almost every environment there exists a sequence of constants
n
(bn )n≥0 such that Un√−b
⇒d N (0, 1). The former also establishes this
n
result for the case where the random environment is uniformly ergodic.
Taking our lead from the probability theory, and looking instead at
deterministic dynamical systems, our goal is to establish results of a
similar avour to those obtained for RWREs. Typically these problems
5
are a lot harder in the deterministic setting because very dierent assumptions regarding the dependence/independence of observables obtain to those that are prevalent in the probabilistic setting.
The main results of this thesis are to be found in Chapters 4 and
7. In Chapter 4, we establish general hypotheses under which a deterministic walk in a given xed environment on a nite state space is
transitive, and for which there exist asymptotic occupation times. In
Chapter 7, prove that under certain conditions, a deterministic walk
in a deterministic environment on Z satises a Zero-One law. Specically, we show that it exhibits exactly one of three types of asymptotic
behaviour, each with a probability of 0 or 1.
The thesis is laid out as follows. In Chapter 2, we recall some
important ideas from the theory of stochastic processes and introduce
the main concepts with which this thesis is concerned. In this chapter
we present the classication results of Chung [10] and Solomon [29] for
Markov chains on Z, and for the simple RWRE on Z, respectively.
In Chapter 3, we formally introduce the idea of a deterministic walk
in a deterministic environment on Z (DWDE). We give examples of
this construction, showing how certain cases can be modelled by the
RWRE setup of Chapter 2.
In Chapter 4, we prove recurrence and asymptotic occupation time
results for a deterministic walk on a nite state space. In doing so
we introduce much of the machinery from the realm of deterministic
dynamical systems that will be required in proving the main results of
this thesis. Specically, we introduce Markov maps and their related
properties.
In Chapter 5, we prove a series of results relating to Markov maps
that enjoy a property known as strong distortion. These results are
both of interest in their own right, and are auxiliary to the main results
of Chapter 7.
In Chapter 6, we prove some technical results and establish some
asymptotic properties of a deterministic walk in a xed environment on
general countable state space. We show that these results have some
useful consequences for the case where the state space is Z.
In Chapter 7, we prove our main results for the DWDE on Z. In
6
particular, we show that under certain hypotheses the DWDE satises
a Zero-One law. Specically, we show that the DWDE is either recurrent or transient, and if transient we show that the probability that
it diverges to +∞ is either 0 or 1. An immediate consequence of this
result is that symmetric DWDEs on Z are recurrent.
In Chapter 8, we consider a class of DWDE on Z with inherent
bias. In particular, we show that this class is transient and compute
the direction of divergence.
7
Chapter 2
Random Walks in a Random
Environment
Introduction
In this chapter we present the results that are the motivation for this
thesis. In Section 2.1, we recall some basic ideas from the theory of
stochastic processes, and some basic results from Markov chain theory.
In Section 2.2, we present the rst main result due to Chung [10]. This
result gives a full characterisation of the asymptotic behaviour of a
random walk on Z in a xed, inhomogeneous, environment of transition
probabilities. In Section 2.3, we introduce the concept of a random walk
in a random environment and, under a simplifying assumption, give a
full account of a result due to Solomon [29] that fully classies the
asymptotic behaviour of such systems.
2.1
Markov Chains and Simple Random Walks
Basic notation and terminology. Given a random variable X and set of
possible outcomes A, we denote by {X ∈ A} the event that the random
variable X takes values in the set A, and we denote the probability of
this happening by P ({X ∈ A}).
We will call a sequence of random variables (all taking values on a
common set) of the form X0 , X1 , . . . or . . . , X−1 , X0 , X1 , . . . a stochas-
8
tic process, or more simply a process, and denote it by (Xn ). If for
all n, the random variable Xn takes values in a set S , we say that the
process (Xn ) is S -valued. We call S the state space of the process.
Independent random variables. Two random variables X, Y are independent if for all events {X ∈ A} and {Y ∈ B}
P ({X ∈ A & Y ∈ B}) = P ({X ∈ A})P ({Y ∈ B}).
(2.1)
More generally, a process (Xn ) is independent if
P ({Xi1 ∈ A1 & . . . & Xik ∈ Ak }) = P ({Xi1 ∈ A1 }) . . . P ({Xik ∈ Ak })
(2.2)
for all k ≥ 1, all indices i1 , . . . , ik , all events {Xi1 ∈ A1 }, . . . , {Xik ∈
Ak }.
Identically distributed random variables. Two random variables X, Y
are identically distributed if for all sets of outcomes A
P ({X ∈ A}) = P ({Y ∈ A}).
(2.3)
Denition. We say that a process (Xn ) is i.i.d. as an abbreviation
for saying that it is independent, and that the Xn s are identically distributed.
Stationarity. A process (Xn ) is stationary if
P (Xi1 ∈ A1 & . . . & Xik ∈ Ak ) = P (Xi1 +r ∈ A1 & . . . & Xik +r ∈ Ak )
for all k ≥ 1, all indices i1 , . . . , ik , all sets of outcomes A1 , . . . , Ak and
all r.
Remark. If a process (Xn ) is i.i.d. then it is stationary.
Coin-tossing process. The simplest non-degenerate example of an i.i.d.
process is the coin-tossing process, in which a (possibly biased) coin
9
is tossed repeatedly. The state of such a process at time n is just the
outcome of the n-th coin toss. This process is modelled by an i.i.d. sequence of random variables (Xn )n≥1 taking values in the set {+1, −1} the values +1 and −1 being synonymous with heads and tails, respectively.
Notation. Given events A and B , we denote by P (A|B) the conditional probability that event A occurs given that B occurs - i.e.
P (A|B) := P P(A∩B)
for all B such that P (B) > 0.
(B)
Denition 2.1. Let S be a countable set and let (pij )i,j∈S be a matrix
of non-negative real numbers with rows and columns indexed by S , such
that for all i ∈ S
X
pij = 1.
j∈S
Such a matrix is called a stochastic matrix. Let X0 , X1 , X2 , . . . be an
S -valued process satisfying the condition that
P (Xn+1 = j|X0 = i0 , X1 = i1 , . . . , Xn = in ) = pin ,j
(2.4)
for all n ≥ 0, all j ∈ S and all i0 , . . . , in ∈ S such that P (X0 =
i0 , X1 = i1 , . . . , Xn = in ) > 0. We call such a process a stationary
Markov chain. The set S is the state space of the Markov chain, and
the pij are the transition probabilities.
The stationarity of the Markov chain consists in the fact that the transition probabilities pij depend only on the states i, j , and are independent
of time. From now on stationarity is assumed and we will refer simply
to a Markov chain.
Remark. If a process (Xn )n≥1 is independent then it is Markov.
Equation (2.4) describes the so-called Markov property, and what it
says is that the future behaviour of the process, given its present state,
is independent of its past.
For all n ≥ 1 and j, i1 , . . . , in ∈ S , we let Pj (X1 = i1 , . . . , Xn = in )
10
denote the conditional probability P (X1 = i1 , . . . , Xn = in |X0 = j). It
follows from (2.4) that Pj (X1 = i1 , . . . , Xn = in ) = pji1 pi1 i2 . . . pin−1 in .
Denition 2.2. Given a Markov chain on a state space S and states
i, j ∈ S , we say that i communicates with j if
Pi (Xn = j, for some n ≥ 1) > 0.
We say that states i and j intercommunicate if i communicates with j
and j communicates with i.
We dene the communication class of a state i to be the set of all
states with which i intercommunicates. We say that the communication class of state i is closed if, for every j , i communicates with j
only if j communicates with i.
We say that a Markov chain is irreducible if its entire state space S
forms a communication class - i.e. if every pair of states intercommunicate.
We introduce some fundamental notions for a general process (Xn )
on a discrete state space S .
Denition 2.3. Given a process (Xn ) on a countable state space S ,
we say that the state i ∈ S is recurrent if
Pi (Xn = i, for some n ≥ 1) = 1.
We say that a state i is transient if it is not recurrent.
We say that the process (Xn ) is transitive on S if for all i, j ∈ S ,
Pi (Xn = j, for some n ≥ 1) = 1.
These denitions, which are central to this thesis, have some important properties in the Markov chain setting. (Proofs of Propositions
2.4-2.8 and Theorem 2.9 below can be found in Chapter 8 of [4].)
Proposition 2.4. Given a Markov chain with state space S , a state
i ∈ S is recurrent if and only if
P∞
n=1
11
Pi (Xn = i) = ∞.
Notation. Given a process (Xn ), we write {Xn = i i.o.} to denote the
event that state i is visited innitely often.
Proposition 2.5. Given a Markov chain with state space S , if a state
i ∈ S is recurrent then Pi (Xn = i i.o.) = 1, whereas if i is transient
then Pi (Xn = i i.o.) = 0.
Proposition 2.6. For any communication class in a Markov chain,
either every state in the class is recurrent, or every state in the class
is transient.
It follows from Proposition 2.6 that we may say that an irreducible
Markov chain is itself either recurrent or transient.
Proposition 2.7. If a Markov chain is irreducible and recurrent then
it is transitive.
Aperiodicity. We dene the period of a state i ∈ S to be gcd{n : p(n)
ii >
0}. If a Markov chain is irreducible it is readily shown that every state
has the same period, in which case it is natural to speak of the period
of the Markov chain itself. We say that an irreducible Markov chain is
aperiodic if it has a period of 1.
Proposition 2.8. For every irreducible aperiodic Markov chain with
nite state space S and transition matrix M := (pij )i,j∈S , there exists
a unique distribution π on S such that for every initial distribution v
on S
lim vM n = π.
(2.5)
n→∞
The simple random walk on Z. Given a coin-tossing process (Xn )n≥1
taking values ±1, we dene the simple random walk on Z to be
Un :=
n
X
i=1
12
Xi .
(2.6)
The simple random walk is not i.i.d. (for example, U2 clearly depends
on U1 ). However, by construction, the simple random walk has independent increments and it is straightforward to show that it therefore
has the Markov property, and is thus a Markov chain on Z.
Theorem 2.9. The simple random walk satises the following trichotomy.
(i) If P (X1 = 1) >
1
2
then P (limn→∞ Un = ∞) = 1.
(ii) If P (X1 = 1) <
1
2
then P (limn→∞ Un = −∞) = 1.
(iii) If P (X1 = 1) = 21 then P (lim supn→∞ Un = ∞ & lim inf n→∞ Un =
−∞) = 1.
An immediate consequence of this result is that if cases (i) or (ii) hold,
then the simple random walk is transient, whereas if case (iii) holds,
then it is recurrent and, by irreducibility, transitive.
2.2
Random Walks on
Z
in a Fixed Environment
In this section we will be concerned with the asymptotic behaviour of
a random walk on Z in a general environment of transition probabilities.
A sequence (. . . , α−1 , α0 , α1 , . . .) of real numbers in (0, 1) denes a
Markov chain (Un )n≥1 on Z with transition probabilities given by
pk,k+1 = αk
and pk,k−1 = 1 − αk .
We call the sequence of transition probabilities (. . . , α−1 , α0 , α1 , . . .) the
environment of the Markov chain. Throughout this section we assume
the environment to be xed. We also assume that 0 < αi < 1 for all
i ∈ Z, thereby ensuring that the Markov chain is irreducible.
In preparation for the main result of this section we introduce the
following notation.
13
Notation. For all n ∈ Z dene
1 − αn
αn
σn :=
(2.7)
and
(
ρn :=
σ0 σ1 . . . σn , n ≥ 0
σ0 σ−1 . . . σn , n < 0.
(2.8)
Since the Markov chain is irreducible, it follows from Proposition 2.6
that it is either recurrent or transient. Moreover, we have the following
result due to Chung (see p.65-71 of [10]).
Theorem 2.10 (Chung). For a xed environment (. . . , α−1 , α0 , α1 , . . .)
we have:
(a) If
P∞
n=1 (ρ−n )
−1
= ∞ and
P∞
n=1
ρn < ∞ then
P ( lim Un = ∞) = 1.
n→∞
(b) If
P∞
n=1 (ρ−n )
−1
< ∞ and
P∞
n=1
ρn = ∞ then
P ( lim Un = −∞) = 1.
n→∞
(c) If
P∞
n=1 (ρ−n )
−1
= ∞ and
P∞
n=1
ρn = ∞ then
P (lim inf Un = −∞ & lim sup Un = +∞) = 1.
n→∞
(d) If n=1 (ρ−n )−1 < ∞ and
exists p ∈ (0, 1) such that
P∞
n→∞
P∞
n=1
ρn < ∞ then for each i ∈ Z there
P ( lim Un = ∞|U0 = i) = p & P ( lim Un = −∞|U0 = i) = 1−p.
n→∞
n→∞
Remark. For example, if pi,i+1 = 43 for all i ≥ 0 and pi,i−1 =
i < 0, then case (d) of Theorem 2.10 holds.
3
4
for all
Before proving Theorem 2.10 we establish an intermediate result (Lemma
2.11 below), for which we require the following notation.
Notation. For i, j ∈ Z, we will use the abbreviation fij := Pi (Xn =
j, for some n ≥ 1).
14
Lemma 2.11. If i > j then
P∞
n=i ρn
,
fij = P∞
n=j ρn
and if i < j then
Pi
fij = Pjn=−∞
n=−∞
ρ−1
n
ρ−1
n
(2.9)
.
(2.10)
Proof. We establish (2.9) as the proof of (2.10) is similar. Without loss
of generality we suppose that k ≥ i ≥ j ≥ 0. Given integers i, j, k
dene
kfij
:= Pi (∃n ≥ 0 : Un = j & Ur 6= k, 0 ≤ r ≤ n).
(2.11)
Thus, kfij denotes the probability that the random walk started in state
i eventually enters state j before it enters state k . For i ∈ Z, dene
βi := 1 − αi . The probability kfij constitutes the unique solution to the
system of equations


 αi ui+1 + βi ui−1 i 6= j, k
ui =
1
i=j

 0
i = k.
(2.12)
Equation (2.12) is solved by recursion. Rearranging we obtain for j <
r<k
αr (ur+1 − ur ) = βr (ur − ur−1 )
from which it follows that
ur+1 − ur =
βr . . . βj+1
(uj+1 − 1).
αr . . . αj+1
(2.13)
Using the convention that the empty product is 1, from (2.13) we obtain
k−1
k−1
X
X
βr . . . βj+1
−1 =
(ur+1 − ur ) = (1 +
)(uj+1 − 1)(2.14)
,
α . . . αj+1
r=j
r=j+1 r
ui − 1 =
i−1
i−1
X
X
βr . . . βj+1
(ur+1 − ur ) = (1 +
)(uj+1 − 1)(2.15)
.
α
.
.
.
α
r
j+1
r=j
r=j+1
It is immediate from (2.14) that
−1 = (uj+1 − 1)[1 + σj+1 + (σj+1 σj+2 ) + . . . + (σj+1 . . . σk−1 )] (2.16)
15
Subtracting (2.14) from (2.15) we obtain
ui = (1 − uj+1 )[(σj+1 . . . σi ) + . . . + (σj+1 . . . σk−1 )].
(2.17)
Equation (2.14) also implies that
(1 − uj+1 ) = [1 + σj+1 + (σj+1 σj+2 ) + . . . + (σj+1 . . . σk−1 )]−1 . (2.18)
Substituting (2.18) back into (2.17) we obtain
Pk−1
(σj+1 . . . σi ) + . . . + (σj+1 . . . σk−1 )
n=i σj+1 . . . σn
ui =
= Pk−1
.
1 + σj+1 + (σj+1 σj+2 ) + . . . + (σj+1 . . . σk−1 )
σ
.
.
.
σ
j+1
n
n=j
(2.19)
Multiplying equation (2.19) through above and below by σ0 . . . σj we
obtain
Pk−1
(σ0 . . . σi ) + . . . + (σ0 . . . σk−1 )
n=i ρn
.
= Pk−1
k fij =
(σ0 . . . σj ) + . . . + (σ0 . . . σk−1 )
n=j ρn
(2.20)
Since 0 < αn < 1, for all n ∈ Z, it follows from the Markov property
(recall equation (2.4)) that
P (−∞ < inf Un & sup Un < ∞) = 0.
n≥0
n≥0
Thus, given i and j such that i > j , we see that kfij → fij as k → +∞.
Therefore, letting k → ∞ in equation (2.20) we obtain (2.9). This
establishes the claim.
Proof of Theorem 2.10. Since 0 < αr < 1 for all r ∈ Z, it follows from
the Markov property (2.4) that for all r ∈ Z, P (lim inf n→∞ Un = r) = 0
and P (lim supn→∞ Un = r) = 0. Hence
P (| lim inf Un | = ∞ & | lim sup Un | = ∞) = 1,
n→∞
n→∞
(2.21)
and so for almost every realisation of the random walk, it either diverges
to the left, to the right, or it is transitive. It follows from Lemma 2.11
P
P
−1
that if ∞
= ∞ and ∞
n=1 (ρ−n )
n=1 ρn < ∞ then for all i < j , fij = 1
and fji < 1, from which it now follows that P (limn→∞ Un = ∞) = 1.
This establishes part (a). Parts (b) and (c) are established similarly.
P
P
−1
In proving part (d) observe that if ∞
< ∞ and ∞
n=1 (ρ−n )
n=1 ρn <
∞, then by Lemma 2.11 for all i, j ∈ Z such that i 6= j , 0 < fij < 1 and
16
0 < fji < 1. It follows from the Markov property that for each i ∈ Z,
fii = αi fi+1,i + (1 − αi )fi−1,i . Thus, 0 < fii < 1 and so i is transient.
It follows from Proposition 2.5 that for all i ∈ Z, Pi (Un = i i.o.) = 0.
Since jumps are ±1 it follows from (2.21) that
Pi ( lim Un = −∞ or
n→∞
lim Un = ∞) = 1.
n→∞
From equations (2.9) and (2.10) it is clear that 0 < Pi (limn→∞ Un =
∞) < 1, and that this probability depends on i.
2.3
Random Walks on
Z in a Random Environment
In this section we consider a more general problem than that in the
previous section. Rather than the environment of transition probabilities being xed, we suppose that it is generated randomly. Once the
environment has been generated, the random walk is then run as before. We call such a process a random walk in a random environment
(RWRE).
Before stating the main result of this chapter, we introduce some prerequisite concepts and some useful notation.
Markov chain measure. Since the results of this section do not depend on the starting position of the RWRE we shall adopt the convention that U0 = 0. Associated with any given xed environment
α := (. . . , α−1 , α0 , α1 , . . .) is a natural measure on the space of all possible realisations of the random walk. We call this measure the Markov
chain measure, and denote it by Pα . The measure is constructed naturally by dening it rst on sets of paths specied by a nite number of
initial coordinates. For example, Pα (U1 = 1 & U2 = 0) = α0 (1 − α1 ).
Standard arguments then apply to show that Kolmogorov's Extension
Theorem can be used to extend this measure to the sigma-algebra of
all sets of paths that are generated by such cylinders. Details of this
construction can be found in [29].
Random environment on Z. Suppose that (αn )n∈Z is an i.i.d. sequence
17
of random variables, each taking values in (0, 1). Since each realisation
of this process is a xed environment on Z as per Section 2.2, we call
the process (αn )n∈Z a random environment on Z.
Denoting by µ the probability distribution on (0, 1) associated with
the random variable α0 , the i.i.d. process (αn )n∈Z induces the product
probability measure Q := . . . µ × µ × µ . . . on the space of all possible
environments (0, 1)Z .
Denition. In the spirit of Section 2.2, for each n ∈ Z we may also
dene the random variable σn as per equation (2.7), and the random
variable ρn as per equation (2.8). Since (αn )n∈Z is i.i.d. it follows that
the process (σn )n∈Z is also i.i.d. and hence that there exists a random
variable σ whose distribution is shared by each of the σn 's.
Notation. We let α denote an arbitrary xed environment, and ω
denote an arbitrary realisation of the random walk.
We are now in a position to state the main result of this section.
Theorem 2.12 (Solomon). For a simple random walk in a random
environment such that ln σ ∈ L1 (Q), we have the following trichotomy.
(i) If E(ln σ) < 0 then lim Un = ∞ for Q − a.e. α and Pα − a.e. ω .
n→∞
(ii) If E(ln σ) > 0 then lim Un = −∞ for Q − a.e. α and Pα − a.e. ω .
n→∞
(iii) If E(ln σ) = 0 then lim inf Un = −∞ and lim sup Un = +∞ for
n→∞
n→∞
Q − a.e. α and Pα − a.e. ω .
Remark. Solomon's result does not in fact require that ln σ is integrable, and shows that under more general hypotheses a similar trichotomy holds. We refer the reader to Theorem 1.7 of [29] for further
details.
We prove (i) and (iii). The proof of (ii) is analogous to that of (i).
18
Proof of Theorem 2.12(i). Suppose that E(ln σ) = −c < 0 (c > 0).
For all n ≥ 1 dene
Sn := ln σ1 + . . . + ln σn .
Since (ln σi )i∈Z is i.i.d. it follows from the Strong Law of Large Numbers
that for almost every environment there exists N ≥ 1 such that for all
n>N
Sn
− (−c) ≤ c .
n
2
(2.22)
Hence, for almost every environment there exists N ≥ 1 such that for
all n > N
−cn
Sn ≤
< 0.
(2.23)
2
From (2.23) we obtain that for almost every environment there exists
N such that
∞
∞
∞
X
n=N
Thus
ρn =
X
eSn ≤
n=N
∞
X
X
(e
−c
2
)n < ∞.
n=N
ρn < ∞, Q − a.e.
(2.24)
n=1
Since (αn )n∈Z is i.i.d. it follows that {ρn }n≥1 = {ρ−n }n≥0 in distribution. From (2.24), it follows that for almost every environment we that
1
limn→∞ ρn = 0, and hence that limn→∞ ρ1n = ∞ and limn→∞ ρ−n
= ∞.
It is now immediate that
∞
X
1
= ∞,
ρ
−n
n=1
Q − a.e.
(2.25)
From (2.24) and (2.25) it is follows that case (a) of Theorem 2.10 holds
for Q-a.e. environment. This completes the proof of (i).
The proof of Theorem 2.12(iii) requires some preparation. We present
the main steps in the argument as Lemmas 2.13 - 2.18 below. Lemmas
2.13 - 2.18 concern a general i.i.d. sequence (Xn )n≥1 of non-degenerate
(i.e. non-constant) R-valued random variables. We assume the process
to be dened on some probability space (Ω, F, P ), in the sense that
for all n ≥ 1 the random variable Xn is P -measurable. More formally,
19
for all n ≥ 1 we assume that the sigma-algebra Fn := Xn−1 (B(R)) is
contained in F , where B(R) denotes the Borel sigma-algebra generated
by open subsets of R.
Given a process (Yi )i≥0 dened on a probability space Ω, we say that
E ⊂ Ω is a permutable event if it is invariant under all nite permutations of the indices of the process.
Example. Let Y1 , Y2 , . . . be a sequence of real-valued random variables
and c ∈ R ∪ {−∞, ∞}. Dene Tn := Y1 + . . . + Yn . Since the event
{lim supk→∞ Tk = c} is independent of the order in which the Yi 's occur, it is therefore a permutable event.
We have the following Lemma, a proof of which can be found in [15].
Lemma 2.13 (Hewitt-Savage Zero-One Law). If Y1 , Y2 , . . . is an i.i.d.
process then every permutable event has probability 0 or 1.
Returning to the case of a general i.i.d. process of real-valued random
variables (Xn )n≥1 , dene
Sn :=
n
X
Xi ,
i=1
and
t+ := inf{n ≥ 1 : Sn > 0}
&
t+
0 := inf{n ≥ 1 : Sn ≥ 0}
t− := inf{n ≥ 1 : Sn < 0}
&
t−
0 := inf{n ≥ 1 : Sn ≤ 0}.
and
Lemma 2.14. Suppose that P (X1 > 0) > 0 and P (X1 < 0) > 0. Then
(a) P (lim supn→∞ Sn = ∞) = 1 or P (limn→∞ Sn = −∞) = 1, and
(b) P (limn→∞ Sn = ∞) = 1 or P (lim inf n→∞ Sn = −∞) = 1.
Proof. We prove (a) as the proof of (b) is identical. By assumption
there exists c > 0 such that P (X1 > c) > 0. For all k ∈ Z, the event
20
{kc ≤ lim supn→∞ Sn < (k + 1)c} is permutable, and it follows from
Lemma 2.13 that since (Xi )i≥1 is i.i.d. then
P ({kc ≤ lim sup Sn < (k + 1)c}) = 0 or 1.
n→∞
But since (Xi )i≥1 is i.i.d. and P (X1 > c) > 0, it follows that P ({kc ≤
lim supn→∞ Sn < (k+1)c}) = 0, and hence that P (−∞ < lim supn→∞ Sn <
∞) = 0. Since the event {lim supn→∞ Sn = ∞} is also permutable, the
result now follows.
Lemma 2.15.
+
(a) If P (t+
0 < ∞) < 1 then P (lim sup Sn = −∞) = 1, but if P (t0 <
∞) = 1 then P (lim sup Sn = ∞) = 1.
−
(b) If P (t−
0 < ∞) < 1 then P (lim inf Sn = ∞) = 1, but if P (t0 <
∞) = 1 then P (lim inf Sn = −∞) = 1.
Proof. We prove (a) as the proof of (b) is similar. Suppose that P (t+
0 <
+
∞) < 1. It follows that P (t0 = ∞) > 0. Hence P (supn≥0 Sn < ∞) >
0, and P (lim supn→∞ Sn < ∞) > 0, and it now follows from Lemma
2.14(a) that P (lim sup Sn = −∞) = 1.
Conversely, suppose that P (t+0 < ∞) = 1. It follows that P (t+0 =
∞) = 0 and hence that P (∃n ≥ 1 : Sn ≥ 0) = 1. Since (Xn )n≥1 is i.i.d.
it is stationary, and it follows that P (lim supn→∞ Sn ≥ 0) = 1. Hence,
by Lemma 2.14(a), P (lim supn→∞ Sn = ∞) = 1.
Lemma 2.16.
(a) E(t+ ) =
P (t−
0
1
= ∞)
and
(b) E(t− ) =
P (t+
0
1
.
= ∞)
Proof. We prove (a) as the proof of (b) is similar. For 0 ≤ m ≤ n we
dene the sets
Anm := {0 ≥ Sm , S1 ≥ Sm , . . . , Sm−1 ≥ Sm , Sm < Sm+1 , . . . , Sm < Sn }.
The event Anm consist the set of walks that realise their minima between
time 0 and n for the last time at time m, and hence by the total law
of probability
n
X
P (Anm ) = 1.
m=0
21
(2.26)
Since (Xn )n≥1 is an independent process we have
P (Anm ) = P ({0 ≥ Sm , S1 ≥ Sm , . . . , Sm−1 ≥ Sm })P ({Sm < Sm+1 , . . . , Sm < Sn }).
(2.27)
Since (Xn )n≥1 is i.i.d. we have
P ({Sm < Sm+1 , . . . , Sm < Sn }) = P (t−
0 > n − m).
(2.28)
From the denition of Sn , and from the fact that (Xn ) is i.i.d. it follows
that
P (0 ≥ Sm , S1 ≥ Sm , . . . , Sm−1 ≥ Sm ) =
P (0 ≥ X1 + . . . + Xm , 0 ≥ X2 + . . . + Xm , . . . , 0 ≥ Xm ) =
P (0 ≥ Xm + . . . + X1 , 0 ≥ Xm−1 + . . . + X1 , . . . , 0 ≥ X1 ) =
P (0 ≥ Sm , 0 ≥ Sm−1 , . . . , 0 ≥ S1 ) = P (t+ > m).
(2.29)
From (2.26), (2.27), (2.28) and (2.29) it follows that for all n ∈ N
n
X
P (t+ > m)P (t−
0 > n − m) = 1.
(2.30)
m=0
Since
∞
X
P (t+ > m) =
∞
X
mP (t+ = m) = E(t+ )
m=0
m=0
we complete the proof by using (2.30) to establish that
P (t−
0
= ∞)
∞
X
P (t+ > m) = 1.
(2.31)
m=0
We consider separately the cases E(t+ ) = ∞ and E(t+ ) < ∞.
Suppose that E(t+ ) = ∞. Since (P (t−0 > n))n≥1 is a monotonically
decreasing sequence of probabilities it follows that for all n ∈ N,
P (t−
0
= ∞)
n
X
m=0
+
P (t > m) ≤
n
X
P (t+ > m)P (t−
0 > n − m) = 1.
m=0
Letting n → ∞ we have P (t−0 = ∞)E(t+ ) ≤ 1, from which it is
immediate that P (t−0 = ∞) = 0, as required.
22
Suppose instead that E(t+ ) < ∞. In completing the proof we rst
show that P (t−0 = ∞) > 0. Suppose to the contrary that P (t−0 =
∞) = 0. It follows that for all > 0 there exists N such that (i) for
P
+
all n ≥ N , P (t−0 > n) < and (ii) ∞
> m) < . Letting
m=N P (t
< (E(t+ ) + 1)−1 and N large enough
2N
X
N
X
P (t+ > m)P (t−
0 > 2N − m) =
m=0
P (t+ > m)P (t−
0 > 2N − m)
m=0
2N
X
+
P (t+ > m)P (t−
0 > 2N − m)
m=N +1
≤ (
N
X
P (t+ > m)) + m=0
+
≤ (E(t ) + 1) < 1.
contradicting (2.30). Thus P (t−0 = ∞) > 0. By similar argument to
the above it follows that for all > 0 there exists N such that (i) for all
P∞
−
+
n ≥ N , P (t−
0 > n) < (1 + )P (t0 = ∞) and (ii)
m=N P (t > m) < .
Hence, for all > 0 there exists N such that
P (t−
0
= ∞)
2N
X
m=0
+
P (t > m) ≤
2N
X
P (t+ > m)P (t−
0 > 2N − m) = 1
m=0
≤ (1 +
)P (t−
0
= ∞)
2N
X
P (t+ > m) + .
m=0
Let N → ∞ establishes the result.
A stopping time with respect to a process X1 , X2 , . . . is a random variable N taking values in N such that the occurrence or non-occurrence
of the event {N = n} depends only on the values of X1 , . . . , Xn - i.e. is
measurable with respect to the sigma-algebra σ(∪nj=1 Fj ). We have the
following result whose proof can be found in Durrett [15] (see Theorem
4.1.5, p.158-159).
Lemma 2.17 (Wald's Equation). If X1 , X2 , . . . is i.i.d. such that E(|X1 |) <
∞, and N is a stopping time such that E(N ) < ∞, then E(SN ) =
E(X1 )E(N ).
23
Lemma 2.18. If E(X1 ) = 0 then
P (lim sup Sn = ∞ & lim inf Sn = −∞) = 1.
n→∞
n→∞
Proof. Firstly, we show that E(t+ ) = ∞ and E(t− ) = ∞. Suppose to
the contrary that E(t+ ) < ∞. Dene
X1+ := max{X1 , 0} & X1− := max{−X1 , 0}.
For all n ≥ 0, the event {t+ = n} is measurable with respect to
σ{∪nj=0 Fj }, and the random variable t+ is therefore a stopping time.
The random variable St+ (which is the process Sn stopped by the random variable t+ ), denotes the value attained by the process (Sn )n≥0 at
the time it rst exceeds 0. We have that
E(X1+ )
Z
=
Z
X1 · 1{X1 >0} dP =
and
E(St+ ) =
∞ Z
X
S1 · 1{t+ =1} dP
Sn · 1{t+ =n} dP.
(2.32)
(2.33)
n=1
Since Sn · 1{t+ =n} dP ≥ 0 for all n ≥ 0, it follows from (2.32) and
(2.33) that
0 ≤ E(X1+ ) ≤ E(St+ ).
(2.34)
R
Since E(t+ ) < ∞, it follows from Lemma 2.17 that
E(St+ ) = E(X1 )E(t+ ) = 0.
(2.35)
It follows from (2.34) and (2.35) that E(X1+ ) = 0, and hence that
E(X1− ) = 0. Thus E(|X1 |) = 0, and so X1 = 0, contradicting the
assumption that X1 is non-degenerate. Thus, E(t+ ) = ∞. A similar
argument shows that E(t− ) = ∞.
By Lemma 2.16 it is now immediate that
−
P (t+
0 = ∞) = 0 & P (t0 = ∞) = 0.
(2.36)
From (2.36) we have P (t+0 < ∞) = 1 and P (t−0 < ∞) = 1, and the
result now follows from Lemma 2.15.
24
Proof of Theorem 2.12(iii). Dene Sn := ln σ1 + . . . + ln σn . Since
(ln σi )i≥1 is a non-degenerate i.i.d. process it follows from Lemma 2.18
that if E(ln σ) = 0 then
Q(lim inf Sn = −∞ & lim sup Sn = ∞) = 1.
n→∞
n→∞
(2.37)
From (2.37) it is immediate that for almost every environment we have
that
∞
∞
X
n=1
and
ρn =
X
eSn = ∞
(2.38)
n=1
∞
∞
X
X
1
=
e−Sn = ∞.
ρ
n=1 n
n=1
(2.39)
Since {ρn }n≥0 = {ρ−n }n≥0 in distribution it follows from (2.39) that
for almost every environment
∞
X
1
= ∞.
ρ
n=1 −n
(2.40)
From (2.38) and (2.40) it follows that for almost every environment the
conditions of part (c) of Theorem 2.10 are satised, thus completing
the proof.
It is noteworthy that case (d) of Theorem 2.10 disappears in the i.i.d.
random environment setting. This is an issue to which we will return
in the context of deterministic walks in later chapters.
25
Chapter 3
The Deterministic Walk in a
Deterministic Environment
Introduction
In this chapter we introduce the central object of this thesis: the deterministic walk.
In Section 3.1, we introduce some basic measure theoretic properties
of dynamical systems, and dene the deterministic walk.
In Section 3.2, we consider examples of a deterministic walk on a nite
state space. In Section 3.2.1, we illustrate the denition of the deterministic walk by means of an example, showing it to be transitive in
the process. In Section 3.2.2, we show by means of a counter-example
that, even under very nice assumptions, transitivity of a deterministic
walk on a nite state space is not automatic.
In Section 3.3, we consider examples of a deterministic walk on Z.
In Section 3.3.1, we partially generalise Theorem 2.9 to non-i.i.d. settings. In Section 3.3.2, we briey introduce the deterministic analogue
of the RWRE, namely the deterministic walk in a deterministic environment. (A fuller introduction to, and treatment of, this subject is
given in Chapters 7 and 8.) In Section 3.3.3, we show that the RWRE
setup of Chapter 2 is a special case of the deterministic setup.
26
In Section 3.4, we pose the main questions that this thesis attempts to
answer.
3.1
Introducing the Deterministic Walk
Before introducing the deterministic walk, we introduce some prerequisite ideas and notation.
Measure space. We denote a general measure space by (X, F, m), where
X is the space, m is a measure on X , and F is a sigma-algebra of mmeasurable subsets of X . Normally, we omit reference to the underlying
sigma-algebra, and simply write (X, m).
We say that a measure space (X, m) is σ -nite when there exists a
countable collection {Ai }i such that X = ∪i Ai , and m(Ai ) < ∞ for all
i.
When m(X) = 1, we say that (X, m) is a probability space, and that
the measure m is a probability measure.
Measurable transformation. A transformation T : X → X of a measure
space (X, m) is measurable if for every measurable set A, the set T −1 A
is also measurable.
The deterministic walk. Let T be a measurable transformation of a
probability space (X, m), and let S be a countable set such that associated with each element i ∈ S is a measurable function fi : X → S .
We call each fi a transition function, and we call the collection (fi )i∈S
an environment on S .
Dene the skew-product transformation Tf : X × S → X × S by
Tf (x, i) := (T x, fi (x)).
(3.1)
Dene the deterministic walk on S in the xed environment (fi )i∈S to
be
Un := Ui,n (x) := π2 (Tfn (x, i))
(3.2)
27
where π2 (x, y) := y .
We will often refer to the set S as either the state space or the bre. We will also refer to the probability space (X, m) as the base, and
the map T : X → X as the base map.
Measure preserving transformation. We say that a measurable transformation T of a measure space (X, m) is measure preserving if
m(T −1 A) = m(A)
(3.3)
for every measurable set A. Equivalently, if (3.3) holds then we say
that the measure m is T -invariant, or simply invariant.
Ergodic transformation. We say that a measurable transformation
T of a measure space (X, m) is ergodic if for all measurable sets A,
A = T −1 A implies that either m(A) = 0 or m(Ac ) = 0.
We have the following classical result due to Birkho (see [33]).
Theorem 3.1 (Birkho's Ergodic Theorem). Let T : X → X be an er-
godic, measure preserving, transformation of a probability space (X, m)
and let f : X → R be such that f ∈ L1 (m). Then
n−1
1X
lim
f (T j x) =
n→∞ n
j=0
Z
f dm,
for m − a.e. x ∈ X.
X
Denition 3.2. Given a xed environment (fi )i∈S of transition functions on a state space S , for all i, j ∈ S dene
Aij := {x : fi (x) = j}.
We conclude this introductory section with the following elementary
transitivity result.
Proposition 3.3. If S consists of two elements, m(Aii ) < 1 for each
i ∈ S , and T is ergodic and measure-preserving, then the deterministic
walk is transitive on S . (Recall Denition 2.3.)
28
Proof. Since m(Aii ) < 1 for each i ∈ S , and T is ergodic and measure
preserving, it follows from Birkho's Ergodic Theorem that
−n
m(∩∞
Aii ) = 0.
n=0 T
Hence for i, j ∈ S
m({x : Ui,n (x) = j for some n ≥ 1}) = 1.
3.2
Examples of Deterministic Walks on a Finite
State Space
In Section 3.2.1, we give an example of a 3-state deterministic walk,
and ultimately show that it is modelled by an irreducible aperiodic
Markov chain. In Section 3.2.2, we consider another simple example,
but one that in spite of having some nice properties is not transitive.
3.2.1 Example: a Deterministic Markov Chain
Notation. We denote by ([0, 1], λ) the probability space consisting of
the closed unit interval equipped with Lebesgue measure.
Consider the deterministic walk with state space S = {0, 1, 2}, base
map T : ([0, 1], λ) → ([0, 1], λ) dened by
T x := 4x (mod 1),
and transition functions
(
f0 (x) :=
1,
2,
x ∈ [0, 14 ) ∪ [ 34 , 1]
x ∈ [ 41 , 34 )


 0,
f1 (x) :=
1,

 2,
x ∈ [0, 12 )
x ∈ [ 34 , 1]
x ∈ [ 12 , 34 )
29
(3.4)
(3.5)
and
(
f2 (x) :=
0,
2,
x ∈ [0, 43 )
x ∈ [ 34 , 1].
(3.6)
To illustrate how the deterministic walk works in practice we calculate
the itinerary generated by the point x = 193 given that the process is
started in state 2. We see that
3
19
3
(mod 1), f2 ( 19
)) = ( 12
, 0).
19
3
Tf2 ( 19
, 2) = (4 ×
12
19
10
(mod 1), f0 ( 12
)) = ( 19
, 2)
19
3
, 2) = (4 ×
Tf3 ( 19
10
19
2
(mod 1), f2 ( 10
)) = ( 19
, 0)
19
3
Tf4 ( 19
, 2) = (4 ×
2
19
2
8
(mod 1), f0 ( 19
)) = ( 19
, 1)
3
, 2) = (4 ×
Tf5 ( 19
8
19
8
13
(mod 1), f1 ( 19
)) = ( 19
, 0)
3
, 2) = (4 ×
Tf ( 19
Continuing we obtain
..
.
Thus, the itinerary of the walk U2,n ( 193 ) for time n = 0, . . . , 5 is
2, 0, 2, 0, 1, 0 . . . .
It is natural to ask whether or not this 3-state deterministic walk is
transitive, and whether there exists an asymptotic distribution on S ?
We show that the answer to both questions is "yes". Moreover, we
show that the above example is a model of an irreducible aperiodic
Markov chain. Before doing so we introduce some prerequisite ideas,
and prove some useful results about a particular class of transformations of the unit interval, which we introduce shortly.
Partition. We say that a collection of intervals {I1 , I2 , . . .} is a partition of [0, 1] if the collection is pairwise disjoint, λ(Ij ) > 0 for all
j ≥ 1, and ∪j Ij = [0, 1].
Piecewise linear full-branch maps. Given a transformation T : [0, 1] →
[0, 1] we say that T is piecewise linear if there exists a partition P =
30
{I1 , I2 , . . .} of [0, 1] consisting of intervals such that for all j ≥ 1, T|Ij
is linear. We further say that T is full branch on P if for all j ≥ 1,
T (Ij ) = [0, 1].
Proposition 3.4. Let T be a piecewise linear and full-branch map with
respect to the partition P := {I1 , I2 , . . .}. Then for every measurable
set A, for all n ≥ 1 and all j ≥ 1
λ(Ij ∩ T −n A) = λ(Ij )λ(A).
(3.7)
Proof. In the case n = 1, for all measurable A ⊂ [0, 1] the inverse
branch TI−1
: [0, 1] → Ij scales A uniformly such that λ(TI−1
A) = λ(Ij ∩
j
j
−1
n
T A) = λ(Ij )λ(A). Since the mapping T satises the hypotheses
with respect to the partition
Pn :=
n−1
_
T −i P = {a0 ∩ T −1 a1 ∩ . . . ∩ T −n+1 an−1 : a0 , . . . , an−1 ∈ P},
i=0
the result follows by induction on n ≥ 2,
Corollary 3.5. A piecewise linear and full-branch map T preserves
Lebesgue measure.
Proof. For each measurable set A, λ(T −1 A) = λ(∪j≥1 Ij ∩ T −1 A). Since
the Ij 's are disjoint it follows from (3.7) that λ(∪j≥1 Ij ∩ T −1 A) =
P
j≥1 λ(Ij )λ(A) = λ(A).
Proposition 3.6. Let S be the state space and suppose that the base
map T : [0, 1] → [0, 1] is piecewise linear and full-branch on a partition
P consisting of intervals. If each transition function in the environment
(fi )i∈S is constant on elements of P then (Un )n≥0 is a Markov chain
with transition probabilities pij = λ(Aij ). (Recall Denition 3.2.)
Proof. By construction, for all n ≥ 1, and all j, i1 , . . . , in ∈ S ,
Pj (U1 = i1 , . . . , Un = in ) = λ(Aj,i1 ∩ T −1 Ai1 ,i2 ∩ . . . ∩ T −n+1 Ain−1 ,in ).
(3.8)
By assumption each Aij is the union of partition elements in P . It
follows from Proposition 3.4 and Corollary 3.5 that
λ(Aj,i1 ∩T −1 Ai1 ,i2 ∩. . .∩T −n+1 Ain−1 ,in ) = λ(Aj,i1 )λ(Ai1 ,i2 ) . . . λ(Ain−1 ,in )
(3.9)
31
It is immediate from (3.8) and (3.9) that for all n ≥ 1, and all j, ii , . . . , in+1 ∈
S,
Pj (Un+1 = in+1 |U1 = i1 , . . . , Un = in ) = λ(Ain ,in+1 )
= Pj (Un+1 = in+1 |Un = in ).
Thus, (Un )n≥0 is a Markov chain with a transition matrix whose entries
are given by pij = λ(Aij ).
It follows from Proposition 3.6 and equations (3.4), (3.5) and (3.6)
that the above example of a 3-state deterministic walk is a model of
the Markov chain whose transition matrix is

0

(pij )i,j∈S :=  12
3
4
1
2
1
4
0

1
2
1
.
4
1
4
(3.10)
Since the matrix in (3.10) is irreducible and aperiodic, it follows that
the deterministic walk is indeed transitive and, by Proposition 2.8,
there does exist a unique stable distribution on the S .
3.2.2 Counter-example to Transitivity
In light of Proposition 3.3 it might be hoped that if S is nite, T is
ergodic and measure preserving, and m({x ∈ X : fi (x) = j}) > 0
for all i, j ∈ S , then (Un )n≥0 is necessarily transitive. As the following
counter-example shows, this is not the case: Consider the deterministic
walk on the state space S = {0, 1, 2} driven by the transformation
T : [0, 1] → [0, 1], where T x := 2x mod 1, and with transition functions
f0 , f1 and f2 given by

1 3

 0, x ∈ [ 4 , 4 )
f0 (x) =
1, x ∈ [0, 14 )

 2, x ∈ [ 3 , 1]
4

1

 0, x ∈ [0, 4 )
f1 (x) =
1, x ∈ [ 34 , 1]

 2, x ∈ [ 1 , 3 )
4 4
32
and

3

 0, x ∈ [ 4 , 1]
f2 (x) =
1, x ∈ [ 14 , 43 )

 2, x ∈ [0, 1 ).
4
Dening β := {[0, 12 ), [ 21 , 1)}, by inspection we see that the skew-product
Tf : [0, 1]×S → [0, 1]×S decomposes the product state space β ×S into
two closed communication classes given by {[0, 12 ) × {2} , [ 12 , 1] × {1}}
and {[0, 21 ) × {0} , [ 12 , 1] × {0} , [0, 21 ) × {1} , [ 12 , 1] × {2}}.
Since the rst communication class does not project onto the whole
bre S , (in particular, the state 0 is not represented in this class) it
follows that the deterministic walk is not transitive on S . In fact, it is
easy to show that Tf induces a Markov chain on the state space β × S
(with respect to the product measure Lebesgue × counting measure),
and it follows that any walk started in the second communication class
necessarily visits all three states innitely often.
3.3
Examples of Deterministic Walks on
Z
In Section 3.3.1, we consider simple examples of deterministic walks on
Z, and show how these partially generalise Theorem 2.9 to the non-i.i.d.
setting. In Section 3.3.2, we give a brief introduction to the concept of
a deterministic walk in a deterministic environment. This topic will be
dealt with in greater detail in Chapters 7 and 8. In Section 3.3.3, we
give an example of a deterministic walk on Z that serves as a model of
the RWRE setup in Chapter 2.
3.3.1 Deterministic Walks in a Homogeneous Environment
An equivalent, but more convenient, way of representing a deterministic
walk on Z is to dene
Tf (x, i) := (T x, i + fi (x)) & Un := Ui,n (x) := π2 (Tfn (x, i))
where (fi )i∈Z is an environment of measurable functions of the form
fi : X → Z.
33
We say that an environment (fi )i∈Z is homogeneous if for all i ∈ Z,
fi = h, for some function h : X → Z. The following result, which
generalises parts (i) and (ii) of Theorem 2.9, follows immediately from
Theorem 3.1.
Proposition 3.7. Suppose that T is ergodic and measure preserving,
(fi )i∈Z is homogeneous such that fi = h for all i ∈ Z, and that h ∈
L1 (m). Then:
R
(a) If X hdm > 0 then m({x : limn→∞ Ui,n (x) = ∞}) = 1, for all
i ∈ Z.
R
(b) If X hdm < 0 then m({x : limn→∞ Ui,n (x) = −∞}) = 1, for all
i ∈ Z.
In the ergodic setting, unlike the i.i.d. setting, the case where X hdm =
0 is indeterminate, and we need a stronger property than ergodicity to
be able to say something.
R
Tail eld. Given a measurable transformation T : X → X of a measure
space (X, F, m), and denoting the sigma-eld Fn := T −n F for n ≥ 0,
we dene the tail eld to be ∩n≥0 Fn . If A ∈ ∩n≥0 Fn then we say that
A is a tail event.
Exact transformation. We say that a measurable transformation T
of a probability space (X, m) is exact if every tail-event has measure 0
or 1.
It can be shown that if T is exact then it is ergodic.
Proposition 3.8. Suppose that the hypotheses
of Proposition 3.7 hold,
R
and, moreover, that T is exact and
the following holds:
X
hdm = 0. Then exactly one of
(a) m({x : limn→∞ Ui,n (x) = ∞}) = 1, for all i ∈ Z.
(b) m({x : limn→∞ Ui,n (x) = −∞}) = 1, for all i ∈ Z.
34
(c) m({x : lim supn→∞ Ui,n (x) = ∞ &
1, for all i ∈ Z.
lim inf n→∞ Ui,n (x) = −∞}) =
(d) For all i ∈ Z and m-a.e. x ∈ X , the orbit {Ui,n (x)}n≥0 is bounded.
Proof. The result follows from the fact that the events {lim inf n→∞ Un =
+∞}, {lim inf n→∞ Un = −∞}, {lim supn→∞ Un = +∞} and {lim supn→∞ Un =
−∞} are all tail events, and therefore, since T is exact, have probability
of 0 of 1.
3.3.2 Deterministic Walks in a Deterministic Environment
In Section 2.3 we introduced random walks in a random environment.
We consider an analogous setup for the deterministic walk where the
environment of transition functions on the state space must rst be
generated before the deterministic walk is run.
In Chapters 7 and 8 our primary interest in such systems will be
in the situation where the state space is Z, and where environments
(. . . , f−1 , f0 , f1 , . . .) of transition functions are generated by an i.i.d.
or, more generally, an ergodic and stationary process. Because such
processes can be described deterministically, we call such a system a
deterministic walk in a deterministic environment on Z (DWDE).
3.3.3 A Deterministic Version of a Random Walk on Z
In this section we show how the random walk in an arbitrary xed environment on Z considered in Section 2.3 models a deterministic walk,
and we give an example of a DWDE that can be modelled by a RWRE.
Denition. We dene the partition
n
n
2 −1
P0 := {[0, 12 ), [ 21 , 34 ), [ 43 , 78 ), . . . , [ 2 2−2
n , 2n ), . . .},
and let
T0 : [0, 1] → [0, 1]
be a piecewise linear and full-branch map with respect to P0 .
Since every real number in [0, 1] can be expressed as the countable
35
sum of dyadic fractions, it follows that for every p ∈ [0, 1] there exists
a set Ap such that λ(Ap ) = p, and Ap is expressible as the union of
intervals in the partition P0 .
For each p ∈ (0, 1) we dene the function
(
fp (x) :=
1, x ∈ Ap
−1, otherwise.
(3.11)
The following proposition shows how we can construct a deterministic
walk that is modelled by a random walk on Z in an arbitrary xed
environment, as per Section 2.2.
Proposition 3.9. Given a sequence of probabilities . . . , α−1 , α0 , α1 , . . .,
the deterministic walk on Z with base transformation T0 , and environment
(. . . , fα−1 , fα0 , fα1 , . . .),
is modelled by the random walk on Z in the environment
(. . . , α−1 , α0 , α1 , . . .).
Proof. Fix n ≥ 1 and k0 , . . . , kn+1 ∈ Z. We have
P (U1 = k1 , . . . , Un+1 = kn+1 |U0 = k0 ) = λ(∩nj=0 T0−j Bj )
where


 Aαj , if kj+1 − kj = 1
Bj :=
Acαj , if kj+1 − kj = −1

 ∅, otherwise
(3.12)
for j = 0, . . . , n.
Since the sets Aαj , Acαj are unions of intervals in the partition P0 ,
and since T0 is piecewise linear full-branch on such intervals it follows
that from Proposition 3.6 that the deterministic walk is modelled by a
Markov chain on Z, and hence that
P (Un+1 = kn+1 |U0 = k0 , . . . , Un = kn ) = P (Un+1 = kn+1 |Un = kn ).
36
Moreover
P (Un+1


 λ(Aαn ) = αn , if kn+1 − kn = 1
= kn+1 |Un = kn ) =
λ(Acαn ) = 1 − αn , if kn+1 − kn = −1 (3.13)

 0, otherwise.
This completes the proof.
An example of a deterministic version of a RWRE. Suppose that the
environment of transition functions is generated by an i.i.d. process
by choosing a function g1 with probability 13 , and a function g2 with
probability 23 , where
(
1, x ∈ [ 34 , 1]
−1, x ∈ [0, 43 )
(3.14)
1, x ∈ [0, 12 ) ∪ [ 78 , 1]
−1, x ∈ [ 21 , 78 ).
(3.15)
g1 (x) :=
and
(
g2 (x) :=
Since both g1 and g2 are constant on elements of P0 , it follows from
Proposition 3.9 that for every choice of environment the deterministic
walk is modelled by the random walk on Z. Since
E(ln σ) = 31 ln
1− 14
1
4
+ 32 ln
1− 58
5
8
≈ 0.026 > 0,
it follows from Theorem 2.12 that P (limn→∞ Un = −∞) = 1.
3.4
Main Questions
Having dened, and given examples of, deterministic walks we are now
in a position to state the principal questions with which this thesis is
concerned. These are as follows:
1. In the setting of a deterministic walk on a nite bre in an arbitrary xed environment, what hypotheses ensure that:
(i) The deterministic walk is transitive?
37
(ii) The deterministic walk has an asymptotic occupation time
distribution?
2. In the setting of the DWDE on Z, how can we extend results of the
avour of Chung (Theorem 2.10) and Solomon (Theorem 2.12) to
systems in which the deterministic walk in a typical environment
does NOT have the Markov property (as dened in equation
(2.4))? Moreover, how can these results be achieved for DWDEs
in which jumps sizes other than ±1 are possible? In particular
we ask:
(i) For a deterministic walk in a given xed environment, what
hypotheses ensure that the probability that the deterministic
walk is recurrent is either 0 or 1?
(ii) What hypotheses ensure that the probability that the DWDE
is recurrent is either 0 or 1? Moreover, in the case where it
is transient with probability 1, what additional conditions
ensure that the DWDE diverges to +∞ with probability 0
or 1?
(iii) While in general there does not exist a sharp criterion for
classifying the asymptotic behaviour of the DWDE, we may
ask if we can construct non-trivial examples in which the
recurrence or transience of the DWDE is computable, and if
transient, decide whether it is to +∞ or −∞ that the process
diverges?
Questions 1 is the subject of Chapter 4, while Question 2 is the subject
of the remaining chapters.
38
Chapter 4
Properties of a Deterministic
Walk on a Finite State Space
Introduction
This chapter is principally concerned with establishing general results
about the behaviour of a deterministic walk on a nite state space.
In Section 4.1, we introduce a class of dynamical systems called Markov
maps. This class of transformations and their associated properties are
of central importance to the rest of the thesis.
In Section 4.2, we present the main result of the chapter, in which
we establish sucient conditions for the deterministic walk to be transitive on a nite state space. Having established this result, we show
that transitivity on a nite state space is in some sense robust to perturbations of the environment.
In Section 4.3, we introduce the key notion of distortion, and show
how deterministic walks driven by Markov maps with this property
admit asymptotic occupation times.
En route to the main results of this chapter we establish some auxiliary results that will also be of importance in later chapters.
39
4.1
Markov maps
Non-singular map. A measurable transformation T of a measure space
(X, m) is said to be non-singular if for all measurable sets A, m(A) = 0
implies that m(T −1 A) = 0.
Partition. Given a measure space (X, m), we say that a collection
β , consisting of subsets of X of positive measure, is a partition of X if
(i) for all a, b ∈ β either a = b or m(a ∩ b) = 0 and
(ii) ∪a∈β a = X .
Markov map. A non-singular transformation T of a measure space
(X, m) is said to be Markov, with a measurable Markov partition β of
X , if
(i) for all a ∈ β , T (a) is the union of elements of β ,
(ii) T|a : a → T (a) is a bijection.
Cylinder set. Given a Markov map T with with Markov partition
β , for all n ≥ 1 and all sequences a0 , . . . , an−1 ∈ β , we call the set
−j
a := ∩n−1
aj a cylinder set of rank n or an n-cylinder. We denote
j=0 T
by βn the collection of all n-cylinders. We say that a cylinder set a is
admissible if m(a) > 0.
Notation. We denote the rank of a cylinder set a by |a|.
Denition 4.1. We will sometimes use the following standard cylinder
set notation
[a0 , . . . , an−1 ] :=
n−1
\
T −j aj .
j=0
We say that a Markov partition β separates points in X if for all x, y ∈
X such that x 6= y , there exists n ≥ 1 and a, b ∈ βn , such that a∩b = ∅,
x ∈ a and y ∈ b.
40
In Chapter 2, we recalled various notions for Markov chains. Here
we introduce analogous ideas for Markov maps. It will be apparent
denitions 4.2 and 4.3 below are analogues of denitions 2.2 and 2.3
for Markov chains, respectively. Throughout we let T be a Markov
transformation of a σ -nite measure space (X, m) with Markov partition β .
Denition 4.2. Given a, b ∈ β we say that a communicates with b if
m({x ∈ a : T n x ∈ b for some n ≥ 1}) > 0.
Given a, b ∈ β , we say that a and b intercommunicate if a communicates with b and vice versa.
Given a ∈ β we dene the communication class of a to be the set
of all b ∈ β with which it intercommunicates. (We observe that intercommunication is an equivalence relation.)
We say that a communication class C is closed if for all a ∈ C and all
b ∈ β , a communicates with b, only if b communicates with a. (When
β is nite, the existence of closed communication classes is automatic.)
We say that β is irreducible if β is itself a communication class under
T.
By analogy with the concept of a recurrent/transient state and the
concept of a transitive process as given in Dention 2.3, the following
denes what it means for a partition element to be recurrent/transient,
and for a communication class to be transitive.
Denition 4.3. We say that a partition element a ∈ β is recurrent if
for a.e. x ∈ a there exists n ≥ 1 such that T n x ∈ a.
We say that a ∈ β is transient if it is not recurrent.
We say that a communication class C is transitive if for all a, b ∈ C , and
41
for a.e. x ∈ a, there exists n ≥ 1 such that T n x ∈ b. We say that T is
transitive if the Markov partition β forms a transitive communication
class.
Remark. We note that if T is transitive on a communication class C ,
then since T is non-singular it follows that for all a, b ∈ C and for a.e.
x ∈ a, T n x ∈ b innitely often.
Theorem 4.4. Let T be a Markov transformation of a probability space
(X, m) with Markov partition β . Given a ∈ β , if its communication
class C is not closed, then a is transient.
Proof. Since C is not closed, it follows that there exists b ∈ β such
that a communicates with b, but not vice versa. In particular, there
exist d1 , . . . , dk ∈ β such that di 6= a, i = 1, . . . k, and [a, d1 , . . . , dk , b]
is admissible. Since b does not communicate with a, it follows that
m({x ∈ a : T r x ∈
/ a, ∀r ≥ 1}) ≥ m([a, d1 , . . . , dk , b]) > 0. Thus a is
transient.
Theorem 4.5. Let T be a Markov transformation of a probability space
(X, m) with Markov partition β . If a ∈ β is recurrent, then T r x ∈ a
i.o. for a.e. x ∈ a.
Proof. Let a0 := {x ∈ a : T r x ∈
/ a, ∀r ≥ 1}. By assumption, m(a0 ) = 0.
Dening E := {x ∈ a : T n x ∈ a i.o.}, it follows that E = a\(a ∩
−n 0
∪∞
a ). Since T is non-singular it follows that m(E) = m(a) −
n=0 T
∞
m(a ∩ ∪n=0 T −n a0 ) = m(a).
4.2
Transitivity of a Deterministic Walk on a Finite
State Space
In this section, we assume that a deterministic walk on a nite state
space S is driven by a Markov transformation T of a probability space
(X, m), with Markov partition β . In proving the main result of the
section, Theorem 4.7 below, it will also be convenient to assume that
transition functions are constant on elements of β , but as we note later,
this assumption is not entirely necessary.
42
Denition. Dene the probability measure
µ := m ×
counting measure
.
#S
(4.1)
The next result, while elementary, is of central importance in the immediate, and in later chapters.
Proposition 4.6. The skew-product Tf : X × S → X × S is a Markov
transformation of the probability space (X ×S, µ) with Markov partition
β × S.
Proof. Since T is non-singular with respect to m, it follows that Tf is
non-singular with respect to the probability measure µ. For all a ∈ β
and i ∈ S , since by assumption fi is constant on a, there exists j ∈ S
such that
Tf (a × {i}) = T a × {j}.
(4.2)
Since T is one-one on elements of β it follows from (4.2) that Tf is
one-one on elements of β × S . Also, since T is Markov, it follows that
T a is the union of elements of β . This completes the proof.
Notation. Dene β̃ := β × S and, for n ≥ 1, β̃n :=
Wn−1
j=0
Tf−j β̃ .
It is an immediate consequence of the hypotheses stated at the outset of this section that for all n ≥ 1 and all v ∈ β̃n there exists a ∈ βn ,
i, k ∈ S such that
v = a × {i} & Tfn (v) = T n a × {k}.
(4.3)
Theorem 4.7. Suppose that the state space S is nite and that the
base map T is ergodic and invariant, with nite Markov partition β .
Then the deterministic walk is transitive on S if and only if every closed
communication class in β × S projects onto the whole of S .
Proof. By Proposition 4.6, Tf : X × S → X × S is a Markov map with
Markov partition β̃ . We observe that the "only if" direction of the proof
is trivial, in that if there exists a closed communication class that does
43
not project onto the whole of S then, clearly, the deterministic walk
cannot be transitive. In proving the "if" part of the result we in fact
prove something stronger.
(i) Tf is transitive on closed communication classes in β̃ .
(ii) If a partition element a × {i} ∈ β̃ does not lie in a closed communication class then µ{(y, j) ∈ X × S : Tfn (y, j) ∈ a × {i} i.o.} = 0.
Since S and β are nite, so is β̃ , and it follows from (i) and (ii) that
with probability 1, an orbit under the skew-product dynamics must
eventually end up in a closed communication class C ⊂ β̃ on which the
orbit is transitive thereafter. Hence, the deterministic walk is transitive
on S provided that every closed communication class projects onto the
whole of S .
Proof of (i): Since T is Markov with Markov partition β , and the
environment of transition functions (fi )i∈S are constant on elements
of β , it follows that for all n ≥ 1, every admissible n-cylinder x :=
[x0 , . . . , xn−1 ], and all r0 ∈ S , there is a unique itinerary on S
r0 , r1 , . . . , rn−1
such that fri (xi ) = ri+1 for i = 0, . . . , n − 2. We thus dene
t(x, r0 ) := rn−1 .
Fix a × {j}, b × {k} ∈ β̃ such that both lie in the same closed communication class C . Let
a × {j1 }, . . . , a × {jl } & b × {i1 }, . . . , b × {in }
(4.4)
be enumerations of all elements in C whose β -coordinates are a and b
respectively. Since T is ergodic and measure preserving, such collections are non-empty. We show that if we start in the communication
class C then we must eventually visit b × {k}.
Since b × {i1 }, . . . , b × {in } all lie in the closed communication class
C , then for each s = 1, . . . , n there exists a cylinder w(is ) of the form
[x1 , x2 , . . . , b] such that
t([b, w(is )], is ) := k,
44
(4.5)
and [b, w(is )] is admissible (where in an abuse of notation we have
written [b, w(is )] as shorthand for [b, x1 , x2 , . . . , b]). Equation (4.5) says
that if the skew-product visits the set [b, w(is )] × {is }, then it must
eventually visit the partition element b × {k}.
Since a × {j1 }, b × {k} ∈ C , there exists an admissible cylinder E1
of the form [a, . . . , b] such that
t(E1 , j1 ) = k.
For r = 1, . . . , l − 1 we may inductively dene admissible cylinders
Er+1 = [Er , w(t(Er , jr+1 ))].
(4.6)
For each r = 1, . . . , l the cylinder Er is of the form [a, . . . , b]. Thus,
t(Er , jr+1 ) is the S -coordinate at time |Er | − 1 of every point in Er ×
{jr+1 }. Hence, every point in Er × {jr+1 } visits the partition element
b × {t(Er , jr+1 )} at time |Er | − 1. It follows by construction that every
point in the set Er+1 × {jr+1 } as dened in (4.6) will visit the partition
element b × {k} at time |Er+1 | − 1.
Since m(El ) > 0 it follows from Birkho's Ergodic Theorem that
the base dynamics must eventually visit El . Since we are in the closed
communication class C , and since El ⊂ a, it follows that when the
base dynamics visit the set El , the deterministic walk visits one of the
states j1 , . . . , jl . But since El ⊂ El−1 ⊂ . . . ⊂ E1 it follows that we
must eventually visit the partition element b × {k}. This completes the
proof of (i).
Proof of (ii): Fix a × {i} lying in a communication class C that is
not closed, and let a × i1 , . . . , a × il be all the other partition elements
in C whose β -coordinate is a. Fix b × {k} ∈ β̃ with which a × {i}
communicates, but which does not communicate with a × {i}. Let A
denote the set of partition elements that intercommunicate with a×{i}
whose β -coordinate is b. Also, let A0 denote set of partition elements
whose β -coordinate is b, with which a × {i} communicates, but that do
not communicate with a × {i}. Clearly, b × {k} ∈ A0 .
By a similar argument to that given in the proof of part (i), for
all b × {j} ∈ A there exists a cylinder w(j) = [x1 , x2 , . . . , b] such that
b × {t([b, w(j)], j)} ∈ A0 .
45
Since a × {i} communicates with b × {k} there exists an admissible
cylinder E0 of the form [a, . . . , b] such that
t(E0 , i) = k.
For r = 0, . . . , l − 1 we may inductively dene admissible cylinders
(
Er+1 :=
Er , if b × {t(Er , ir+1 )} ∈ A0
[Er , w(t(Er , ir+1 ))], if b × {t(Er , ir+1 )} ∈ A.
(4.7)
Let a × {s1 }, . . . , a × {st } be the set of all partition elements in β̃ whose
β -coordinate is a, and with which a × {i} communicates, but which do
not communicate with a × {i}.
Assuming that the skew-product dynamics start in a × {i} then by
Birkho's Ergodic Theorem, the base dynamics must (with probability
1) eventually make a rst visit to the set El ⊂ a, at which point, the
deterministic walk must be in one of the states i, i1 , . . . , il , s1 , . . . , st . If
it is one of the si (for i = 1, . . . , t), then we cannot return to a × {i}
by assumption. If instead it is in one of the states i, i0 , . . . , il then,
since El ⊆ El−1 ⊆ . . . ⊆ E0 , by the same argument as in part (i), it
must either eventually visit the state b × {k} or some other element of
the set A0 . By denition, it cannot return from any of these partition
elements to the partition element a × {i}. This completes the proof of
(ii).
For immediate purposes it will be convenient to say that x ∈ X is
transitive if the associated orbit of the deterministic walk {Ui,n (x)}∞
n=0
visits every state in S at least once, and that a set A ⊂ X transitive
if for all x ∈ A, x is transitive. (Note that this notion of transitivity
is distinct from those given Denition 2.3 and Denition 4.3, and is
not used again beyond this section.) Under the hypotheses of Theorem
4.7 it is immediate its proof that if the deterministic walk is transitive on the state space S , then there must exist a transitive cylinder
set a of positive measure. (Recall the cylinder set El in the proof of
Theorem 4.7.) Any perturbation (fi0 )i∈S of the environment (fi )i∈S of
transition functions that preserves the existence of a transitive positive
measure subset a0 ⊂ a will, by Birkho's Ergodic Theorem, preserve
the transitivity of the deterministic walk. Therefore, we may say that
46
the transitivity of the deterministic walk on a nite bre is robust to
perturbations of the environment. It follows that the hypothesis that
transition functions are constant on elements of the Markov partition
is not a necessary condition for the deterministic walk to be transitive
on a nite state space.
4.3
Asymptotic Occupation Times for a Deterministic Walk on a Finite State Space
In this section, we consider a deterministic walk on a nite state space
S that is driven by a Markov transformation T of a probability space
(X, m) with a Markov partition β (not necessarily nite), and we assume that transition functions are constant on elements of β . We
establish conditions for the existence of asymptotic occupation times.
The following denition is given in Chapter 4 of [1].
Denition 4.8. Let T : X → X be a Markov transformation of a
σ -nite measure space (X, m), with Markov partition β . By virtue of
the local invertibility of a Markov transformation, for all n ≥ 1 and for
each a ∈ βn , we may dene the the bijection
va : T n a → a
x 7→ T −n x.
By the non-singularity of T , we may dene the Radon-Nikodym derivative
−n
d(m ◦ T )
,
dm
such that for every measurable set B
Z
va0 dm = m(a ∩ T −n B).
va0 :=
T n a∩B
(4.8)
We say that T has Strong Distortion, with distortion constant D ≥ 1,
if for all n ≥ 1 and for all a ∈ βn , and for a.e. x, y ∈ T n a,
va0 (x)
≤ D.
va0 (y)
47
(4.9)
Example 4.9. We call a C 2 , uniformly expanding Markov transfor-
mation T : [0, 1] → [0, 1], an Adler transformation if (i) it has Markov
partition β consisting of intervals, such that for all a ∈ β , T |a is strictly
monotonic and T |a extends to a C 2 function on a, (ii) T 0 (x) 6= 0 for
00 (x)|
all x ∈ [0, 1] and (iii) supa∈β supx∈a |TT 0 (x)
2 < ∞. It can be shown that
Adler transformations have the Strong Distortion Property. (In Example 5.7 we consider an example of a Markov map that does not have
the Strong Distortion Property.)
Proposition 4.10. Suppose that the map T has the Strong Distortion
Property. Then the skew-product Tf : X × S → X × S has the Strong
Distortion Property with respect to the measure µ.
Proof of Proposition 4.10. We show that µ has the same distortion under Tf with respect to the partition β̃ as the measure m has under T
with respect to β . Since for all n ≥ 1 and a ∈ βn , T n |a : a → T n a is a
bijection, it follows that for all ã ∈ β̃n , Tfn |ã : ã → Tfn (ã) is also a bijection, and, therefore, that the inverse map vã : Tfn (ã) → ã exists. Since
Tf is non-singular with respect to µ, the Radon-Nikodym derivative
vã0 :=
d(µ ◦ Tf−n )
dµ
exists. By (4.3) we may write ã = a × {i} and Tfn (ã) = T n a × {k}, for
some i, k ∈ S . For every m-measurable set B we have
Z
T n a∩B
va0 (x)dm(x) = m(a ∩ T −n B)
= µ(a ∩ T −n B × {i})
Z
=
vã0 (x, k)dµ(x, k).
(4.10)
T n a∩B×{k}
Since vã0 (x, k) is independent of k it follows that vã0 (x, k) = g(x), for
1
some g : T n a → R. Denoting counting measure × #S
by γ it follows
from the denition of m that
Z
T n a∩B×{k}
vã0 (x, k)dµ(x, k)
Z
g(x)dm × dγ
=
T n a∩B×{k}
Z
=
g(x)dm(x).
T n a∩B
48
(4.11)
Since the set B was arbitrary, it follows from (4.10) and (4.11) that
g = va0 . Thus, for m-a.e. x ∈ T n a ∩ B
vã0 (x, k) = va0 (x).
This completes the proof.
Equivalent measures. Let (X, m) be a measure a space. A subset
A ⊂ X is called a null-set of m if m(A) = 0. Two measures, m and
m0 , on X are said to be equivalent if they have the same null-sets, and
we denote this by m ≈ m0 .
Finite images. We say that a Markov transformation T of probability space (X, m), with Markov partition β , has Finite Images if
#{T a : a ∈ β} < ∞.
The following result combines Lemma 4.4.1 and Theorem 4.6.3 in [1].
(We note that [1] uses the term topologically transitive instead of irreducible.)
Proposition 4.11. Let T be an irreducible Markov transformation T
of probability space (X, m), with Markov partition β , and with Strong
Distortion and Finite Images. Then there exists an ergodic, invariant,
probability measure m0 ≈ m.
We may now state the main result of this section.
Theorem 4.12. Suppose that the state space S is nite and that the
map T is a Markov map, with Markov partition β , that has Strong
Distortion and Finite Images. Suppose also that β × S is irreducible
under Tf . Then there exists a distribution (πi )i∈S such that for all
i, j ∈ S
n−1
1X
lim
1i (Uj,r (x)) = πi
n→∞ n
r=0
for m − a.e. x.
(4.12)
Proof. Since T has Finite Images and since S is nite, it follows that
Tf also has the Finite Images.
49
By Proposition 4.10, Tf has the Strong Distortion Property with
respect to µ.
By assumption, β × S is irreducible under Tf , and it follows from
Proposition 4.11 that Tf has an ergodic, invariant, probability measure
µ0 ≈ µ. Thus, if for each i ∈ S we dene πi := µ0 (X × {i}), it follows
from Birkho's Ergodic Theorem that
n−1
1X
lim
1i (Uj,r (x)) = πi
n→∞ n
r=0
for µ0 − a.e. (x, j).
Equation (4.12) now follows from (4.13).
50
(4.13)
Chapter 5
Properties of Markov Maps
with Strong Distortion
In this chapter we show that standard properties of Markov chains go
over to Markov maps under suitable distortion conditions. Throughout,
we assume that T : X → X is a Markov transformation of a σ -nite
measure space (X, m) with Markov partition β , such that m(a) < ∞
for all a ∈ β .
The rst two results that we present derive from Aaronson (see Chapter
4 of [1]).
Proposition 5.1. Suppose that T : X → X has the Strong Distortion
Property. Then there exists D ≥ 1 such that all n ≥ 1, for all a ∈ βn ,
and for any measurable set B ,
m(a ∩ T −n B)
≤ Dva0 (x),
n
m(B ∩ T a)
D−1 va0 (x) ≤
(5.1)
for a.e. x ∈ B ∩ T n a.
Proof. By the Strong Distortion Property, for a.e. x ∈ B ∩ T n a
D−1 va0 (x)m(B
n
Z
∩ T a) ≤
B∩T n a
va0 dm ≤ Dva0 (x)m(B ∩ T n a).
(5.2)
The inequality (5.1) now follows immediately from (4.8) and (5.2).
We have the following consequence.
51
Proposition 5.2 (Distortion Proposition). Suppose that T : X → X
has the Strong Distortion Property. Then (i) for all a ∈ β , m(T a) <
∞, and (ii) there exists C ≥ 1 such that for all n ≥ 1, a ∈ βn , and any
measurable set B
C −1
m(B ∩ T n a)
m(a ∩ T −n B)
m(B ∩ T n a)
≤
≤
C
.
m(T n a)
m(a)
m(T n a)
(5.3)
Proof. For arbitrary n ≥ 1, and a ∈ βn , we observe that setting B =
T n a, it follows from Proposition 5.1 that for a.e. x ∈ T n a
D−1 (va0 (x))−1 ≤
m(T n a)
≤ D(va0 (x))−1 .
m(a)
(5.4)
In the case of n = 1, we see that (i) now follows immediately from the
assumption that m(a) < ∞.
We establish (ii) as follows. Taking a ∈ βn , from (5.1) we have that
for a.e. y ∈ B ∩ T n a
D−1 (va0 (y))−1 ≤
m(B ∩ T n a)
≤ D(va0 (y))−1 .
m(a ∩ T −n B)
(5.5)
Letting x = y and dividing (5.5) into (5.4) we obtain
D−2
n
m(a ∩ T −n B)
m(B ∩ T n a)
2 m(B ∩ T a)
≤
≤
D
.
m(T n a)
m(a)
m(T n a)
Letting C = D2 completes the proof.
In the special case where the distortion constant D = 1, it is immediate
from Proposition 5.2 that T is Markov chain with state space β , such
m(b)
that for all a, b ∈ β the transition probability pab = m([a,b])
= m(T
.
m(a)
a)
An example of such a map is a Markov transformation T : ([0, 1], λ) →
([0, 1], λ), with Markov partition β consisting of intervals {I1 , I2 , . . .}
such that T|Ij is linear, for all j ≥ 1.
Theorem 5.3. Let T : X → X satisfy the Strong Distortion Property,
and suppose that a ∈ β is transient. Then m({x ∈ X : T n (x) ∈
a i.o.}) = 0.
52
−n c
Proof. Let a0 := {x ∈ a : T n x ∈
/ a, ∀n ≥ 1} = a ∩ ( ∞
a ).
n=1 T
0
Thus, the set a contains those points in a whose orbits immediately
leave a and never return. Clearly, a0 is measurable and, by assumption,
m(a0 ) > 0. For all r ≥ 1, we dene sets
T
Er := {x ∈ a : ∃ 0 < n1 < . . . < nr : T nj (x) ∈ a, for j = 1, . . . , r}.
Thus, Er denotes the set of points x ∈ a that make at least r return
visits to a.
Consider a cylinder w = [w0 , . . . , wn−1 ] that contains exactly r + 1
occurrences of the symbol a and for which w0 = wn−1 = a. We dene
Dr to be the collection of all such cylinders. It follows that Er =
∪w∈Dr w, and that Er is therefore measurable. Since T a0 ⊂ T |w| w, it
follows from Proposition 5.2 that there exists C ≥ 1 such that for each
cylinder w ∈ Dr ,
0
m(w ∩ T −n (T a0 ))
m(T a0 )
−1 m(T a )
≥ C −1
=
C
.
m(w)
m(T n w)
m(T a)
(5.6)
0
m(T a )
We dene δ := Cm(T
. From Proposition 5.2(i) we have that m(T a) <
a)
0
∞, and since m(a ) > 0, and T is non-singular, it follows that m(T a0 ) >
0, and so δ > 0. From (5.6) it follows that for all cylinders w ∈ Dr ,
m(w ∩ (Er+1 )c ) ≥ δm(w). Taking unions over all cylinders w ∈ Dr , we
have
m(Er \Er+1 ) = m(
[
w ∩ (Er+1 )c ) ≥ δm(
w∈Dr
[
w) = δm(Er ). (5.7)
w∈Dr
Since Er+1 ⊂ Er , it follows that
m(Er ) − m(Er+1 ) ≥ δm(Er ).
This establishes
m(Er+1 ) ≤ (1 − δ)m(Er ).
(5.8)
For all r ∈ 1, {x : T n (x) ∈ a i.o.} ⊂ Er+1 ⊂ Er , and so
m({x ∈ a : T n (x) ∈ a i.o.}) ≤ m(Er+1 ) ≤ (1−δ)m(Er ) ≤ (1−δ)r m(a),
Thus m({x ∈ a : T n (x) ∈ a i.o.}) = 0. It follows from the nonsingularity of T that for all b ∈ β , m({x ∈ b : T n (x) ∈ a i.o.}) = 0,
and therefore that m({x ∈ X : T n (x) ∈ a i.o.}) = 0.
53
Theorem 5.4. If T : X → X satises the Strong Distortion Property,
and a ∈ β is recurrent, then for all b that lie in the same communication
class as a, b is recurrent.
Proof. Suppose that a is recurrent, and x b such that a and b intercommunicate. Dene
E := {x ∈ a : T n (x) ∈ a i.o. & ∀n ≥ 1, T n (x) ∈
/ b}.
We rst show that
(5.9)
m(E) = 0.
Since a communicates with b there exists a cylinder
u := [d1 , . . . , dk , b]
such that a ∩ T −1 u is admissible. For r ≥ 1, dene the sets
Er := {x ∈ a : ∃ 0 < n1 < . . . < nr : T nj (x) ∈ a, for j = 1, . . . , r,
&
s
T (x) ∈
/ b, for s = 0, . . . , nr )}.
Thus Er denotes the set of points x ∈ a that make at least r return
visits to a before visiting b, if ever. For r ≥ 1, recall from the proof of
Theorem 5.3 the sets Dr and dene the collection of cylinder sets
Dr,b := {w ∈ Dr : wj 6= b, ∀ j = 0, . . . , |w| − 1}.
Observe that Er = ∪w∈Dr,b w. Therefore, Er is measurable. It follows
from Proposition 5.2 that there exists C ≥ 1 such that for each w ∈ Dr,b
m(w ∩ T −|w| u)
m(u)
≥ C −1
.
m(w)
m(T a)
(5.10)
m(u)
, a similar argument to that employed in TheDening δ := C −1 m(T
a)
orem 5.3 establishes that m(Er+1 ) ≤ (1 − δ)m(Er ). For all r ≥ 1,
E ⊂ Er+1 ⊂ Er , and so
m(E) ≤ m(Er+1 ) ≤ (1 − δ)m(Er ) ≤ (1 − δ)r m(a),
from which (5.9) now follows.
Letting D := {x ∈ a : T n (x) ∈ a i.o. & ∃N ∀n ≥ N, T n (x) ∈/ b}, it
−n
follows that D ⊂ a ∩ (∪∞
E). Since T is non-singular it follows
n=0 T
54
from (5.9) that m(D) = 0. Since a is recurrent it follows from Theorem
4.5 that T r x ∈ b i.o. for a.e. x ∈ a.
Since b communicates with a, there is a positive measure subset of
b whose orbits enter a, and therefore return to b innitely often. By
Theorem 5.3 it is now immediate that b is recurrent.
Corollary 5.5. Let T : X → X satisfy the Strong Distortion Property, and C be a communication class. Then either (i) every b ∈ C is
recurrent, or (ii) every b ∈ C is transient.
In consequence of Corollary 5.5 we may say that a communication class
is either recurrent or transient.
Corollary 5.6. Let T : X → X satisfy the Strong Distortion Property.
If a nite communication class C is closed, then it is recurrent.
Proof. Suppose to the contrary that C is transient. Then by Theorem
5.3, for all b ∈ C and a.e. x ∈ X , the orbit of x visits b only nitely
often. But since C is closed and nite, this is impossible, and so C is
recurrent.
Example 5.7. The Strong Distortion Property is essential to Corollary
5.6. By way of counter-example, let T : [0, 1] → [0, 1] be a map such
that T |[0, 12 ] is a continuous, monotonic, bijection onto [0, 1] such that
there exists c ∈ (0, 21 ) for which T (x) < x for all x ≤ c, while for all
x ∈ ( 21 , 1], T (x) = 2x − 1. It is immediate that 0 is an attracting xed
point such that for Lebesgue almost every x ∈ [0, 1], T r (x) → 0 as
r → ∞. While the map T is an irreducible Markov map with respect to
the nite Markov partition {[0, 21 ], [ 12 , 1)}, it does not have the Strong
Distortion Property, and it is not recurrent.
From the proof of Theorem 5.4, we have the following corollary.
Corollary 5.8. Let T : X → X satisfy the Strong Distortion Property.
Then a communication class is recurrent if and only if it is transitive.
Remark. We observe that while Theorem 4.7 and Corollary 5.8 have
55
similar conclusions, they rest on quite dierent assumptions, the former
applying to skew-products with an ergodic and invariant base transformation, and the latter to a general Markov map with Strong Distortion.
56
Chapter 6
Asymptotic Properties of a
Deterministic Walk in a Fixed
Environment
This chapter is concerned with asymptotic properties of a deterministic
walk in an arbitrary xed environment. In particular, we establish conditions that ensure that the transience or recurrence of communication
classes in the partition β̃ under the skew-product dynamics go over to
the deterministic walk under projection to the bre. This result holds
both in the nite and innite bre cases, and has a useful corollary in
the case of the bre Z.
Denition 6.1. We say that a Markov transformation T of probability
space (X, m), with Markov partition β , has Big Images if inf a∈β m(T a) >
0.
Standing hypotheses for the deterministic walk. For the rest of this
thesis we assume that the deterministic walk is driven by a Markov
transformation T of a probability space (X, m), with Markov partition
β , that has Strong Distortion and Big Images, and that transition functions are constant on elements of β .
57
For the rest of the thesis we let µ denote the σ -nite measure
m × counting measure.
It follows from the assumption that transition functions are constant
on elements of the Markov partition β that for all n ≥ 1 and all v ∈ β̃n
there exists a ∈ βn , and i, k ∈ S such that
v = a × {i} & Tfn (v) = T n a × {k}.
(6.1)
It follows from (6.1) that for all cylinders v
|v|
µ(Tf v) ≤ 1.
(6.2)
It also follows from (6.1) and from the Big Image property (as per
Denition 6.1) that
µ(Tfn v) = m(T n a) ≥ > 0
(6.3)
where := inf a∈β m(a).
Bounded jumps. We say that the deterministic walk has bounded jumps
if f (X) is a nite subset of S , for all transition functions f .
Remark. In the case where #β < ∞, it follows from the hypothesis that transition functions are constant on elements of β that the
deterministic walk has bounded jumps.
Notation. For A ⊂ β̃ , we dene the projection of A,
π2 (A) := {i ∈ S : ∃a ∈ β : a × {i} ∈ A}.
Denition. Given K ⊂ S , i ∈ S , and x ∈ X , we say that that the orbit
(Ui,n (x))n≥0 is transitive on K if it visits every element of K innitely
often. Conversely, we say that the orbit (Ui,n (x))n≥0 is transient on K
if it visits every element of K at most nitely often.
Theorem 6.2. If a deterministic walk in a xed environment has
bounded jumps, then for any communication class C ⊂ β̃ , either (i)
for all a ∈ C and µ-a.e. (x, i) ∈ a the orbit (Ui,n (x))n≥0 is transitive on
π2 (C), or (ii) for all a ∈ C and µ-a.e. (x, i) ∈ a the orbit (Ui,n (x))n≥0
is transient on π2 (C).
58
Proof. It follows from Proposition 4.10 and Corollary 5.5 that the communication class C is either recurrent or transient.
Suppose that C is recurrent. Then by Corollary 5.8, C is transitive,
and it follows that for all a ∈ C and µ-a.e. (x, i) ∈ a, (Ui,n (x))n≥0 is
transitive on π2 (C).
Conversely, suppose that C is transient. We deal separately with the
cases where #β < ∞ and #β = ∞.
Suppose that #β < ∞. An immediate consequence of Proposition
4.10 and Theorem 5.3 is that for all a ∈ C and µ-a.e. (x, i) ∈ a, the
orbit of (x, i) under Tf visits each partition element in C at most nitely
often. For each i ∈ S , X × {i} contains only nitely many partition
elements, and the result follows.
Suppose, instead, that #β = ∞. Fix i ∈ S such that
Ci := {a ∈ C : a ⊂ X × {i}} =
6 ∅.
Dene
Di := {x : ∃a ∈ Ci , x ∈ a}.
To complete the proof we show that
µ({(x, i) ∈ Di : Ui,n (x) = i i.o.}) = 0.
(6.4)
It then follows from the non-singularity of Tf that for all a ∈ C and
µ-a.e. (x, i) ∈ a, the orbit (Ui,n (x))n≥0 is transient on π2 (C), as required.
Since the deterministic walk has bounded jumps it follows that there
exists a nite set J ⊂ S such that for all x ∈ Di , π2 (Tf (x)) ∈ J . For
each l ∈ J , let {al,n }n∈N be an enumeration of all the partition elements
of β such that al,n × {l} ⊂ Tf (Di ). By (6.3) there exists > 0 such
that for all a ∈ β̃ , µ(Tf (a)) ≥ . For each l ∈ J dene
Ml := inf{k : m(∪∞
n=k al,n ) < }.
2
Dene
l
Λ := {(x, l) ∈ X × S : l ∈ J & x ∈ a, for some a ∈ {al,n }M
n=1 }.
59
Given an n-cylinder v ∈ β̃n , we let vj denote the j th component of v ,
for 0 ≤ j ≤ n − 1. For n ≥ 1, q ≥ 0 and i ∈ S dene
Bq,n := {v ∈ β̃n : ∃0 = r0 < r1 < . . . < rq = n − 1, vrk ∈ Ci , for k = 0, . . . , q,
& vt ∩ Λ = ∅, for t = 0, . . . , n − 1}.
Thus, each v ∈ Bq,n is an n-cylinder contained in the set Di , each of
whose elements make at least q return visits to Ci before visiting Λ, if
ever.
It follows from the Proposition 4.10 and Proposition 5.2 that there
exists C ≥ 1 such that for all n ≥ 1 and q ≥ 0, and for all v :=
[c0 × {i}, . . . , cn−1 × {i}] ∈ Bq,n , we have
µ(v ∩ Tf−n (Λ))
µ(Tfn (v) ∩ Λ))
−1
≥C
.
µ(v)
µ(Tfn (v))
(6.5)
By (4.3), Tfn (v) = Tf (cn−1 × {i}) = T cn−1 × {k}, where k := fi (cn−1 ).
From the denition of µ and Λ we have
k
k
µ(Tfn (v) ∩ Λ)
µ(T cn−1 × {k} ∩ (∪M
m(T cn−1 ∩ ∪M
j=1 ak,j × {k}))
j=1 ak,j )
=
=
.
µ(Tfn (v))
µ(T cn−1 × {k})
m(T cn−1 )
k
m(T cn−1 ∩ ∪M
j=1 ak,j )
k
≥ m(T cn−1 ∩ ∪M
= .
j=1 ak,j ) ≥ −
m(T cn−1 )
2
2
(6.6)
(6.7)
It now follows from (6.5), (6.6) and (6.7) that
µ(v ∩ Tf−n (Λ))
≥ C −1 > 0.
µ(v)
2
(6.8)
Having established (6.8), a similar argument to that given in the proof
of Theorem 5.3 shows that the measure of walks that start in Di and
that make q returns to Ci , without visiting the set Λ, decays exponentially in q at a rate of at least 1 − C −1 2 . Dening
A := {(x, i) ∈ Di : Ui,n (x) = i i.o. & ∀n ≥ 1, Tfn (x, i) ∈
/ Λ},
it follows immediately that
µ(A) = 0.
60
(6.9)
Dening
/ Λ},
B := {(x, i) ∈ Di : Ui,n (x) = i i.o. & ∃N ∀n ≥ N, Tfn (x, i) ∈
it follows that B ⊂ Di ∩(∪n≥0 Tf−n A). Since Tf is non-singular it follows
from (6.9) that
µ(B) = 0.
(6.10)
Since the deterministic walk has bounded jumps #Λ < ∞. It follows
that since C is transient, for µ-a.e. (x, i) ∈ Di there exists N such that
for all n ≥ N , Tfn (x, i) ∈/ Λ. Equation (6.4) now follows from (6.10).
This completes the proof.
We have the following analogue of Theorem 4.7.
Corollary 6.3. Suppose that S is nite. Then the deterministic walk
in a given xed environment is transitive if and only if for every closed
communication class C ⊂ β̃ , π2 (C) = S .
Proof. Since S is nite, the deterministic walk automatically has bounded
jumps. By Corollary 5.5, a closed communication class C ⊂ β̃ is either
recurrent or transient. If it is transient, then by Theorem 6.2 for all
a ∈ C and µ-a.e. (x, i) ∈ a, (Ui,n (x))n≥0 is transient on π2 (C), but since
S is nite, this is impossible. Therefore, the deterministic walk is transitive if and only if π2 (C) = S , for all closed communication classes
π2 (C).
The following result will be of use later on.
Corollary 6.4. Suppose that S = Z, the deterministic walk in a given
environment has bounded jumps, and that for every closed communication class C ∈ β̃ , π2 (C) = Z. Then either
(i) the deterministic walk is recurrent and lim supn→∞ Ui,n (x) = ∞
and lim inf n→∞ Ui,n (x) = −∞, for all a ∈ C and µ-a.e. (x, i) ∈ a,
or
(ii) limn→∞ |Ui,n (x)| = ∞, for all a ∈ C and µ-a.e. (x, i) ∈ a.
61
Proof. It is immediate from Theorem 6.2 that for each closed communication class C ⊂ β̃ , either (Ui,n (x))n≥0 is recurrent on π2 (C) = Z for
all a ∈ C and µ-a.e. (x, i) ∈ a, or (Ui,n (x))n≥0 is transient on π2 (C) = Z
for all a ∈ C and µ-a.e. (x, i) ∈ a. The recurrent case implies that (i)
holds, whereas the transient case implies that (ii) holds.
Remark. In the transient case, the limit ±∞ may depend on x. For
example, in the case of a Markov chain on Z with jumps of ±1 such
that pi,i+1 = 34 and pi,i−1 = 41 for all i ≥ 0, and pi,i+1 = 14 and pi,i−1 = 34
for all i < 0, then the walk diverges both to the left and to the right
with positive probability.
62
Chapter 7
Zero-One Laws for a
Deterministic Walk on
Z
in a
Deterministic Environment
Introduction
In Section 3.3.2, we introduced the idea of a deterministic walk in a
deterministic environment on Z. In this chapter our goal is to establish
Zero-One laws for the recurrence vs transience of the DWDE on Z.
In Section 7.1, we provide the setup that will be assumed throughout
this chapter.
In Section 7.2, we establish some auxiliary results regarding the
measurability of certain sets of environments. These results will be of
use subsequently.
In Section 7.3, we present the rst main result of this chapter, Theorem 7.3, in which we show that under certain hypotheses the DWDE
exhibits exactly one of four types of asymptotic behaviour with probability 1. Specically, the DWDE is either (a) transitive on Z almost
surely, (b) diverges to +∞ almost surely, (c) diverges to −∞ almost
surely, or (d) for almost every environment ω there exists p ∈ (0, 1)
such that the deterministic walk in ω diverges to +∞ with probability
p, and diverges to −∞ with probability 1 − p.
In Section 7.4, we introduce a local connectivity property for the
DWDE, which we call the Linkage property. We show that this prop-
63
erty is naturally satised by a large class of DWDEs and establish a
useful consequence of this property.
In Section 7.5, we state and prove the main result of this chapter, in
which we show that if in addition to the hypotheses of Theorem 7.3 the
DWDE also satises the Linkage Property and has uniformly bounded
jumps (i.e. jump sizes are uniformly bounded), then case (d) above
occurs with probability zero, and the DWDE exhibits exactly one of
the three types of asymptotic behaviour found in Theorem 2.12. We
conclude by showing that an immediate corollary of this result is that
if the DWDE has a symmetric (in a sense to be made precise) i.i.d.
environment, then the DWDE is recurrent, and therefore transitive.
7.1
A Deterministic Walk on
Z
in a Deterministic
Environment
In this section we introduce the class of DWDEs on Z with which we
will be concerned in this chapter. Throughout we continue to assume
that the standing hypotheses obtain, and that the Markov partition β
separates points (Recall Denition 4.1).
Deterministic environment. We assume that the process by which the
environment is generated is ergodic and stationary. More precisely, we
consider the following setup.
As per the standing hypotheses, we suppose that the set of transition functions from which environments are constructed is some set
G ⊆ {f : X → Z : f is constant on elements of β}.
We suppose that η : Ω → Ω is an invertible measurable transformation
of a probability space (Ω, P ), and that φ : Ω → G is a measurable
function. Each point ω ∈ Ω encodes an environment (fi )i∈Z such that
fi := φ ◦ η i (ω),
for all i ∈ Z.
We call the process (φ ◦ η i )i∈Z a deterministic environment on Z.
64
(7.1)
We shall be interested in the cases where where the deterministic environment is either i.i.d. or, more generally, ergodic and stationary (i.e.
η is ergodic and measure preserving).
A deterministic walk in a deterministic environment on Z.
Associated to every ω ∈ Ω is a function
f (ω) : X × Z → Z
(x, i) 7→ fi (x)
where fi := φ(η i (ω)). For ω ∈ Ω, we thus dene the skew-product
Tf (ω) : X × Z → X × Z such that
Tf (ω) (x, i) = (T x, i + fi (x)).
(7.2)
Typically, we will suppress reference to ω , the dependence being tacit,
and simply write Tf instead of Tf (ω) .
We dene a deterministic walk in a deterministic environment on
Z (DWDE) to be the process Un := π2 ◦ Tfn , for n ≥ 0. Specically, for
i ∈ Z, x ∈ X , and ω ∈ Ω, we dene
Un := Ui,n (x, ω) := π2 (Tfn(ω) (x, i)).
(7.3)
It follows that Ui,0 (x, ω) := i and for all n ≥ 0
Ui,n+1 (x, ω) := Ui,n (x, ω) + f (ω)(T n (x), Ui,n (x, ω)).
Hence, Ui,n+1 (x, ω) =
7.2
Pn
r=0
(7.4)
f (ω)(T r (x), Ui,r (x, ω)).
Measurability Lemmas
In this section we establish some technical results regarding the measurability of certain sets of environments.
Given the set of transition functions
G ⊆ {g : X → Z : g constant on elements of β},
65
(7.5)
if #β < ∞ then it follows from (7.5) that G is necessarily countable, in
which case G is equipped with the discrete topology and we assume that
φ : Ω → G is measurable with respect to the sigma-algebra generated
by that topology. If instead #β = ∞, and {an }n≥1 is an enumeration
of the Markov partition β , dene the function s : G × G → N such that
s(f, g) = max{n : f (an ) = g(an )}.
(7.6)
Dening the metric d : G × G → [0, 1] such that
d(f, g) = ( 21 )s(f,g)
(7.7)
we assume that φ : Ω → G is measurable with respect to the sigmaalgebra generated by the topology given by d.
Lemma 7.1. Given n ≥ 0, k0 , . . . , kn−1 ∈ Z and r ∈ [0, 1], the set {ω :
m({x : U0,s (x, ω) =
Ps−1
i=0
ki , for s = 1, . . . , n}) > r} is measurable.
Proof. Fix n ≥ 0, k0 , . . . , kn−1 ∈ Z, and r ∈ [0, 1]. Fix a ∈ βn and
dene
Πa := {ω : U0,s (x, ω) =
s−1
X
ki , ∀x ∈ a & s = 0, . . . , n}.
i=0
Firstly, we show that Πa is measurable. In both the case where #β < ∞
and #β = ∞ it is apparent that for all k ∈ Z and all b ∈ β , {g : g(b) =
k} is an open set, and hence φ−1 {g : g(b) = k} is measurable. Since
η : Ω → Ω is measurable it follows that for all i, k ∈ Z and all b ∈ β the
set {ω : (φ(η i (ω))(b) = k} is measurable. If a := [a0 , . . . , an−1 ], letting
P
Rs := s−1
i=0 ki since
Πa = {ω : φ(η Rs ω)(as ) = ks , for s = 0, . . . , n − 1}
it follows that Πa is measurable.
A given ω ∈ Ω satises
m({x : U0,s (x, ω) = Rs , for s = 0, . . . , n}) > r
(7.8)
if and only if there exists a nite collection K of n-cylinders a such
that m(∪a∈K a) > r and ω ∈ ∩a∈K Πa . Letting ∆ denote the set of all
66
nite collections K of n-cylinders such that m(∪a∈K a) > r it follows
that
{ω : m({x : U0,s (x, ω) = Rs , for s = 0, . . . , n}) > r} =
[ \
Πa .
K∈∆ a∈K
(7.9)
Since ∆ is countable, it follows that ∪K∈∆ ∩a∈K Πa is measurable, and
so (7.9) establishes the result.
As a consequence of Lemma 7.1 we have the following corollary.
Corollary 7.2. Given n ≥ 1, l, k ∈ Z, and r ∈ [0, 1], the set {ω :
m({x : Ul,j (x, ω) ≤ k, for some 1 ≤ j ≤ n}) > r} is measurable.
7.3
A 4-case Zero-One Law for the DWDE
Before stating and proving the main result of this section, we introduce
some prerequisite notions.
Irreducibility of the DWDE. We say that the DWDE is irreducible if
for P -a.e. ω ∈ Ω, the Markov partition β̃ := β × Z is irreducible under
the skew-product Tf (ω) .
Recurrence and transience of the DWDE. We say that the DWDE is
recurrent if for P -a.e. ω ∈ Ω, the deterministic walk in the environment
(. . . , φ(η −1 ω), φ(ω), φ(ηω), . . .) is recurrent. Similarly, we say that the
DWDE is transient if for P -a.e. ω ∈ Ω, the deterministic walk in the
environment (. . . , φ(η −1 ω), φ(ω), φ(ηω), . . .) is transient.
Recall that the DWDE has bounded jumps if for all g ∈ G, the set
g(X) is nite.
Theorem 7.3. If the DWDE is irreducible with bounded jumps and
an ergodic and stationary environment, then it is either recurrent or
transient. Moreover, in the transient case, exactly one of the following
holds.
67
(i) limn→∞ Ui,n (x, ω) = +∞ for all i ∈ Z, for P -a.e. ω ∈ Ω and for
m-a.e. x ∈ X .
(ii) limn→∞ Ui,n (x, ω) = −∞ for all i ∈ Z, for P -a.e. ω ∈ Ω and for
m-a.e. x ∈ X .
(iii) For P -a.e. ω ∈ Ω, and for all i ∈ Z,
m({x : lim Ui,n (x, ω) = ∞})+m({x : lim Ui,n (x, ω) = −∞}) = 1,
n→∞
n→∞
and 0 < m({x : limn→∞ Ui,n (x, ω) = ∞}) < 1.
Proof. Let R+ := {ω : m({x : limn→∞ U0,n (x, ω) = +∞}) = 0}. Thus,
R+ denotes the set of environments for which the measure of the set
of walks that diverge to +∞ when started in state 0, is zero.
We rst establish that R+ is P -measurable. For all ω ∈ Ω, n, k ≥ 1,
we dene sets
Rkn (ω) := {x ∈ X : U0,r (x, ω) ≤ −k, for some 1 ≤ r ≤ n}.
For n, k, r ≥ 1, dene
R(n, k, r) := {ω : m(Rkn (ω)) > 1 − 1r }.
It follows from Corollary 7.2 that for all n, k, r ≥ 1, the set R(n, k, r)
is P -measurable. Since, by assumption, the DWDE is irreducible, it
follows From Corollary 6.4 that
R+ = {ω : m({x : lim inf U0,n (x, ω) = −∞}) = 1},
n→∞
and therefore R+ = ∩r≥1 ∩k≥1 ∪n≥1 R(n, k, r). Hence, R+ is P -measurable.
We next show that if ω ∈ R+ then for all k ∈ Z,
m({x : lim Uk,n (x, ω) = ∞}) = 0.
n→∞
(7.10)
Suppose to the contrary that ω ∈ R+ and that there exist k 6= 0 such
that
m({x : lim Uk,n (x, ω) = ∞}) > 0.
n→∞
68
Then there exists a ∈ β and a set A ⊂ a, such that m(A) > 0 and for
all x ∈ A
lim Uk,n (x, ω) = ∞.
n→∞
By the irreducibility of β̃ , for each b ∈ β there exists an admissible
cylinder [b × {0}, . . . , a × {k}]. It follows from the non-singularity of
Tf , that µ([b × {0}, . . . , A × {k}]) > 0, but this contradicts the fact
that ω ∈ R+ , and so we have established (7.10).
Using the fact that for all k ∈ Z and for all x ∈ X
Uk,n (x, ω) = U0,n (x, η k (ω)) + k
(7.11)
it is now immediate from (7.10) and (7.11) that if ω ∈ R+ then η k (ω) ∈
R+ for all k ∈ Z. In particular, ω ∈ R+ if and only if η(ω) ∈ R+ , from
which it follows that η −1 (R+ ) = R+ . Since the η -invariant measure P
is ergodic, it follows from Birkho's Ergodic Theorem that P (R+ ) =
0 or 1.
An identical argument shows that the set
R− := {ω : m({x : lim U0,n (x, ω) = −∞}) = 0}
n→∞
has P -measure of 0 or 1.
If P (R+ ) = P (R− ) = 1, then since the DWDE has bounded jumps
it follows from Corollary 6.4(i) that the DWDE is recurrent. Alternatively, in the transient cases we have the following
P (R+ ) = 0, P (R− ) = 1 ⇒
case (i)
P (R+ ) = 1, P (R− ) = 0 ⇒
case (ii)
P (R+ ) = 0, P (R− ) = 0 ⇒
case (iii).
This concludes the proof.
7.4
The Linkage Property
Under two additional hypotheses, we will show that transient case (iii)
of Theorem 7.3 has zero probability. In this section we introduce the
69
rst of these conditions: the Linkage Property. (The second condition
- Uniformly Bounded Jumps - will be introduced in the next section.)
Notation. Given a collection of cylinder sets v1 , v2 , . . . , vk we dene
v1 · v2 · . . . · vk := v1 ∩ T −|v1 | (v2 ∩ T −|v2 | (. . . (T −|vk−1 | (vk )) . . .)).
Linkage property. We say that a DWDE has the Linkage Property
if it is irreducible and there exists r > 0 such that for almost every
environment and all a, b ∈ β̃ such that |π2 (a) − π2 (b)| = 1, either
b ⊂ Tf (a), or there exists a cylinder c such that
(a) µ(c) ≥ r, and
(b) the cylinder a · c · b is admissible.
In the following example we show that under relatively simple assumptions the DWDE has the Linkage Property.
Full-branch maps. We say that a Markov transformation T of probability space (X, m), with Markov partition β , has full branches (or is
full-branch ) if for all a ∈ β , T a = X .
Example. Suppose that the DWDE satises (i) 2 ≤ #β < ∞ and
(ii) for all g ∈ G, g(X) = {+1, −1}, and that the base transformation
T is full-branch. We show that the Linkage Property holds.
We x an environment (fi )i∈Z , and x a = a0 ×{i} ∈ β̃ , and suppose
that fi (a0 ) = +1. The case where fi (a0 ) = −1 is identical. If b =
b0 × {i + 1} then, since T is full-branch, it is immediate that Tf (a) =
X × {i + 1} ⊃ b. Thus, we assume that b = b0 × {i − 1}. By (ii) there
exists c1 = c01 × {i + 1}, c2 = c02 × {i} ∈ β̃ such that fi+1 (c01 ) = fi (c02 ) =
−1. Since T is full-branch it follows that a · c1 · c2 · b is admissible.
Thus, for all a, b ∈ β̃ such that |π2 (a) − π2 (b)| = 1 either Tf (a) ⊃ b
or there exists c ∈ β̃2 such that a · c · b is admissible. Since #β < ∞ and
T is full-branch we may take r := min{m(d) : d ∈ β2 } > 0, thereby
establishing that the Linkage Property holds.
70
Proposition 7.4. If the DWDE has the Linkage Property, then there
exist positive numbers {rk }k≥0 such that for every environment, and
for all a, b ∈ β̃ such that |π2 (a) − π2 (b)| ≤ k , there exists a cylinder c
such that µ(c) ≥ rk and a · c · b is admissible.
Proof. Fix an arbitrary environment. Given a ∈ β̃ let π2 (a) denote the
Z-component of a, and a0 denote the β -component of a. Since the case
|π2 (a) − π2 (b)| = 1 is immediate from the Linkage Property, we deal in
turn with the cases where π2 (b) − π2 (a) = 0 and π2 (b) − π2 (a) = k ≥ 2
- the case where π2 (a) − π2 (b) = k ≥ 2 is proven similarly. Fix d0 ∈ β
and for k ∈ Z dene dk := d0 × {k}. Dene κ := m(d0 ).
Let π2 (b) − π2 (a) = 0 and suppose that fπ2 (a) (a0 ) = +1. The case
where fπ2 (a) (a0 ) = −1 is identical. By the Linkage Property there exist
cylinders v1 , v2 such that µ(v1 ) > r and µ(v2 ) > r, and a·v1 ·dπ2 (a)+1 ·v2 ·b
is admissible. By Proposition 4.10, Proposition 5.2, and equation (6.2)
µ(v1 · dπ2 (a)+1 · v2 ) ≥ C −1 µ(v1 )µ(dπ2 (a)+1 · v2 )
≥ C −2 µ(v1 )µ(dπ2 (a)+1 )µ(v2 )
≥ C −2 r2 κ > 0.
This establishes the result for the case π2 (b) − π2 (a) = 0.
Assume that π2 (b) − π2 (a) = k ≥ 2. By a similar argument to the case
where π2 (b)−π2 (a) = 0 above, there exist cylinders v1 , . . . , vk , such that
µ(vi ) > r for i = 1, . . . , k , and a·v1 ·dπ2 (a)+1 ·v2 ·. . .·vk−1 ·dπ2 (a)+k−1 ·vk ·b
is admissible. As before, by Proposition 4.10, Proposition 5.2 and (6.2)
we have
µ(v1 · dπ2 (a)+1 · . . . · vk−1 · dπ2 (a)+k−1 · vk ) ≥ (C −1 )2k−2 µ(v1 ) . . . µ(vk )κk−1
≥ (C −1 )2k−2 rk κk−1 > 0.
Taking rk := (C −1 )2k−2 rk κk−1 establishes the result.
7.5
A 3-case Zero-One Law for the DWDE
Uniformly bounded jumps. We say that the DWDE has uniformly
bounded jumps if there exists a nite set J ⊂ Z such that for all g ∈ G,
71
g(X) ⊂ J .
We are now in a position to state and prove the main result of this
chapter.
Theorem 7.5. Suppose that the DWDE satises the Linkage Property
and that it has uniformly bounded jumps and an ergodic and stationary
environment. If the DWDE is transient then either case (i) or case (ii)
of Theorem 7.3 holds.
Before outlining our strategy for proving Theorem 7.5 we introduce the
following prerequisite denitions.
Notation. Let M := max{|g(x)| : x ∈ X, g ∈ G}, and for j ∈ Z
dene
Λj := {(x, n) ∈ X × Z : jM ≤ n ≤ (j + 1)M − 1}.
(7.12)
Clearly, for all i ∈ Z, µ(Λi ) = M .
Notation. Dene
D+ := {ω ∈ Ω : µ({(x, i) ∈ Λ0 : lim Ui,n (x, ω) = +∞}) > 0}.
n→∞
The set D+ denotes the set of environments for which the deterministic
walk diverges to +∞ with positive probability. Thus, D+ = (R+ )c and
so D+ is also P -measurable.
Analogously, dene
D− := {ω ∈ Ω : µ({(x, i) ∈ Λ0 : lim Ui,n (x, ω) = −∞}) > 0}.
n→∞
By identical reasoning, D− = (R− )c and so D− is P -measurable. Moreover, since the hypotheses of Theorem 7.5 imply those of Theorem 7.3
it follows that P (D+ ) = 0 or 1 and P (D− ) = 0 or 1.
Notation. In the fashion of Chung [10], for all i, j, k ∈ Z, n ≥ 0,
and ω ∈ Ω dene taboo-hitting-time sets
n
j Ai,k (ω)
:= {x ∈ Λi : Tfn (x) ∈ Λk , & Tfr (x) ∈
/ Λk ∪Λj for r = 1, . . . , n−1},
72
Also, for all ω ∈ Ω dene
j Ai,k (ω)
:= ∪n≥1 (j Ani,k (ω)).
Under the standing hypotheses, for all ω ∈ Ω the set j Ani,k (ω) is a union
of cylinders in ∪r≥1 β̃r , and is therefore µ-measurable. By extension, for
all ω ∈ Ω the set j Ai,k (ω) is also µ-measurable.
Notation. Dene
E + := {ω ∈ Ω : ∃p > 0, s.t. for infinitely many k ≥ 1, µ(−k A−k,0 (ω)) ≥ p}.
It is immediate from Corollary 7.2 that for d, k ≥ 1, {ω : µ(−k A−k,0 (ω)) ≥
d−1 } is P -measurable. Since, by denition of E + , we have
−1
E + = ∪d≥1 ∩k≥1 ∪∞
N =k {ω : µ(−k A−k,0 (ω)) ≥ d }
(7.13)
it follows that E + is P -measurable.
Strategy for proving Theorem 7.5. The proof of Theorem 7.5 contains
two main parts. Firstly, we prove (in Lemma 7.7) that E + ∩ D− = ∅.
Secondly, we show (in Lemma 7.8) that if P (D+ ) = 1 then P (E + ) > 0.
The result then follows from Theorem 7.3. Suppose to the contrary
that case (iii) of Theorem 7.3 holds. Then P (D+ ) = 1 and P (D− ) = 1,
and it follows from Lemma 7.8 that P (E + ) > 0. But by Lemma 7.7
it follows that P (E + ) = 0, which is the desired contradiction. Hence,
case (iii) cannot hold, establishing the result.
It remains to prove Lemma 7.7 and 7.8. Firstly, we establish the following useful result.
Lemma 7.6. Suppose that the DWDE is irreducible and that it has
uniformly bounded jumps. Then
(a) ω ∈ D− if and only if µ(∩k≥1 (0 A0,−k (ω))) > 0, and
(b) ω ∈ D+ if and only if µ(∩k≥1 (0 A0,k (ω))) > 0.
73
Proof. We prove (a) as the proof of (b) is similar. The (⇐) part
follows directly from Corollary 6.4. For the (⇒) part, assume that
µ(∩k≥1 (0 A0,−k (ω))) = 0. Since Tf is non-singular with respect to µ, it
follows that
[
Tf−j (∩k≥1 (0 A0,−k (ω)))) = 0.
µ(
j≥0
But
{(x, i) ∈ Λ0 : lim Ui,n (x, ω) = −∞} ⊂
n→∞
[
Tf−j (∩k≥1 (0 A0,−k (ω))).
j≥0
This completes the proof.
Lemma 7.7. Suppose that the DWDE satises the Linkage Property
and that it has uniformly bounded jumps. Then E + ∩ D− = ∅.
Proof. We show that for each ω ∈ E + there exists a strictly increasing
sequence (nj )j≥1 of natural numbers, and q ∈ (0, 1), such that
µ(0 A0,−nj (ω)) ≤ M q j .
(7.14)
The result then follows from Lemma 7.6(a).
For d ≥ 1, dene
Ed+ = {ω : µ(−n A−n,0 (ω)) ≥ d−1 for infinitely many n ≥ 1}.
By (7.13) E + = ∪d≥1 Ed+ . For the remainder of the proof we x d ≥ 1
and x ω ∈ Ed+ , and write j Ai,k instead of j Ai,k (ω).
Since ω ∈ Ed+ there exists an innite, strictly increasing, sequence
of natural numbers (nj )j≥1 such that for each j ≥ 1
µ(−nj A−nj ,0 ) ≥ d−1 .
(7.15)
Since the Markov partition β separates points, it follows that for all
> 0 there exists n ≥ 1 such that for all a ∈ βn , m(a) < . To see
this, suppose to the contrary that there exists > 0 such that for all
n ≥ 1 there exists a ∈ βn such that m(a) > . It follows that there
exists an innite sequence of partition elements {an : an ∈ β, n ≥ 1}
74
such that m( n≥1 an ) > , and so there exists a positive measure set
of points that are not separated by the partition β , contradicting the
assumption that β separates points.
By extension, the partition β̃ separates points in X × Z, and from
this it is immediate that for all > 0 there exists n ≥ 1 such that
µ(c) < for all c ∈ β̃n . It follows from Proposition 7.4 that there exists
r > 0 such that for all a, b ∈ β̃ for which |π2 (a) − π2 (b)| ≤ M , there
exists a cylinder set c such that µ(c) > r and a · c · b is admissible. Let
k0 be such that for all n ≥ k0 and all a ∈ β̃n
T
(7.16)
µ(a) < r.
Since (nj )j≥1 is a strictly increasing sequence we may suppose, after
passing to a subsequence if necessary, that for all j ≥ 1
nj+1 − nj − 1 > k0 .
(7.17)
It follows from (7.17) that for all cylinders c such that
c⊆
−nj A−nj ,−nj+1
or c ⊆
−nj A−nj ,−nj−1 ,
we have
µ(c) < r.
(7.18)
Claim. There exists δ > 0 such that for all j ≥ 1, and for each cylinder
b satisfying
|b|
b ⊂ 0 A0,−nj & Tf b ⊂ Λ−nj ,
(7.19)
we have
−|b|
µ(b ∩ Tf
(−nj+1 A−nj ,0 )) ≥ δµ(b).
(7.20)
Assuming the above claim we complete the proof as follows. Taking
unions over all cylinders b satisfying (7.19), it follows from (7.20) that
µ(0 A0,−nj \0 A0,−nj+1 ) ≥ δµ(0 A0,−nj ).
(7.21)
Since 0 A0,−nj+1 ⊂ 0 A0,−nj , from (7.21) we have
µ(0 A0,−nj+1 ) ≤ (1 − δ)µ(0 A0,−nj ).
75
(7.22)
Taking q = 1 − δ ∈ (0, 1), establishes (7.14).
Proof of Claim. Fix j ≥ 1. Let {an }n≥1 be an enumeration of all
partition elements a ∈ β̃ such that a ⊂ Λ−nj . We rst show that there
exists γ > 0 such that for all a ∈ {an }n≥1
µ(a ∩
−nj+1 A−nj ,0 )
(7.23)
≥ γµ(a).
Fix a ∈ {an }n≥1 . By the Linkage Property and Proposition 7.4, it
follows that for each t ∈ N, there exists a cylinder Dt := [d0 , . . . , dk−1 ],
such that di 6= a or at for i = 0, . . . , k − 1, a · Dt · at is admissible, and
(7.24)
µ(Dt ) > r.
For t ∈ N dene
Bt := at ∩
and
pt :=
(7.25)
−nj A−nj ,0 ,
µ(Bt )
.
µ(−nj A−nj ,0 )
(7.26)
Since the partition element a ⊂ Λ−nj and Tf (a) ⊃ Dt , it follows that
the cylinder
Dt ⊂ Λ−nj −1 ∪ Λ−nj ∪ Λ−nj +1 .
Furthermore, from (7.18) and (7.24) it follows that for all i = 0, . . . , k −
1,
di ∩ (Λ−nj−1 ∪ Λ−nj+1 ) = ∅
and hence for all x ∈ a · Dt ,
(7.27)
Tfi (x) ∈
/ Λ−nj−1 ∪ Λ−nj+1 for i = 0, . . . , k.
Thus, for all t ≥ 1
a·Dt ·at ⊂
−nj+1 A−nj ,−nj
\
−nj−1 A−nj ,−nj
⊂
−nj+1 A−nj ,−nj
From (7.25) and (7.28) we have
a · Dt · Bt ⊂
−nj+1 A−nj ,0 .
\
0 A−nj ,−nj .
(7.28)
(7.29)
Moreover, it follows from (7.27) and (7.29) that a · Dt · Bt consists
of points in Λ−nj that start in a, and whose last visit to Λ−nj before
76
eventually visiting Λ0 occurs in the partition element at . It follows that
for all t 6= s
a · Dt · Bt ∩ a · Ds · Bs = ∅.
(7.30)
It follows from Proposition 4.10, Proposition 5.2 and equation (6.2)
that for all t ≥ 1
µ(Tf (a) ∩ Dt · Bt )
µ(a · Dt · Bt )
≥ C −1
≥ C −1 µ(Tf (a) ∩ Dt · Bt ).
µ(a)
µ(Tf (a))
(7.31)
Since Tf (a) ⊃ Dt we have
µ(Tf (a) ∩ Dt · Bt ) = µ(Dt · Bt ).
(7.32)
From T |Dt | Dt ⊃ at ⊃ Bt it follows from Proposition 4.10, Proposition
5.2 and equation (6.2)
µ(Dt · Bt )
µ(Bt )
≥ C −1
≥ C −1 µ(Bt ).
|Dt |
µ(Dt )
µ(Tf Dt )
(7.33)
It now follows from (7.31), (7.32) and (7.33) that
µ(a · Dt · Bt )
≥ C −2 µ(Dt )µ(Bt )
µ(a)
(7.34)
From (7.24), (7.26) and (7.34) we obtain
µ(a · Dt · Bt )
≥ C −2 rpt µ(−nj A−nj ,0 ) ≥ C −2 rpt d−1 > 0.
(7.35)
µ(a)
S
P
It follows from (7.30) that µ( t∈N (a · Dt · Bt )) = t∈N µ(a · Dt · Bt ).
Taking unions over t ∈ N, it follows from (7.35) that
[
X
µ( (a · Dt · Bt )) ≥ µ(a)
C −2 rpt d−1 = (C −2 rd−1 )µ(a) > 0. (7.36)
t∈N
t∈N
Taking γ := C −2 rd−1 > 0 it follows from (7.29) that
µ(a ∩−nj+1 A−nj ,0 ) ≥ µ(
[
(a · Dt · Bt )) ≥ γµ(a).
(7.37)
t∈N
This establishes (7.23).
The set 0 A0,−nj can be expressed as the union of cylinders satisfying
77
(7.19). We x such a cylinder b. Since Tf|b| b is the union of partition
elements in β̃ , it is immediate from (7.23) that
|b|
|b|
µ(Tf b ∩−nj+1 A−nj ,0 ) ≥ γµ(Tf b).
(7.38)
From Proposition 4.10, Proposition 5.2 and (7.38) we have that
−|b|
µ(b ∩ Tf
|b|
µ(Tf b ∩−nj+1 A−nj ,0 )
(−nj+1 A−nj ,0 ))
≥ C −1
≥ C −1 γ.
|b|
µ(b)
µ(Tf b)
(7.39)
Letting δ := C γ > 0 establishes (7.20), thereby proving the claim.
−1
Remark. In the special case where T is full-branch and for all g ∈ G,
g(X) = {−1, +1}, then the conclusion to Lemma 7.7 stills holds when
the assumption that the DWDE has the Linkage Property is replaced
with the weaker assumption that it is irreducible.
Lemma 7.8. If the DWDE is irreducible with uniformly bounded jumps
and a stationary environment, then P (D+ ) = 1 implies that P (E + ) >
0.
Proof. By Lemma 7.6(b), P (D+ ) = 1 implies that for some c ≥ 1,
P ({ω : µ(∩k≥1 (0 A0,k (ω))) ≥ c−1 }) ≥ c−1 .
(7.40)
For k, c ≥ 1, dene the sets D(k, c) := {ω : µ(0 A0,k (ω)) ≥ c−1 }. By
Corollary 7.2 the sets D(k, c) are P -measurable. It follows immediately
from (7.40) that if P (D+ ) = 1 then there exists c ≥ 1 such that for all
k≥1
P (D(k, c)) ≥ c−1 .
(7.41)
For k, c ≥ 1 dene the sets E(k, c) := {ω : µ(−k A−k,0 (ω)) ≥ c−1 }. By
Corollary 7.2 the sets E(k, c) are P -measurable. It follows from (7.41)
and the η -invariance of P that for all k ≥ 1,
P (E(k, c)) ≥ c−1 .
From (7.42), it is immediate that for all k ∈ N
P (∪N ≥k E(N, c)) ≥ c−1
78
(7.42)
and that
P (∩k≥1 ∪N ≥k E(N, c)) ≥ c−1 .
Since E + ⊃ ∩k≥1 ∪N ≥k E(N, c), it follows that P (E + ) ≥ c−1 > 0.
We have the following corollary of Theorem 7.5.
Corollary 7.9. Suppose that the DWDE satises the hypotheses of
Theorem 7.5, and that the deterministic environment is i.i.d. such that
P (fi = g) = P (fi = −g),
(7.43)
for all transition functions g ∈ G and all i ∈ Z. Then the DWDE is
transitive on Z.
Proof. It follows from the symmetry of (7.43) that P (R+ ) = P (R− ).
By Theorems 7.3 and 7.5, it follows that P (R+ ) = P (R− ) = 1, and
hence the DWDE is transitive on Z.
79
Chapter 8
Transient DWDEs on
Z
At the end of Chapter 7 we showed that under certain symmetry conditions, the DWDE on Z is recurrent. In this chapter we consider a
class of examples for which the asymptotic behaviour of the DWDE
on Z is transient and establish hypotheses from which it is possible to
determine whether case (i) or (ii) of Theorem 7.5 holds.
Symbolic metric. Let T be a Markov transformation of a measure
space (X, m), with Markov partition β that separates points in X .
The partition β coordinatises the space X , in that for all x ∈ X we
may dene xn to be the unique partition element containing T n x (for
n ≥ 0). Any sequence (x0 , x1 , . . .) uniquely identies a point in X , and
for all x, y ∈ X we may therefore dene the separation time
S(x, y) := max{n : xn = yn }.
(8.1)
Given θ ∈ (0, 1) we dene the symbolic metric dθ on X such that for
all x, y ∈ X
dθ (x, y) := θS(x,y) .
(8.2)
Gibbs property. Let T be a Markov transformation of a probability
space (X, m) with Markov partition β that separates points. Given
the Radon-Nikodym derivative
g :=
dm
,
dm ◦ T
we say that T has the Gibbs Property if for some θ ∈ (0, 1) and all
a ∈ β , log g|a is Lipschitz continuous with respect to the symbolic met-
80
ric dθ . We say that the measure m is a Gibbs measure with potential
− log g .
It can be shown (see [3]) that if T has the Gibbs Property then is
has the Strong Distortion Property.
Gibbs-Markov maps. We say that a Markov map, with Markov partition β , is Gibbs-Markov if β separates points, and T has both the Big
Image and Gibbs properties.
It can be shown (see [3]) that Gibbs-Markov maps have the property
that there exists C ≥ 1 such that for all n ≥ 1, all a ∈ βn and all x ∈ a
C −1 gn (x) ≤ m(a) ≤ Cgn (x),
(8.3)
where gn (x) := g(x)g(T x) · · · g(T n−1 x).
Example. It can be shown (see [3]) that Markov transformation T :
[0, 1] → [0, 1] that has Big Images and is Adler (recall Example 4.9) is
Gibbs-Markov with an ergodic and invariant measure that is equivalent
to Lebesgue measure.
Theorem 8.1. Suppose that the DWDE has an ergodic, stationary
environment, and is driven by a full-branch Gibbs-Markov transformation T of a probability space (X, m), with nite Markov partition β and
Gibbs potential h. Suppose further that for some 1 ≤ r ≤ #β − 1, and
all transition functions f ∈ G,
#{a ∈ β : f (a) = +1} = r
&
#{a ∈ β : f (a) = −1} = #β − r.
Then:
(i) If inf h > 12 ln 4r(#β − r) and r >
#β
2
then
lim Ui,n (x, ω) = +∞, for all i ∈ Z, P − a.e. ω & m − a.e. x.
n→∞
(ii) If inf h > 21 ln 4r(#β − r) and r <
#β
2
then
lim Ui,n (x, ω) = −∞, for all i ∈ Z, P − a.e. ω & m − a.e. x.
n→∞
81
The hypotheses of Theorem 8.1 are a special case of those of Theorem
7.5. In particular, as observed in Section 7.4, since T is full-branch
Gibbs-Markov with a nite Markov partition β , and since f (X) =
{+1, −1} for all transition functions f ∈ G, the DWDE has the Linkage Property.
Before proceeding to the proof of the this result, it will be necessary to
introduce some machinery.
Conservativity and dissipativity. A non-singular transformation S of a
probability space (Y, q) is conservative if for all measurable sets A ⊂ Y ,
and a.e. x ∈ A, there exists n such that T n x ∈ A. It follows that if
S is conservative then for all measurable sets A, T n x ∈ A i.o. for a.e.
x ∈ A. A non-singular transformation S of a probability space (Y, q)
is (totally) dissipative if for all measurable sets A ⊂ Y , and a.e. x ∈ A,
there exists N such that T n x ∈/ A for all n ≥ N .
In the case of Markov maps, it follows that if S is conservative then it is
recurrent (as dened in Section 4.1), and that if S is totally dissipative
then is it transient (as dened in Section 4.1).
The following result appears as Theorem 4.4.3 in [1].
Theorem 8.2 (Aaronson). Let S be an irreducible Markov transforma-
tion of a probability space (Y, q) that has the Strong Distortion Property.
Then either S is conservative or it is totally dissipative. Moreover, for
all sets A of positive measure, if S is conservative then
∞
X
q(S −j A) = ∞,
(8.4)
j=0
and if S is totally dissipative then
∞
X
q(S −j A) < ∞.
(8.5)
j=0
Under the hypotheses of Theorem 8.2 it follows from Corollary 5.5 that
82
S is conservative if and only if it is recurrent (as dened in Section 4.1),
and that S is totally dissipative if and only if it is transient (as dened
in Section 4.1). We have the following corollary.
Corollary 8.3. If S satises the hypotheses of Theorem 8.2 then for
any positive measure set A, if S is recurrent then
∞
X
q(A ∩ S −j A) = ∞,
(8.6)
j=0
whereas if S is transient then
∞
X
q(A ∩ S −j A) < ∞.
(8.7)
j=0
Proof. If S is transient then it is totally dissipative and (8.7) follows
immediately from (8.5). If S is recurrent, then suppose with a view
to a contradiction that there exists a positive measure set A for which
(8.7) holds. Then there exists N ≥ 1 such that
−n
q(A ∩ (∪∞
A)) < q(A),
n=N S
from which it follows that
−n
q(A\ ∪∞
A) > 0.
n=N S
(8.8)
But since S is recurrent it is conservative, and it follows that S r x ∈ A
i.o. for a.e. x ∈ A, contradicting (8.8). This completes the proof.
Denition. Fix θ ∈ (0, 1). Let ν denote the weighted product measure
on X × Z such that for all m-measurable sets A and all i ∈ Z,
ν(A × {i}) = θ|i| m(A).
(8.9)
Clearly, ν is a nite measure on X × Z.
Proposition 8.4. For all ω ∈ Ω, the skew-product transformation Tfω
of the measure space (X × Z, ν) has the Strong Distortion Property.
83
Proof. We x an environment ω , and n ≥ 1, and let ã ∈ β̃n be an
admissible n-cylinder. By equation (4.3), there exist a ∈ βn , i, k ∈ Z
such that ã = a × {i}, and Tfn (ã) = T n a × {k}. By a similar argument
to that given in the proof of Proposition 4.10 we have that for ν -a.e.
(x, k) ∈ Tfn (ã)
d(ν ◦ Tf−n )
d(m ◦ T −n )
(x, k) = θ|i|−|k|
(x) = θ|i|−|k| va0 (x).
dν
dm
(8.10)
Suppose that T has distortion constant D ≥ 1. From (8.10) we obtain
that for ν -a.e. (x, k) & (y, k) ∈ Tfn (ã) we have
d(ν ◦ Tf−n )
(x, k) ·
dν
!−1
d(ν ◦ Tf−n )
v 0 (x)
(y, k)
≤ D.
= a0
dν
va (y)
(8.11)
This concludes the proof.
Proof of Theorem 8.1. We prove part (i) of the theorem as the proof
of part (ii) is similar. In particular, we x an arbitrary environment
(fi )i∈Z satisfying the hypotheses of Theorem 8.1 and show that (a) the
skew product transformation Tf : (X × Z, ν) → (X × Z, ν) is transient,
and (b) that
lim m(0 A0,−k ) = 0,
(8.12)
k→∞
where for k ≥ 1 dene
n
0 A0,−k
:= {x ∈ X : U0,n (x) = −k, and −k < U0,j (x) < 0, for 1 ≤ j ≤ n−1},
n
and dene 0 A0,−k := ∪∞
n=1 0 A0,−k . The result then follows from Theorem 7.5.
We rst show that the deterministic walk in the environment (fi )i∈Z
is transient. Fix a partition element a × {0} ∈ β̃ . Since ν is a nite
measure on X × Z and ν ≈ µ, by Corollary 8.3 and Proposition 8.4, it
suces to show that
∞
X
ν(a × {0} ∩ Tf−n (a × {0})) < ∞.
n=0
84
(8.13)
Since the deterministic walk is restricted to jumps of +1 and -1, it can
only return to the partition element a×{0} in an even number of steps.
From (8.9) it follows that for all 2n + 1-cylinders that start and end in
a × {0} we have
ν([a×{0}, a1 ×{k1 }, . . . , a2n−1 ×{k2n−1 }, a×{0}]) ≤ m([a, a1 , . . . , a2n−1 , a]).
(8.14)
Given that
! we return to a × {0} at time 2n, for each n ≥ 1, there
2n
are
numbers of ways choosing the timing of n leftward and n
n
rightward jumps. Since T is full branch, and since every transition
function takes the value +1 on exactly r elements of β , and the value
−1 on the remaining
! #β − r elements of β , it follows that there are
rn (#β − r)n
2n
n
cylinder sets of rank 2n + 1 that start and end in
the partition element a × {0}.
Recalling the Radon-Nikodym derivative g =
dm
d(m◦T )
, and dening
M := sup g = e− inf h
it follows from (8.3) that there exists C > 0 such that for all n ≥ 1,
(8.15)
sup m(a) ≤ CM n .
a∈βn
It follows from (8.14) and (8.15)
∞
X
ν(a×{0}∩Tf−n (a×{0})) ≤ C
n=0
∞
X
rn (#β−r)n
n=0
!
2n
M 2n+1 . (8.16)
n
By Stirling's formula it follows that for large n
2n
n
!
1
≈ (2πn)− 2 4n .
(8.17)
It follows from (8.16) and (8.17) that for some constant C 0 > 0
∞
X
ν(a × {0} ∩
Tf−n (a
× {0})) ≤ C
n=0
0
∞
X
(4r(#β − r)M 2 )n .
(8.18)
n=0
It follows from the hypotheses that M 2 = e−2 inf h < (4r(#β − r))−1
and so
4r(#β − r)M 2 < 1.
(8.19)
85
From (8.19) it follows that the right hand side of (8.18) converges as
required. This establishes that the deterministic walk is transient.
We now establish (8.12). Given that the deterministic walk can only
take jumps of +1 and -1, let cn,k denote the number of ways of choosing
the timing n+k leftward jumps and n rightward jumps so that the rst
time the walk visits state −k is at time 2n + k, and that at no time before does it return to state 0. It follows from the hypotheses that there
2n+k
are cn,k rn (#β − r)n+k cylinder sets a ∈ β2n+k such that a ⊂ 0 A0,−k
.
From (8.15) we obtain
m(0 A0,−k ) ≤ C
∞
X
cn,k rn (#β − r)n+k M 2n+k
(8.20)
n=0
= C(#β − r)k M k
∞
X
cn,k (r(#β − r)M 2 )n .
n=0
It follows from (8.19) that
1
4
(8.21)
(#β − r)M < 12 .
(8.22)
r(#β − r)M 2 <
and since r > #β − r we also have that
From (8.21) and (8.22), there exists a positive integer R such that
r(#β − r)M 2 <
R(R−1)
(2R−1)2
<
(8.23)
1
4
and
R−1
2R−1
(#β − r)M <
(8.24)
< 12 .
It follows from (8.20), (8.23), (8.24) that
m(0 A0,−k ) ≤ C
R−1
2R − 1
k X
∞
n=0
cn,k
R(R − 1)
(2R − 1)2
n
.
(8.25)
Consider the deterministic walk on Z dened by
Vn (x) :=
n−1
X
j=0
86
f ◦ T j (x)
(8.26)
where the transformation T : ([0, 1], λ) → ([0, 1], λ) is dened by
T x := (2R − 1)x (mod 1),
and
(
R
)
+1 if x ∈ [0, 2R−1
R
−1 if x ∈ [ 2R−1 , 1].
f (x) :=
Clearly, T is a piecewise linear and full-branch Markov map, with
Markov partition P , consisting of 2R − 1 intervals of equal length.
Also the transition function f satises
#{a ∈ P : f (a) = +1} = R and #{a ∈ P : f (a) = −1} = R − 1.
(8.27)
It follows from (8.27) and Proposition 3.6 that the deterministic walk Vn
dened by (8.26) is a model of a simple random walk whose probability
R−1
of a leftward jump at any given time is 2R−1
, and whose probability of
R
a rightward jump at any given time is 2R−1 . It is immediate that this
process diverges to the right with probability 1, and hence that
(8.28)
lim λ(0 B0,−k ) = 0,
k→∞
where λ denotes Lebesgue measure and
0 B0,−k
:= {x ∈ [0, 1] : Vn (x) = −k, and −k < Vj (x) < 0, for 1 ≤ j ≤ n−1}.
For all n ≥ 1 and all a ∈ Pn , λ(a) =
λ(0 B0,−k ) =
R−1
2R − 1
k X
∞
n=0
n
1
2R−1
cn,k
, and it follows that
R(R − 1)
(2R − 1)2
n
.
(8.29)
Equation (8.12) now follows from (8.25), (8.28) and (8.29). This completes the proof.
Example. Consider a DWDE for which the base transformation is a
full-branch Gibbs-Markov map T : ([0, 1], λ) → ([0, 1], λ), with Markov
partition β := {[0, 13 ), [ 13 , 32 ), [ 23 , 1]}, such that λ has Gibbs potential
h : [0, 1] → R+ and inf h > 12 ln 8. Suppose the deterministic environment is ergodic and stationary and that the transition functions f ∈ G
87
satisfy #{a ∈ β : f (a) = +1} = 2 and #{a ∈ β : f (a) = −1} = 1. It
follows from Theorem 8.1 that limn→∞ Ui,n (x, ω) = +∞ for all i ∈ Z,
P -a.e. ω and m-a.e. x.
Remark. In the context of the above example, it is clear that the
fastest rate at which measures of cylinders can decay is 13 . Theorem
8.1 says that as long as cylinders decay at a rate that is faster than √18
then the DWDE will still diverge to the right.
88
Future Work
The results described in Chapters 7 and 8 naturally give rise to several questions and possible future directions for the current work. At
present the author's main interests lie in:
(i) Extending the Zero-One Law (Theorem 7.5) to the non-uniformly
bounded jump case.
(ii) Extending the sharp classication results of Solomon (Theorem
2.12) and Bolthausen & Goldsheid (Theorem 2 of [5]) to the setting described in Chapter 7.
(iii) Showing that in the situation where transition functions are uniformly fair (i.e. E(f ) = 0 for all f ∈ G) the DWDE is recurrent,
and moreover satises the Central Limit Theorem.
(iv) Showing that in the recurrent regime of Theorem 7.5, the growth
rate Snn exists almost surely (and is therefore equal to 0), and
establishing limit laws for the DWDE. In particular, establishing
whether or not Sinai's (ln n)2 law holds in certain situations.
(v) Extending the main results of Chapters 7 and 8 to situations
where the base dynamics is non-Markov, and thereby developing
techniques that are more broadly of use in physical applications
such as the random dynamical billiards as described in Chapter
1 (see Lenci et al [11, 12, 21], Dolgopyat et al [14], and Simula &
Stenlund [26]).
89
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