Ergodic theory of mulitidimensional random dynamical systems

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Random Dynamical Systems

by

Li-Yu Shelley Hsieh

B.Sc, National Chengchi University, 1998 M.Sc, National Taiwan University, 2001

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of

MASTER OF SCIENCE

in the Department of Mathematics and Statistics

c

Li-Yu Shelley Hsieh, 2008 University of Victoria

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Ergodic Theory of Multidimensional

Random Dynamical Systems

by

Li-Yu Shelley Hsieh

B.Sc, National Chengchi University, 1998 M.Sc, National Taiwan University, 2001

Supervisory Committee

Dr. Christopher Bose, Supervisor (Department of Mathematics and Statistics)

Dr. Roderick Edwards, Member (Department of Mathematics and Statistics)

Dr. Anthony Quas, Member (Department of Mathematics and Statistics)

Dr. Arthur Watton, Outside Member (Department of Physics)

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Supervisory Committee

Dr. Christopher Bose, Supervisor (Department of Mathematics and Statistics)

Dr. Roderick Edwards, Member (Department of Mathematics and Statistics)

Dr. Anthony Quas, Member (Department of Mathematics and Statistics)

Dr. Arthur Watton, Outside Member (Department of Physics)

Dr. Arno Berger, External Examiner (University of Albetra)

Abstract

Given a random dynamical system T constructed from Jab lo´nski transformations, consider its Perron-Frobenius operator PT. We prove a weak form of the

Lasota-Yorke inequality for PT and thereby prove the existence of BV- invariant densities

for T . Using the Spectral Decomposition Theorem we prove that the support of an invariant density is open a.e. and give conditions such that the invariant density for T is unique. We study the asymptotic behavior of the Markov operator PT, especially

when T has a unique absolutely continuous invariant measure (ACIM). Under the assumption of uniqueness, we obtain spectral stability in the sense of Keller. As an application, we can use Ulam’s method to approximate the invariant density of PT.

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List of Figures

2.1 _{f : A → R . . . .} 14 2.2 VA∪Bf > VAf + VBf . . . 15 2.3 Rectangular Partition . . . 23 4.1 τ : I1∪ I2∪ I3 → [0, 1] . . . 46 4.2 τ : I1∪ I2 → [0, 1] . . . 54 4.3 τ : [0, 1] → [0, 1]. . . 73

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Acknowledgements

In this opportunity, I would like to express my gratitude to many individuals involved in completing this thesis for providing me different sorts of support.

To Dr. Chris Bose, my academic supervisor. I am indebted to Chris for everything he does for me. I appreciate his unlimited help for supporting me, guiding me and leading me to the right direction. Not only in my academics but also in my spirit, Chris is a great consultant whom I trust, believe in and rely on. Without him, I would never have finished this thesis.

To Dr. Rod Edwards, Dr. Anthony Quas, and many other professors who have taught me. I feel grateful to be their student to learn much more than I expected. Thank you for answering my endless questions.

To Dr. Arno Berger, my external examiner. Thank you for the useful comments so that my thesis is getting better.

To Mahsa Allahbakhshi, Angus Argyle, Magdalena Georgescu, Linghong Lu, Joshua Sorge, Susana Wieler, and everyone whom I know in the Math and Stat Department at UVic. Thank you for encouraging me when I feel depressed and companying with me while I feel lonely. I do not get homesick because of our firm friendship.

To Ani and Ray Merriman, my host family. Thank you for doing such a good job to look after me and providing me with all the information I need.

To Yue-Gui Li, my mother. Thank you for supporting me and allowing me to take a break from my duties.

To Li-Mei and Li-Hsiang, my sisters. Thank you for encouraging me to follow my dream and never give up.

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Introduction

Ergodic theory is the field of mathematics which studies dynamical systems from the point of view of statistical behavior of orbits under a transformation. The basic ingre-dients are a state space (a measurable space), a measurable transformation acting on points in the state space and an invariant measure on the state space. Asymptotics are investigated with respect to the transformation and the invariant measure. For example, one can ask what fraction of the orbit of a single point, under the action of the transformation, will lie in a given measurable set. A very powerful theory can be brought to bear on questions of this type through Ergodic theory.

In a natural way, the action of a measurable transformation induces an action on measures supported on the state space. For instance, invariant measures are fixed points for this action. More generally, we can study the evolution or orbits of measures under this action.

In the case of absolutely continuous measures on the state space (measures which can be obtained by integrating a density function with respect to some natural refer-ence measure on the space) this evolution of measures is facilitated by examining the evolution of their densities under the action of a linear operator, the so-called Perron-Frobenius operator. (See, for example Lasota and Mackey [29].) As an illustration, absolutely continuous invariant measures (ACIM) correspond to fixed densities for the Perron-Frobenius operator. In general, the main advantage of this method is that systems may exhibit very chaotic orbits in the phase space but still may have stable behavior when one views the related evolution of densities on the phase space. For example, there may be a global attracting fixed point for this “functional dynami-cal system”, the fixed point representing the density of the stable statistics for the system.

Another advantage of this functional dynamical viewpoint is to give a precise formulation of the notion of a random dynamical system. This is a special type of Markov process which may also exhibit stable statistics when treated with the machinery of Ergodic theory.

In this thesis we review the framework for studying a random dynamical system T , constructed from a family of nonsingular transformations {τk}k=1,··· ,q on a phase

space X and a probability vector p = (p1, · · · , pq). We analyze its associated

Perron-Frobenius operator PT acting on L1(X, BX, ν). To keep things manageable, we focus

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measure mn, as the reference measure. The phase space is denoted as (In, BIn, m_n).

Our transformations are also restricted to a special class of multidimensional maps called Jab lo´nski transformations [21].

This setting has been studied extensively in papers such as Boyarsky, G´ora and Lou [7] and Kamthan and Mackey [22]. In particular the latter reference contains most of the results to be found in this thesis. Unfortunately, a key inequality early in that paper (the Lasota-Yorke inequality, Equation (5) in Proposition 3.1) does not hold in the generality claimed by the authors. This defect has been noted in the literature (See Math. Rev. MR1345800 (96g:58112) ) but has not yet been addressed in published work. In this thesis, we prove instead a weak-Lasota-Yorke inequality which allows us to recover most of the results of [22]. The main technical difference is in our use of the Spectral Decomposition Theorem (Theorem 4.4) instead of the Ionescu-Tulcea and Marinescu Theorem (Theorem 5.10) to obtain the results.

The presentation is organized as follows.

In Chapter 2, we give the background required to establish our main results. We present and discuss Tonelli variation in n dimensions (see Clarkson and Adams [13]). We indicate the difference between Tonelli variation and classical, one-dimensional variation and its relation to the more modern notion of Generalized Variation (see Giusti [17]). We introduce the Perron-Frobenius and the Koopman operators related to a nonsingular transformation, and the class of piecewise C2 _{Jab lo´}_nski

transforma-tions. While these maps may seem very specialized, they form a natural basis for investigation since, if we are given a piecewise C2 transformation τ on a rectangular partition of the n-dimensional cube, then τ can be approximated by a sequence of piecewise C2 _{Jab lo´}_{nski transformations (see [7]). The approximation holds in the}

following sense. Let τ be a piecewise C2 transformation on a rectangular partition of In_{. Assume τ is expanding in every coordinate direction. Then there is a sequence}

{τη}∞η=1 of C2 expanding Jab lo´nski transformations which converges pointwise to τ .

Moreover, if for each η we have invariant densities fη (Pτηfη = fη), and if f

∗ _{is any}

weak limit point of the {fη}∞η=1, then Pτf∗ = f∗. In particular, if τ has a unique

ACIM, f∗dmn, then fη → f∗ weakly.

In Chapter 3, we give an intuitive description of a random dynamical system T on the probability space (In_{, B}

In, m_n). The ingredients are a finite set of transformations

from In _{to I}n_{, F = {τ}

1, τ2, · · · , τq} and a probability vector p = (p1, p2, . . . pq), that

is, each pk ≥ 0 and Pq_k=1pk = 1. Define T at each point x in In by choosing τk

with probability pk and sending x to τk(x ). In some studies the probability vector

p is allowed to vary with the point x , or is dependent on the iteration number of the transformation. We deal only with constant p in this thesis. In Section 3.2 we explain how to view T as a Markov process, and we study T by using its Perron-Frobenius operator PT. We obtain a weak Lasota-Yorke inequality, Equation

(3.8), under an expanding-on-average condition. This inequality is sufficient to obtain invariant densities for T : if a random dynamical system T satisfies Equation (3.8), then PT has at least one invariant density (Theorem 3.12).

In Chapter 4, we give the definitions of Markov operator and constrictive operator. We introduce the Spectral Decomposition Theorem (see [29], Komornik and Lasota

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[26] or Lasota, Li and Yorke [28]), which allows us to analyze the set of invariant densities for the operator P . We get some results from the Spectral Decomposition Theorem directly and reprove the uniqueness theorem from Boyarsky and Lou [9] by using this theorem instead of using Ionescu-Tulcea and Marinescu Theorem, (see our Theorem 4.46) as the latter would require the strong Lasota-Yorke inequality. A more detailed description about the uniqueness of the invariant density of P can be found in Section 4.4. The main result of this chapter is shown in Theorem 4.50: if any one of the individual transformations of the random dynamical system T has a unique invariant ACIM with τN-ergodic for all positive integers N , then so does T . This theorem is modified from [22, Theorem 3.3]. In Section 4.5, we consider the specific random dynamical system T constructed from piecewise linear Markov trans-formations with respect to the partition P. In this case, a piecewise linear Markov transformation preserves a finite-dimensional subspace of L1(X, BX, ν) so that we

have the following property. The Perron-Frobenius operator PT is represented by

a Markov matrix MT with respect to T . Therefore, we can apply another useful

tool, the (matrix) Perron-Frobenius Theorem, Theorem A.6 in Appendix, to indi-cate whether PT has a unique invariant density. The main theorem in this section,

Theorem 4.56 is from [22, Theorem 3.4]. However, here we deal with the expanding-on-average condition instead of the more restrictive individually expanding condition required in [22] .

In Chapter 5, we study the asymptotic behavior of a Markov operator PT. The

analysis depends on the following representation (Spectral Decomposition of PT):

We assume there exists an integer r > 0, densities {gi}ri=1, functionals λi : L1(In) →

R, i = 1, 2, · · · , r, a permutation σ on {1, 2, · · · , r} and an operator Q : L1(In) → L1(In) such that for every f ∈ L1,

P f =

r

X

i=1

λi(f )gσ(i)+ P Qf.

In addition, it is assumed that kPN_{Qf k}

1goes to zero as N goes to infinity. Then such

sequence {PN_}

N ∈N is called asymptotically periodic. Furthermore, given a density

f , we examine whether the dynamical system approaches to an invariant density f∗ dependent on f . If the approximation holds for each density f (i.e. f∗ is independent of f ), then such sequence {PN_}

N ∈N is called asymptotically stable, in Section 5.1, (see

[29]). For the last two sections of this thesis, we bring back the Ionescu-Tulcea and Marinescu Theorem from [20] and study spectral stability. When the given operator is quasi-compact, we can consider the perturbation by Ulam’s method and calculate the error bounded by Keller’s method in [24]. (See [37] or Li [31] for original articles on this well-known numerical scheme). In this chapter, we confirm [22, Theorem 5.1 and Theorem 7.1] in our Theorem 5.7 and 5.28.

In conclusion, we provide a self-contained and rigorous treatment of the existence of invariant densities for random transformations T constructed from piecewise C2

Jab lo´nski transformations. We study the set of invariant densities and give conditions under which this set contains a unique element, leading to a unique ACIM. We

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study the asymptotic properties of the related Perron-Frobenius operator PT and it’s

stability in the sense of Keller. Finally, as an application we show convergence of Ulam’s method of approximation for this class of transformations.

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Preliminaries

Let I = [0, 1] denote the unit interval in R and In = [0, 1]n _{the unit cube in R}n. Let BIn denote the Borel σ-algebra1 on In, and µ be a measure on (In, B_In). By a

transformation τ : In _{→ I}n_{, we mean a function}

τ (x ) = (ϕ1(x ), · · · , ϕn(x )),

where x ≡ (x1, · · · , xn) ∈ In and ϕi(x ) ∈ [0, 1] for i = 1, 2, · · · , n. Such ϕi’s are

called the components of τ . Definition 2.1. (finite partition)

P = {D1, · · · , Dm} is a finite partition of In if m is finite, m

∪

j=1Dj = I

n _{with measure}

zero boundaries of each Dj, and Dj’s are measurable and pairwise measurably disjoint.

That is, for each pair Di and Dj, the measure of Di∩ Dj is equal to zero.

We say τ is piecewise C2 if

(1) there exists a finite partition P = {Dj}mj=1 of In;

(2) for each Dj ∈ P, τj ≡ τ |Do

j maps D

o

j, the interior set, to its image bijectively,

and all τj, τ

0

j and τ

00

j can be continuously extended to Dj.

We denote by mn the Lebesgue measure on Rn and L1(In) = L1(In, BIn, m_n) the

collection of all integrable functions on In. On the measure space (In, BIn, µ), we

say τ is measurable if τ−1(BIn) ⊆ B_In. It means for any measurable set A ∈ B_In,

each component ϕi satisfies ϕ−1i (A) ∈ BIn.

Here are some general definitions in ergodic theory (see Walters [38]). We list them below for convenience.

Definition 2.2. For a measure space (In_{, B}

In, µ) and a measurable transformation

τ : In_{→ I}n_{, define that}

(1) µ is τ -invariant if for all A ∈ BIn, µ(τ−1A) = µ(A). Equivalently, we say τ is

measure preserving with respect to µ. Moreover, define the measure µ ◦ τ−1 as

µ ◦ τ−1(A) = µ(τ−1A).

1_B

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(2) µ is τ -ergodic if for all A ∈ BIn, τ−1A = A implies µ(A) = 0, or µ(In\ A) = 0.

(3) µ is τ -exact if for all A ∈ BIn with µ(A) > 0, lim

r→∞µ(τ

r_{A) = 1.}

(4) τ is nonsingular with respect to µ if µ ◦ τ−1 is absolutely continuous with respect to µ and denoted as µ ◦ τ−1 << µ. That is, for every A ∈ BIn, µ(A) = 0 implies

µ ◦ τ−1(A) = 0.

Here are some simple examples of the above definition. In our examples, take µ to be the Lebesgue measure m in R and the transformation τ to be defined on the unit interval I into itself.

Example 2.3. (1) For all x ∈ I, let τ (x) = 2x mod 1.

(2) For an irrational number α and for all x ∈ I, define τ by τ (x) = x + α mod 1. (3) For a real number a > 1 and for all x ∈ I, let τ (x) = ax mod 1.

(4) For all x ∈ I, if τ (x) = x2_{, then τ is nonsingular. On the other hand, any}

constant or simple function is not a nonsingular transformation. For example, for all x ∈ I, let ˜τ (x) = 1₃. If A = {1₃}, then m(A) = 0; but

m◦ ˜τ−1(A) = m(˜τ−1A) = m([0, 1]) = 1.

Such ˜τ is not nonsingular. This means that µ ◦ ˜τ−1 has a singular part with respect to the Lebesgue measure.

2.1 Koopman and Perron-Frobenius Operators

In this section, introduce two special operators related to τ , the Koopman operator acting on L∞(X, BX, µ) and the Perron-Frobenius operator acting on L1(X, BX, µ).

The Koopman operator was first defined by Koopman in 1931 [27]. In this article, we deal with the special case on the unit cube in n dimensions. Denote L∞(In) ≡ L∞(In_{, B}

In, m_n) and L1(In) ≡ L1(In, B_In, m_n). The Koopman operator is the dual

of the Perron-Frobenius operator which is presented in Proposition 2.9. Definition 2.4. (Koopman operator)

Let τ : In_{→ I}n _{be a nonsingular transformation with respect to the Lebesgue measure}

mn. The Koopman operator, Kτ : L∞(In) → L∞(In), with respect to the

transfor-mation τ is defined as follows. For all g ∈ L∞(In_),

Kτg = g ◦ τ.

In the definition of the Koopman operator, the condition of a nonsingular trans-formation is essential; otherwise, Kτ is not well-defined. For instance, take τ (x) = 3₄,

for all x ∈ I = [0, 1]. Define g and ˜g ∈ L∞(I) by

g(x) = 1 for all x ∈ I and ˜g(x) = g(x), x ∈ [0, 1] \ {

3 4}

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Then ˜g = g almost everywhere (we abbreviate as a.e.). However, for all x ∈ I, Kτg(x) = g ◦ τ (x) = g(3₄) = 1,

Kτ˜g(x) = ˜g ◦ τ (x) = ˜g(3₄) = 0.

It implies Kτg 6= Kτ˜g. Thus, the operator Kτ is not well-defined.

Definition 2.5. (density and its support)

(1) For every f ∈ L1_(In_{), f is called a density if f ≥ 0 and kf k}

1 = 1, where k · k1

is L1- norm.

(2) For every f ∈ L1_(In_{), supp f = {x ∈ I}n _{: f (x) > 0}.}

Note that the support of a given L1 function is not a closed set. More precisely it is an element of the measure algebra of classes of sets under the equivalence relation of equality up to sets of measure zero.

The following construction gives us the idea of what the Perron-Frobenius operator is. Let τ : In_{→ I}n _{be nonsingular. For each f ∈ L}1_(In_{) and any set A ∈ B}

In, define dµ = f dmn◦ τ−1 as follows µ(A) = Z f dmn◦ τ−1(A) ≡ Z τ−1_(A) f dmn. (2.1)

Since τ is nonsingular with respect to mn,

mn(A) = 0 implies mn(τ−1(A)) = 0.

By Equation (2.1),

mn(τ−1(A)) = 0 implies µ(A) = 0.

Therefore, we get µ << mn. By the Radon-Nikodym Theorem (see Theorem A.4 in

Appendix), there exists a unique g ∈ L1(In) such that

dµ = f dmn◦ τ−1 = gdmn.

Define an operator Pτ : L1(In) → L1(In) such that Pτf = g. Hence, for any A ∈ BIn,

Z A Pτf dmn = Z A gdmn = Z A dµ = µ(A) = Z f dmn◦ τ−1(A) ≡ Z τ−1_(A) f dmn.

Hence, for a nonsingular transformation τ , such operator Pτ defined as above satisfies

Z A Pτf dmn= Z τ−1_(A) f dmn. (2.2)

From Equation (2.2), we know that Pτ is positive, preserves integrals, bounded and

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(a) For every set A ∈ BIn, any positive L1 function f implies Z A Pτf dmn= Z τ−1_(A) f dmn≥ 0. Hence Pτf ≥ 0.

(b) By Equation (2.2) and τ−1(In_{) = I}n_{, we have the fact that}

Z In Pτf dmn= Z τ−1_(In₎ f dmn= Z In f dmn.

Therefore, Pτ preserves integrals. Moreover, if f ≥ 0, then

kPτf k1 = Z In |Pτf | = Z In Pτf = Z In f = Z In |f | = kf k1. (c) For − |f | ≤ f ≤ |f |, −Pτ|f | ≤ Pτf ≤ Pτ|f | implies |Pτf | ≤ Pτ|f |. Thus, kPτf k₁ = Z In |Pτf | dmn≤ Z In Pτ|f | dmn= Z In |f | dmn= kf k₁. Therefore, Pτ is bounded.

(d) Given f1 and f2 in L1(In), let Pτf1 = g1 and Pτf2 = g2 and for a fixed A ∈ BIn,

Z A Pτ(f1 + f2)dmn = Z τ−1_(A) (f1+ f2)dmn = Z τ−1_(A) f1dmn+ Z τ−1_(A) f2dmn = Z A Pτf1dmn+ Z A Pτf2dmn= Z A (Pτf1 + Pτf2)dmn.

Since this equation is true for all A ∈ BIn,

Pτ(f1+ f2) = Pτf1+ Pτf2.

Hence, Pτ is linear.

Lemma 2.6. Pτ defined as above is unique.

Proof. Assume both operators Pτ and ˜Pτ satisfy Equation (2.2). For all A in BIn

and for all f in L1_(In_),

Z A Pτf dmn= Z τ−1_(A) f dmn = Z A ˜ Pτf dmn.

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Since R

A(Pτf − ˜Pτf )dmn = 0 is true for all A ∈ BIn and for all f ∈ L 1_(In_),

Pτf − ˜Pτf = 0.

Hence, Pτ = ˜Pτ.

From the above construction we have the following definition of the Perron-Frobenius operator.

Definition 2.7. (Perron-Frobenius operator)

Let τ : In_{→ I}n _{be a nonsingular transformation. Suppose P}

τ : L1(In) → L1(In) is a

bounded linear operator with the property that for every measurable subset A and for every f ∈ L1_(In_{) we have} Z A Pτf dmn= Z τ−1_(A) f dmn.

Then we say that Pτ is the Perron-Frobenius operator associated to τ .

The operator Pτ has the following properties from Pelikan [36] or [29].

Proposition 2.8. Let τ : In _{→ I}n _{be a nonsingular transformation with respect to}

the Lebesgue measure mn on Rn and Pτ : L1(In) → L1(In) be its Perron-Frobenius

operator, then

(a) For every N ∈ N, PτN = P_τN.

(b) For every density f , Pτf = f a.e. if and only if dµ = f dmn is τ -invariant.

Proof. (a) For every N ∈ N, Z A P_τNf dmn = Z τ−1_(A) P_τN −1f dmn= Z τ−2_(A) P_τN −2f dmn = · · · = Z τ−N_(A) f dmn= Z (τN₎−1_(A) f dmn= Z A P_τNf dm_n.

The equation is true for all f ∈ L1_(In_{) and for all A ∈ B}

In, then

PτN = P_τN.

(b) Claim 1: Pτf = f a.e. implies dµ = f dmn is τ -invariant.

Fix a set A ∈ BIn, let µ(A) =

R Af dmn. For Pτf = f , µ(A) = Z A f dmn= Z A Pτf dmn= Z τ−1_(A) f dmn = µ ◦ τ−1(A).

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Claim 2: dµ = f dmn is τ -invariant which implies Pτf = f a.e.. For all A ∈ BIn, Z A Pτf dmn = Z τ−1_(A) f dmn= Z τ−1_(A) dµ = µ ◦ τ−1(A) = µ(A) = Z A dµ = Z A f dmn. Thus, Pτf = f a.e..

By the definition of the Koopman operator Kτ and the Perron-Frobenius operator

Pτ, we establish some relations between Kτ and Pτ.

Proposition 2.9. Let Koopman operator Kτ and Perron-Frobenius operator Pτ be

associated with nonsingular transformation τ . Then (a) For all f ∈ L1_(In_{) and for all g ∈ L}∞_(In_), _{R (P}

τf )gdmn=R f (Kτg)dmn.

(b) Kτ is the dual of Pτ.

Proof. (a) To prove R (Pτf )gdmn=R f (Kτg)dmn is equivalent to prove

Z

(Pτf )gdmn=

Z

f (g ◦ τ )dmn. (2.3)

First, we show that Equation (2.3) holds when g is any characteristic function. Assume g = χA, for a measurable set A. Then

Z (Pτf )gdmn = Z A Pτf dmn= Z τ−1_(A) f dmn, and Z f (g ◦ τ )dmn = Z f (χA◦ τ )dmn = Z τ−1_(A) f dmn.

Hence, Equation (2.3) holds for every simple function. Now we only have to show the simple functions are dense in L∞(In). Let g ∈ L∞(In) with a finite number M > 0 such that kgk∞≤ M . For a given > 0, take a partition

{−M = a0 < a1 < · · · < am = M }

such that aj − aj−1 < , for all j = 1, · · · , m. Denote Aj = g−1([aj−1, aj]). Let

h =Pm

j=1αjχAj be a simple function for some constants α1, · · · , αm. Then

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Thus, simple functions are dense in L∞(In_{). Hence, for all g ∈ L}∞_(In_),

Z

(Pτf )gdmn =

Z

f (g ◦ τ )dmn.

Since Kτg = g ◦ τ , we get the desired result.

(b) Denote the L1− L∞ _{duality, < , >, as below:}

< f, g >= Z

f gdmn.

By the definition of an adjoint operator, for all g ∈ L∞(In_{) and for all f ∈ L}1_(In_),

< f, P_τ∗g > = < Pτf, g >= Z (Pτf )gdmn= Z f (Kτg)dmn = < f, Kτg > .

Therefore, Kτ is the dual of Pτ (i.e. Pτ is the predual of Kτ).

In some special cases, if A ⊂ In is an n-dimensional rectangle, then we get a specific form of Pτf in Equation (2.2) (see Lasota and Mackey [29]). We describe

this as follows.

Lemma 2.10. Assume τ : In → In _{is piecewise C}2 _{on a partition P = {D} j}mj=1

and τ (x) = (ϕ1(x), · · · , ϕn(x)). Let |Jτ|(x) = det

∂(ϕ1(x),··· ,ϕn(x))

∂(x1,··· ,xn)

be the Jacobian determinant. For each x = (x1, · · · , xn) ∈ In, define A(x) =Qn_i=1[0, xi]. Then

(a) For x ∈ In _a.e.,

Pτf (x) = ∂n ∂x1· · · ∂xn Z τ−1_(A(x)) f (y)dmn(y). (b) For x ∈ In _a.e., Pτf (x) = X τ (y)=x f (y) |Jτ|(y) . (2.4)

Proof. (a) By Fubini’s theorem and the definition of the Perron-Frobenius operator, Z xn 0 Z xn−1 0 · · · Z x1 0 Pτf (y )dmn(y ) = Z A(x ) Pτf (y )dmn(y ) = Z τ−1_{(A(x ))} f (y )dmn(y ).

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Take the derivative on both sides in order

∂xn, ∂xn−1, · · · , ∂x1

and apply Lebesgue’s theorem, for x ∈ In a.e.,

Pτf (x ) = ∂n ∂x1· · · ∂xn Z τ−1_{(A(x ))} f (y )dmn(y ).

(b) Since τ is piecewise C2 on P, let τ−1(A(x )) = ∪m

j=1τ −1 j (A(x )) ∩ D o j be a disjoint union. Then, Z τ−1_{(A(x ))} f (y )dmn(y ) = m X j=1 Z τ_j−1A(x )∩Do j f (y )dmn(y ) = X τ−1 j A(x )∩Doj6=φ j∈{1,2,··· ,m} Z A(x )∩τj(Djo) f (τ_j−1z ) |Jτ|(τj−1z ) dmn(z )

by standard change of variables. Note that for x0 ∼ x sufficiently close, the indices in the sum do not depend on x0. Hence we can differentiate both sides

∂n ∂x1···∂xn to obtain Pτf (x ) = X τ−1 j A(x )∩Doj6=φ j∈{1,2,··· ,m} f (τ_j−1x ) |Jτ|(τj−1x )

where we have again used Lebesgue’s theorem. Since the boundaries of Dj have

measure zero, we can, up to measure zero, simplify the right hand side to X τ_j−1A(x )∩Do_j6=φ j∈{1,2,··· ,m} f (τ_j−1x ) |Jτ|(τj−1x ) = X y : τ (y )=x f (y ) |Jτ|(y ) as required.

2.2 _{Tonelli Variation on R}

n

In one-dimensional space, it is not difficult to define the total variation of a function. However, there is a small problem to make the same definition in a higher-dimensional space. Here we describe Tonelli Variation informally as below.

The main idea of the total variation of f on a closed rectangle, A = Qn

i=1[ai, bi],

is shown as following:

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(2) Take the usual total variation of f on the ith _{coordinate, so that we get a total}

variation in one dimension and denote as V[ai,bi]f in Equation (2.6). This is a real

valued function of the remaining (n − 1) variables.

(3) Integrate the rest (n − 1) dimensions and take the infimum over all g = f in L1_(In_{). Therefore,we get a number which we denote by V}

A,if in Equation (2.7).

(4) Go through i = 1, · · · , n and take the maximum value of VA,if . Finally, we get

a total Tonelli Variation and denote as VAf in Equation (2.8).

The following definition of Tonelli Variation is used in Clarkson and Adams [13]. Definition 2.11. (Tonelli Variation)

For each i, define πi to be a projection from Rn to Rn−1 by

πi(x) ≡ πi(x1, · · · , xn) = (x1, · · · , xi−1, xi+1, · · · , xn).

Let A = Qn

i=1[ai, bi] be an n-dimensional rectangle and consider the set of all

parti-tions Si on the ith-coordinate as

Si = {xi0, xi1, · · · , xir| ai = xi0 < xi1 < · · · < xir = bi, r ∈ N}. (2.5)

For a given function f : A → R, define V[ai,bi]f : πi(A) → R by

(V[ai,bi]f )(πix) = sup Si r X c=1 f (xic) − f (xic−1) , (2.6)

where xic ≡ (x1, · · · , xi−1, xic, xi+1, · · · , xn) ≡ (xic, πix). Let mn−1 be the Lebesgue

measure in Rn−1 _{and define}

VA,if = inf ga.e.= f Z πi(A) (V[ai,bi]g)(πix)dmn−1(πix). (2.7) Set VAf = max i=1,··· ,nVA,if. (2.8)

Then VAf is called Tonelli Variation of f on A.

Lemma 2.12. The appearance of inf

ga.e.= f

in the definition ensures that Tonelli Variation is well-defined on elements of L1_(In_{). If f}

1 = f2 a.e., then for any A ∈ BIn,

VAf1 = VAf2. (2.9)

Proof. Consider the ith _{coordinate. If g = f}

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By the definition of Tonelli Variation, VA,if1 = inf ga.e.= f1 Z πi(In) (V[0,1]g)dmn−1, ≥ inf ga.e.= f2 Z πi(In) (V[0,1]g)dmn−1= VA,if2. =⇒ VAf1 ≥ VAf2.

Similarly, we have VAf1 ≤ VAf2. Hence, VInf₁ = V_Inf₂ if f₁ = f₂ a.e.

Figure 2.1: f : A → R

Example 2.13. Consider A = [1₄,3₄] × [1₃,2₃_{] and define a function f : A → R as} f (x1, x2) = f (x) = x2, x ∈ [1₄,3₄] × [1₃,2₃] ∩ Q 1 2x2, x ∈ [ 1 4, 3 4] × [ 1 3, 2 3] \ Q. Then VAf = 1 12. Proof. For the given function f ,

(V_[1 4, 3 4]f )(x2) = 0 (V_[1 3, 2 3]f )(x1) = ∞. Now define g as g(x1, x2) = 1₂x2 on A, so g a.e. = f on A. Moreover, (V_[1 4, 3 4]g)(x2) = 0 (V_[1 3, 2 3]g)(x1) = 1 2( 2 3 − 1 3) = 1 6.

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In this case, a continuous function has the minimum value of variations in the same equivalence class of L1_{(A). Hence,}

VA,1f = inf ˜ fa.e.= f Z 2₃ 1 3 V_[1 4, 3 4] ˜ f = Z 2₃ 1 3 V_[1 4, 3 4]g = 0 VA,2f = inf ˜ fa.e.= f Z 3₄ 1 4 V_[1 3, 2 3] ˜ f = Z 3₄ 1 4 V_[1 3, 2 3]g = 1 12. Therefore, VAf = maxi=1,2VA,if = ₁₂1 .

Definition 2.14. (bounded Tonelli Variation)

For a closed, n-dimensional rectangle, A ⊂ In_{, if f ∈ L}1_{(A) and V}

Af < ∞, then f

is called a function of bounded Tonelli Variation on A. Denote BV (A) as follows BV (A) = {VAf < ∞| f ∈ L1(A), f : A → R} and BV = BV (In).

We list some properties of the Tonelli Variation which are the same as in classical, 1-dimensional variation. Not all the properties are the same when we use Tonelli Variation. For example, a basic identity for the classical, one-dimensional variation is that for a function f and a < c < b,

V[a,b]f = V[a,c]f + V[c,b]f.

Note that this identity fails for the definition of Tonelli Variation because of the appearance of inf

ga.e.= f

. We give an example (in Example 2.15) in one-dimensional case to show VA∪Bf > VAf + VBf . 1/8 1 1 0 x B A 1/2 3/4 3/8 Figure 2.2: VA∪Bf > VAf + VBf

Example 2.15. Let A = [0,1₂) and B = [1₂_{, 1]. Define f : I → R by} f (x) = ₁ 4x, ∀x ∈ [0, 1 2) 3 4x, ∀x ∈ [ 1 2, 1].

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Then, VA∪Bf > VAf + VBf .

Proof. By the definition of Tonelli Variation, when n = 1, VAf = inf

ga.e.= f

VAg.

Define a function g ∈ L1_{(I) by}

g(x) = ₁ 4x, ∀x ∈ (0, 1 2) 3 4x, ∀x ∈ ( 1 2, 1).

It is clear that f = g on [0, 1] a.e. Besides,

VA∪Bf = 1₈ + 1₄ +3₈ = 3₄ VA∪Bg = 1₈ +1₄ + 3₈ = 3₄, which implies VA∪Bf = inf ga.e.= f VA∪Bg = 3 4. In addition, VAf = (1₈ +1₄) = 3₈ VAg = 1₈ implies VAf = inf ga.e.= f VAg = 1₈. V_Bf = 3₈ VBg = 3₈ implies VBf = inf ga.e.= f VBg = 3₈. Therefore, VA∪Bf = 3 4 > 1 8 + 3 8 = VAf + VBf.

Proposition 2.16. Let f, g ∈ L1(In) and A ⊂ In be a closed subrectangle. Assume A =Qn

i=1[ai, bi].

(a) If f ∈ BV (In), then f |A∈ BV (A) and VA(f |A) ≤ VInf .

(b) If f, g ∈ BV (A), then VA(f + g) ≤ VAf + VAg.

(c) If f, g ∈ BV (A) ∩ L∞(In_{), then f g ∈ BV (A) and}

VA(f g) ≤ kgk∞VAf + kf k∞VAg.

(d) Let f |A ∈ BV (A) and l = min

i=1,··· ,n(bi− ai). Assume l = l(A) > 0. Then we have

f χA∈ BV (In) and VIn(f χ_A) ≤ 2V_A(f |_A) + 2 l Z A |f |dmn.

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(e) Assume f ∈ BV (A) and g ∈ C1_{(A) such that for all i = 1, 2, · · · , n,} ∂g(x)

∂xi and g

have continuous extensions to ∂A, the boundary of A. Set constants Γi = sup x∈A ∂g(x) ∂xi

and Γ = max_{i=1,··· ,n}Γi.

Then VA(f g) ≤ kgk∞VAf + Γ Z A |f |dmn. (f ) Suppose B =Qn

i=1[¯ai, ¯bi] is also a closed subrectangle in I

n _{such that A ∪ B ⊂ I}n

is a rectangle and mn(A∩B) = 0. If f ∈ BV (A∪B), then for each i = 1, 2, · · · , n,

VA∪B,if ≥ VA,i(f |A) + VB,i(f |B).

Proof. (a) Fix the ith coordinate. For a given > 0, choose ˜f a.e.= f such that

Z

πi(In)

V[0,1]f (π˜ ix )dmn−1(πix ) ≤ VIn_,if + .

Since ˜f |A= f |A a.e., for all πi(x ) = (x1, · · · , xi−1, xi+1, · · · , xn) ∈ πi(In),

V[ai,bi]f |˜A(πix ) ≤ V[0,1]f (π˜ ix ). Hence, Z πi(A) V[ai,bi]f |˜Admn−1 ≤ Z πi(A) V[0,1]f dm˜ n−1 ≤ Z πi(In) V[0,1]f dm˜ n−1 ≤ VIn_,if + .

Therefore, VA,i(f |A) ≤ VIn_,if + . Take maximum over i = 1, 2, · · · , n. For an

arbitrary , we have

VA(f |A) ≤ VInf.

(b) Fix the ith coordinate. For a given > 0, choose ˜f and ˜g in L1(A) such that ˜

f = f a.e., ˜g = g a.e. and Z πi(A) V[ai,bi]f (π˜ ix )dmn−1(πix ) ≤ VA,if + ; Z πi(A) V[ai,bi]g(π˜ ix )dmn−1(πix ) ≤ VA,ig + .

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one-dimensional variation to obtain Z πi(A) V[ai,bi]( ˜f + ˜g)dmn−1 ≤ Z πi(A) V[ai,bi]f + V˜ [ai,bi]˜g dmn−1 = Z πi(A) V[ai,bi]f dm˜ n−1+ Z πi(A) V[ai,bi]˜gdmn−1 ≤ (VA,if + ) + (VA,ig + ).

=⇒ VA,i(f + g) ≤ VA,if + VA,ig + 2.

Using the same argument as in part (a), we get VA(f + g) ≤ VAf + VAg.

(c) Fix the ith _{coordinate. For a given > 0, choose ˜}_{f and ˜}_{g in L}1_{(A) such that ˜}_f

and ˜g have the same setting as in part (b). If ˆf = min { ˜f , kf k∞}, then

V[ai,bi]f (πˆ ix ) ≤ V[ai,bi]f (π˜ ix ).

Similarly, ˆg = min {˜g, kgk∞} implies V[ai,bi]g(πˆ ix ) ≤ V[ai,bi]g(π˜ ix ). Observe that

ˆ

f = f a.e., ˆg = g a.e. and ˆf ˆg = f g a.e. Then VA,i(f g) ≤ Z πi(A) V[ai,bi]( ˆf ˆg)dmn−1 ≤ kˆgk∞ Z πi(A) V[ai,bi]f dmˆ n−1+ k ˆf k∞ Z πi(A) V[ai,bi]gdmˆ n−1 ≤ kgk∞ Z πi(A) V[ai,bi]f dm˜ n−1+ kf k∞ Z πi(A) V[ai,bi]gdm˜ n−1 ≤ kgk∞(VA,if + ) + kf k∞(VA,ig + ) ,

where we have again used a standard result from classical, one-dimensional vari-ation at the second inequality. Hence, use the same argument as in part (a),

VA(f g) ≤ kgk∞VAf + kf k∞VAg.

(d) Fix the ith _{coordinate. For a given > 0, choose ˜}_{f = f |}

Aa.e. in L1(A) such that

Z

πi(A)

V[ai,bi]f (π˜ ix )dmn−1(πix ) ≤ VA,i(f |A) + ,

where πix = (x1, · · · , xi−1, xi+1, · · · , xn) and x ≡ (xi, πix ). Now recall the

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then exists c ∈ [a, b] such that g(c) ≤ _b−a1 R_abg(y)d(y) and

V[0,1](gχ[a,b]) ≤ V[a,b](g|[a,b]) + |g(a)| + |g(b)|

Set ˆf ∈ L1(In) with ˆf = ˜f on A, ˆf = f on In\ A and l = min

i=1,··· ,n|ai− bi|. Then ˆ f χA= f χA a.e. and VIn_,i(f χ_A) ≤ Z πi(In) V[0,1]( ˆf χA)dmn−1 = Z πi(A) V[0,1]( ˆf χA)dmn−1 ≤ 2 Z πi(A) (V[ai,bi]( ˆf |A))(πix )dmn(πix ) + 2 l Z πi(A) Z bi ai | ˆf |(xi, πix )dm(xi) dmn(πix ) = 2 Z πi(A) V[ai,bi]f dm˜ n−1+ 2 l Z A | ˆf |dmn ≤ 2(VA,i(f |A) + ) + 2 l Z A |f |dmn.

Use the same argument as in part (a) to get VIn(f χ_A) ≤ 2V_A(f |_A) + 2 l Z A |f |dmn.

(e) Fix the ithcoordinate. For a given > 0, choose ˜f = f a.e. as in part (b). We use the well-known fact that in the classical, one-dimensional variation: a function f ∈ BV [a, b] is Riemann integrable on [a, b]. Now, for all πix ∈ πi(A), we have

V[a,b]f (π˜ ix ) < ∞ since the function is mn−1−integrable by choice. For g is C1,

|g(x )| ≤ kgk∞< ∞, ∀x ∈ In.

Following the notation on Equation (2.5) and (2.6), for the same , there exists a partition such that

r X c=1 | ˜f (xic−1)||xic−1− xic| ≤ Z bi ai | ˜f (xi, πix )|dm(xi) +

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and r X c=1 | ˜f g(xic) − f g(x˜ ic−1)| ≤ r X c=1 | ˜f (xic) − ˜f (xic−1)||g(xic)| + r X c=1 | ˜f (xic−1)||g(xic) − g(xic−1)| ≤ kgk∞V[ai,bi]f (π˜ ix ) + r X c=1 | ˜f (xic−1)| ∂g( ˆxic) ∂xi |xic − xic−1| ≤ kgk∞V[ai,bi]f (π˜ ix ) + Γ Z bi ai | ˜f (xi, πix )|dm(xi) + . Hence, V[ai,bi]( ˜f g)(πix ) ≤ kgk∞V[ai,bi]f (π˜ ix ) + Γ Z bi ai | ˜f (xi, πix )|dm(xi) + .

Note that ˜f g = f g a.e., so integration on πi(A) yields

VA,i(f g) ≤ Z πi(A) V[ai,bi]( ˜f g)(πix )dmn−1(πix ) ≤ kgk∞ Z πi(A) V[ai,bi]f (π˜ ix )dmn−1(πix ) + Γ Z πi(A) Z bi ai | ˜f (xi, πix )|dm(xi) + dmn−1(πix ) ≤ kgk∞(VA,if + ) + Γ Z A | ˜f |dmn+ . The same argument as above, we get

VA(f g) ≤ kgk∞VAf + Γ

Z

A

|f |dmn.

(f) Since A ∪ B is a closed rectangle, there exists a unique io such that

[aio, bio] ∪ [¯aio, ¯bio] = [aio, ¯bio] with ¯aio = bio

and

[ai, bi] = [¯ai, ¯bi], ∀i 6= io.

For a given > 0, choose ˜f = f a.e. such that for each i = 1, 2, · · · , n, Z

πi(A∪B)

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Thus, for each coordinate i 6= io, we have πi(A ∪ B) = πi(A) ∪ πi(B) and VA∪B,if + ≥ Z πi(A∪B) V_[a_i_,¯_b_i_]f dm˜ n−1 = Z πi(A) V_[a_i_,¯_b_i_]f dm˜ n−1+ Z πi(B) V_[a_i_,¯_b_i_]f dm˜ n−1 = Z πi(A) V[ai,bi]( ˜f |A)dmn−1+ Z πi(B) V_[¯_a_i_,¯_b_i_]( ˜f |B)dmn−1 ≥ VA,i(f |A) + VB,i(f |B).

Any arbitrary > 0 yields

VA∪B,if ≥ VA,i(f |A) + VB,i(f |B), ∀i 6= io.

For the coordinate io, we have πio(A ∪ B) = πio(A) = πio(B) and

VA∪B,iof + ≥ Z πio(A∪B) V_[a_io_,¯_b_io_]f dm˜ n−1 = Z πio(A∪B) V_[a_io_,b_io_]∪[¯_a_io_,¯_b_io_]f dm˜ n−1 = Z πio(A) V[aio,bio]( ˜f |A)dmn−1+ Z πio(B) V_[¯_a_io_,¯_b_io_]( ˜f |B)dmn−1 ≥ VA,io(f |A) + VB,io(f |B).

Hence, we get the conclusion for each i = 1, 2, · · · , n VA∪B,if ≥ VA,i(f |A) + VB,i(f |B).

2.3 _{Generalized Bounded Variation on R}

n

The notion of Tonelli Variation developed in the previous section is a classical ap-proach to extending the notion of one-dimensional variation to multiple dimensions. There is another, more modern extension based on the well-known formula

V[a,b]f =

Z b

a

|f0(t)| dt

whenever f ∈ C1_{([a, b]; R) and the notion of (weak) generalized derivative.}

Definition 2.17. Let A ⊂ Rnbe an open set, and let C₀1_{(A; R}n) denote the collection of compactly supported smooth vector fields on A. Let div denote the divergence and

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for f ∈ L1_{(A) put} GVA(f ) = sup Z A f (x)div ω(x) dmn(x) : ω ∈ C01(A; R n_{), |ω(x)|} 2 ≤ 1 ∀x ∈ A , where |ω|2 = |(ω1, . . . , ωn)|2 = (Pn_i=1|ωi|2) 1/2

denotes the ordinary vector 2–norm. We use the term Generalized Variation of f over A for this quantity. If GVA(f )

is finite, then f is said to have bounded Generalized Variation over A, and we write f ∈ BGV (A). The set of f ∈ BGV (A) equipped with the norm

kf kBGV = kf k1+ GVA(f )

is a Banach space. More deta can be found in E. Giusti [17]. Generalized Variation was developed for the theory of minimal surfaces.

The notion of Generalized Variation is extended to compact sets A with piecewise smooth boundaries by the following construction. First, assume f is C2_{on the interior}

of A (Ao_{) and vanishing on the boundary ∂A, then (by choosing ω = −∇f /|∇f |} 2

and using Stokes’ theorem)

GVA(f ) := GVAo(f ) = Z A |∇f |2dmn(x ) = Z A n X i=1 ∂f ∂xi (x ) 2!1/2 dmn(x ) (2.10)

(provided that the integral is finite). For a general f ∈ L1(A) we define

GVA(f ) = lim

N GVA(fN),

where fN → f in L1−norm and fN is a sequence of functions on A which are C2

on interior of A and vanishing on ∂A as above. The technical details to justify this construction are also found in [17].

The connection between Tonelli Variation and Generalized Variation, when A is a rectangle, is the following.

Lemma 2.18. Let A ⊆ Rn _{be a closed bounded rectangle and f ∈ L}1_{(A). Then}

GVA(f ) ≤ nVA(f ).

The proof of this is contained in Lemma 4.34 and Corollary 4.35 later in this document.

2.4 Piecewise C

2

Jab lo´

nski Transformation

We introduce special transformations in a higher dimensional space as defined by Jab lo´nski [21]. We first define a rectangular partition and define a Jab lo´nski trans-formation on the rectangular partition.

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(1) A partition P = {Dj}mj=1 (in the sense of Definition 2.1) is called a rectangular

partition where each partition element Dj is an n-dimensional rectangle.

(2) Given two partitions P1 = {D1, · · · , Dm}, P2 = {E1, · · · , El}, we define the join

partition of P1 and P2 by

P ≡ P1∨ P2 = {Di∩ Ej|Di ∈ P1, Ej ∈ P2 where i = 1, · · · , m; j = 1, · · · , l}.

Definition 2.20. (Jab lo´nski transformation)

Let P = {D1, · · · , Dm} be a rectangular partition. For Dj ∈ P, let Dj =

Qn

i=1[aij, bij]

and τj = τ |Do

j. For all x ≡ (x1, · · · , xn) ∈ D

o

j ⊂ In and for i = 1, · · · , n, define

ϕij : (aij, bij) → [0, 1]. If each τj has the following representation

τj(x) = (ϕ1j(x1), ϕ2j(x2), · · · , ϕnj(xn)), (2.11)

then the transformation τ : In _{→ I}n _{is called a Jab lo´}_{nski transformation with respect}

to the rectangular partition P.

Definition 2.21. Using the same notation as in Definition 2.20,

(1) τ is called piecewise C2 _{with respect to the partition P if for i = 1, · · · , n and}

j = 1, · · · , m, each ϕij is C2 bijective on (aij, bij), and ϕij, ϕ

0

ij and ϕ

00

ij have

continuous extensions to [aij, bij].

(2) τ is expanding if for each i = 1, · · · , n, inf

xi∈(aij ,bij )

j=1,··· ,m

|ϕ0_ij(xi)| > 1.

Note: If a Jab lo´nski transformation satisfies condition (1) in Definition 2.21, we call the transformation a piecewise C2 Jab lo´nski transformation.

1 1 1 0 1/2 3/5 2/3 D 2 D 1 D₄ D₃ x2 x

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Example 2.22. Let τ : I2 _{→ I}2 _{be defined on a rectangular partition P of I}2 _as

follows: let P = {D1, D2, D3, D4} and for j = 1, · · · , 4, let τj = τ |Do j. D1 = [0,3₅) × [0,1₂); ∀x ∈ Do1, τ1(x1, x2) = (4₃x1+₁₀1, 2x2 ), D2 = [0,3₅) × [1₂, 1]; ∀x ∈ Do2, τ2(x1, x2) = (4₃x1+₁₀1, 2 − 2x2), D3 = [3₅, 1] × [0,2₃); ∀x ∈ D3o, τ3(x1, x2) = ( x21 , 3 2x2 ), D4 = [3₅, 1] × [2₃, 1]; ∀x ∈ Do4, τ4(x1, x2) = ( x21 , 2x2− 1).

Then τ is a piecewise C2 _{expanding Jab lo´}_{nski transformation.}

Proof. Clearly, τ is piecewise C2 on P satisfying Equation (2.11). By the definition of the transformation τ , we have

|ϕ0₁₁(x1)| = 4₃, |ϕ 0 21(x2)| = 2, |ϕ0₁₂(x1)| = 4₃, |ϕ 0 22(x2)| = 2, |ϕ0₁₃(x1)| = 2x1 ≥ 6₅, |ϕ 0 23(x2)| = 3₂, |ϕ0₁₄(x1)| = 2x1 ≥ 6₅, |ϕ 0 24(x2)| = 2. =⇒        inf j=1,··· ,4 x1∈(0,1) |ϕ0_1j(x1)| = 6₅ > 1, inf i=1,··· ,4 x2∈(0,1) |ϕ0_2j(x2)| = 3₂ > 1.

Therefore, τ is a piecewise C2 _{expanding Jab lo´}_{nski transformation.}

Remark 2.23. A Jab lo´nski transformation τ maps topological rectangles (product of intervals) to measurable rectangles (product of measurable sets). Moreover, τ−1 also has this property.

Following results are simple observations about combinations of Jab lo´nski trans-formations.

Lemma 2.24. (a) Let τ1 and τ2 be Jab lo´nski transformations defined on partitions

P1 and P2 respectively. Then both τ1 and τ2 are also Jab lo´nski transformations

on the join partition of P1 and P2.

(b) Given a Jab lo´nski transformation τ defined on the partition P = {D1, · · · , Dm},

for any integer N ∈ N, τN _{is a Jab lo´}_{nski transformation on the partition P}N ₌

P ∨ τ−1_{P · · · ∨ τ}−N +1_{P. That is}

PN _{≡ {D}

j1 ∩ · · · ∩ τ

−N +1

(DjN) : Dj1, · · · , DjN ∈ P}. (2.12)

Proof. (a) Suppose P1 = {Di}mi=11 and P2 = {Ej}mj=12 , then

P1∨ P2 = {Di ∩ Ej : i = 1, · · · , m1, j = 1, · · · , m2}.

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τ1|Di∩Ej(x ) = τ1|Di(x ) and τ2|Di∩Ej(x ) = τ2|Ej(x ).

Since τ1 is a Jab lo´nski transformation on P1, τ1 on partition P1∨ P2 satisfies the

representation (2.11). Hence, τ1 is also a Jab lo´nski transformation on P1 ∨ P2.

Using the same argument for τ2, then both τ1and τ2are Jab lo´nski transformations

on the partition P1 ∨ P2.

(b) It is enough to prove that τ is a Jab lo´nski transformation on P then τ2 is also a Jab lo´nski transformation on P2_{. For any element D}

i∩ τ−1(Dj) ∈ P2, if for all

x ∈ Di∩ τ−1(Dj), then

τ (x ) ∈ τ (Di∩ τ−1(Dj)) ⊂ (τ (Di) ∩ Dj) ⊂ Dj.

Thus, τ2_|

Di∩τ−1(Dj)(x ) = τ |Dj(x ). That is, τ

2 _{is a Jab lo´}_{nski transformation on}

P2_{. By induction, we get τ}N _{is a Jab lo´}_{nski transformation on P}N_.

We say that a sequence of functions {fN} ⊂ L1(In) is weakly convergent to f if

there exists an f ∈ L1(In) such that for every h ∈ L∞(In)

lim N →∞ Z fNh = Z f h. Definition 2.25. [29] (weakly precompact)

For a subset F in L1_(In_{), F is called weakly precompact if every sequence of}

func-tions {fN}N ∈N in F contains a weakly convergent subsequence {fNk}k∈N that

con-verges to an ¯f in L1(In).

Lemma 2.26. [21] Fix M < ∞. Let

K = {f ≥ 0 | VInf ≤ M, kf k

1 ≤ 1, where f : I n

→ R}, then K is weakly precompact on L1_(In_).

Proposition 2.27. Let f ∈ L1_(In_{) and τ : I}n _{→ I}n _{be a piecewise C}2 _{Jab lo´}_nski

transformation on a rectangular partition P. Suppose A ∈ P and A = Qn

i=1[ai, bi].

Define τ (x) = (ϕ1(x1), · · · , ϕn(xn)) on A. For each i = 1, · · · , n,

(a) τ (A) ⊂ In _{is a rectangle.}

(b) For each i = 1, · · · , n, Vτ (A),i (f |Jτ−1|) ◦ τ −1_{≤ k(ϕ}0 i) −1_k ∞VA,i(f |A) + Γ Z A |f |dmn. (2.13)

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Proof. (a) Since τ is a C2 _{Jab lo´}_{nski transformation on A, τ (A) is a rectangle.}

(b) For each i = 1, 2, · · · , n and for a given > 0, choose ˜f a.e.= f such that

Z

πi(A)

V[ai,bi]( ˜f |A)dmn−1≤ VA,i(f |A) + .

By the definition of τ , we have |Jτ|(x ) =

Qn

i=1|ϕ

0

i(xi)|. From (a), we get a

rectangle τ (A) =Qn i=1Ii. Hence, Vi ≡ Vτ (A),i (f |Jτ−1|) ◦ τ −1_≤ Z πi(τ A) VIi ( ˜f |J_τ−1|) ◦ τ−1_dm n−1 = Z πi(τ A) n Y j=1,j6=i |(ϕ0_j)−1| ◦ τ−1VIi ( ˜f |(ϕ0_i)−1|) ◦ τ−1dmn−1 = Z πi(A) V[ai,bi] ˜f |(ϕ 0 i) −1_| dmn−1.

Since > 0 is arbitrary, for each i = 1, · · · , n, we get Vτ (A),i f |Jτ−1| ◦ τ −1_{≤ k(ϕ}0 i) −1_k ∞VA,i(f |A) + Γ Z A |f |dmn.

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Random Dynamical System

A random dynamical system is constructed from finitely many transformations and a probability vector. An informal description of a random dynamical system T is as follows: Given the collection of transformations F = {τk}qk=1 on In, a point x ∈ In

and the probability vector p = (p1, · · · , pq), we randomly select an image point τj(x )

of x with probability pj. In other words, the image of x under T is a set of q points in

In_{, selected according to the probability law p. In this article, we only consider p} k’s

that are constants. We use the notation T = {τk; pk}k=1,··· ,q for a random dynamical

system constructed from the given transformations in F and the probability vector p. For ~k ∈ {1, · · · , q}K_{, we write}

T~_k(x ) = τkK ◦ τkK−1◦ · · · ◦ τk1(x ) and p~k = pkK · pkK−1· · · pk1. (3.1)

The following definition of a random dynamical system is used in Bahsoun, Bose and Quas [2], Bahsoun and G´ora [3], G´ora and Boyarsky [18] or Kifer [25].

Definition 3.1. (random dynamical system on In for position independent probabil-ity)

For all k = 1, · · · , q, let τk : In → In be given transformations. Let p = (p1, · · · , pq)

be a probability vector (i.e. Pq

k=1pk= 1) with pk> 0 for all k. A random dynamical

system T = {τk; pk}k=1,··· ,q is a Markov process with transition function

P(x, A) =

q

X

k=1

pkχA(τk(x)), (3.2)

where χA is the characteristic function of a measurable set A.

Note that, for ~k ∈ {1, · · · , q}K_,

P~_k(x , A) = q

X

kK,··· ,k1=1

p~_kχA(T~_k(x )). (3.3)

By Equation (3.2), we recognize the transition function as a type of Koopman operator, as in Definition 2.4. We define the Koopman operator with respect to the random dynamical system T as below.

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Definition 3.2. (Koopman operator w.r.t. T )

Let T = {τk; pk}k=1,··· ,q be a random dynamical system constructed from nonsingular

transformations {τk} on In. For all g ∈ L∞(In), define KT : L∞(In) → L∞(In) by

KTg(x) = q X k=1 pk· Kτkg(x), where Kτkg = g ◦ τk.

From the previous chapter, for a nonsingular transformation τ , Kτ is the dual of

Pτ. To get the duality between the Koopman operator KT and the Perron-Frobenius

operator PT with respect to T , there is a natural way to define PT. By Proposition

2.9 (a), R (Pτf )g dmn = R f (Kτg) dmn. By the definition of KT, for all f ∈ L1(In)

and for all g ∈ L∞(In_),

Z f (KTg) dmn = Z f q X k=1 pkKτkg ! dmn= q X k=1 pk Z f (Kτkg)dmn = q X k=1 pk Z (Pτkf )gdmn= Z q X k=1 pkPτkf ! gdmn. Therefore, if we define PT ≡ Pq

k=1pkPτk, we still have the dual relation between

these two operators. That is, P_T∗ = KT.

Definition 3.3. (Perron-Frobenius operator w.r.t. T )

Let T = {τk; pk}k=1,··· ,q be a random dynamical system constructed from nonsingular

transformations {τk}q_k=1 on In. For all f ∈ L1(In), define PT : L1(In) → L1(In) by

PTf = q

X

k=1

pk· Pτkf. (3.4)

Let µ be a measure in the measure space (In_{, B}

In), and T = {τ_k; p_k}q

k=1 be a

random dynamical system defined as above. The measure µ is called T -invariant, in Pelikan [36] or Morita [34], if for all A ∈ BIn,

µ(A) =

q

X

k=1

pkµ(τk−1A).

Lemma 3.4. Let T = {τk; pk}k=1,··· ,q be a random dynamical system and PT be its

Perron-Frobenius operator. For every density f∗, PTf∗ = f∗ a.e. is the same as

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Proof. If PTf∗ = f∗ a.e. and dµ = f∗dmn, then for any measurable set A, µ(A) = Z A dµ = Z A f∗dmn= Z A PTf∗dmn = q X k=1 pk Z A Pτkf ∗ dmn = q X k=1 pk Z τ_k−1(A) f∗dmn = q X k=1 pk Z τ_k−1(A) dµ = q X k=1 pkµ τk−1(A) .

Since A is an arbitrary measurable set, µ is T -invariant. On the other hand, for every measurable set A, if dµ = f∗dmn is T -invariant, then

Z A PTf∗dmn = q X k=1 pk Z A Pτkf ∗ dmn = q X k=1 pk Z τ−1_(A) f∗dmn = q X k=1 pk Z τ−1_(A) dµ = q X k=1 pkµ τ−1(A) = µ(A) = Z A dµ = Z A f∗dmn.

Thus, Pτf∗ = f∗ a.e since A is arbitrary.

3.1 Weak Lasota-Yorke Inequality

We introduce the idea of expanding-on-average for the individual transformations in the random dynamical system T , as described in Boyarsky and Levesque [8], Boyarsky and Lou [10, 11] or Kamthan and Mackey [22]. This condition is weaker than the requirement that each transformation τk in T be strictly expanding. We also present

a weak form of the Lasota-Yorke inequality, in Proposition 3.6. This inequality helps us to control the variation term so that it goes to zero if we iterate T many times. Definition 3.5. (expanding-on-average)

Let T = {τk; pk}k=1,··· ,q be a random dynamical system constructed from piecewise

C2 _{Jab lo´}_{nski transformations (i.e. for k = 1, · · · , q, each τ}

k is defined on a

rect-angular partition Pk = {D1,k, · · · , Dm(k),k}). Define P = {Dj}mj=1 to be the join

partition of {P1, · · · , Pq}. For each Dj ∈ P, let Dj ≡Q_i=1n [aij, bij] and τj,k = τk|Do j.

For i = 1, · · · , n, let ϕij,k : (aij, bij) → I be C2 injective. Since τk is a Jab lo´nski

transformation, for all x ∈ Dj, each τj,k has the representation

τj,k(x) = (ϕ1j,k(x1), · · · , ϕnj,k(xn)).

Define k(ϕ0_i,k)−1k∞ = max

j=1,··· ,m _x sup i∈(aij,bij) ϕ0_ij,k(xi) −1 !

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satisfying 0 < α < 1 such that for each i = 1, · · · , n, q X k=1 pk· k(ϕ 0 i,k) −1_k ∞≤ α, (3.5)

then the random dynamical system T is expanding-on-average.

Before introducing an important theorem in this chapter, we give a weak form of the Lasota-Yorke inequality described in the following proposition. The proof of Proposition 3.6 is similar as in Lasota and York [30], but here we deal with n dimensions instead of one dimension.

Key for the indices: n = space dimension

i = space coordinate, i = 1, 2, · · · , n j = partition element, Dj, j = 1, 2, · · · , m

k = transformation numbers, k = 1, 2, · · · , q ~k ∈ {1, · · · , q}K

; K ∈ N, for different integer K using different ~k for TK p~_k= pkK · pk(K−1)· · · pk2 · pk1 and T~k(x ) = τkK ◦ τkK−1 ◦ · · · ◦ τk1(x ).

Proposition 3.6. Let T = {τk; pk}k=1,··· ,q be a random dynamical system satisfying

Equestion (3.5) which is constructed from piecewise C2 _{Jab lo´}_{nski transformations}

defined on the common partition P. For the same α as in Equation (3.5), con-sider a constant K ∈ N. For ~k ∈ {1, · · · , q}K_{, let T}

~

k’s, as shown on Equation

(3.1), be defined on the partition PK _{= {D}

1, · · · , Dm}, where PK is the join of all

{P, T−1_{P, · · · , T}−K+1_{P} (see Lemma 2.24 (b)). Assume T}

j,~k = T~k|Do

j has the

1 _{real valued function of n variables on D} j. Define constants Γ_ij,~_k = sup x∈Do j ∂g_j,~_k(x) ∂xi and Γ = max i=1,··· ,n; ~k j=1,··· ,m Γ_ij,~_k.

Given Dj =Qn_i=1[aij, bij], define constants

l = min

i=1,··· ,n &

j=1,··· ,m

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Then for every f ∈ L1_(In_),

VInP_TKf ≤ 2αKV_Inf + γ_Kkf k₁. (3.6)

Note: Equation (3.6) is a weak form of the Lasota-Yorke inequality. Usually the inequality

VInP_Tf ≤ αV_Inf + γ kf k₁

is called Lasota-Yorke inequality, in [30].

Proof. Let ~k ∈ {1, · · · , q}K_{. By Equation (3.4) and (3.1),}

PTf = q X k1=1 pk1Pτ_k1f ; P_T2f = q X k2=1 pk2Pτ_k2 q X k1=1 pk1Pτ_k1f ! = q X k2=1 q X k1=1 pk2pk1Pτ_k2 ◦ Pτ_k1f = q X k2,k1=1 pk2pk1Pτ_k2◦τ_k1f. Therefore, P_TKf = q X kK,··· ,k1=1 pkK · · · pk1Pτ_kK◦···τ_k1f ≡ X ~ k p~_kPT~_kf. (3.7)

We use some inequalities in Proposition 2.16 to prove this proposition. The symbol ‘≤(a)_{’ is referred to the inequality in Proposition 2.16 (a) and so on. By Equation}

(3.7) and for each i = 1, · · · , n,

VIn_,i P_TKf (x ) = V_In_,i   X ~_k p~_kPT~kf (x )  ≤(b) X ~_k p~_kVIn_,i P_T ~ kf (x ) = X ~ k p~_kVIn_,i   X T~_k(y )=x f (y ) |JT~_k|(y )  , by Lemma 2.10 (b), ≤(b) m X j=1 X ~_k p~_kVIn_,i f (T−1 j,~kx ) |JT_j,~_k|(T_j,~−1_kx ) · χT_j,~_k(Dj)(x ) ! = m X j=1 X ~_k p~_kVIn_,i f (T−1 j,~kx ) · |J −1 T_j,~_k|(T −1 j,~kx ) · χTj,~k(Dj)(x ) = m X j=1 X ~_k p~_kVIn_,i f |J_T−1 j,~k| ◦ T −1 j,~k(x ) · χTj,~k(Dj)(x ) .

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Thus, by Equation (3.1) and the above inequality, X ~ k p~_kk(ϕ 0 i,~k) −1_k ∞ = q X kK,kK−1,··· ,k1=1 pkKpkK−1· · · pk1k(ϕ 0 i,~k) −1_k ∞ ≤ q X kK,kK−1,··· ,k1=1 pkKpkK−1· · · pk1k(ϕ 0 i,kK) −1_k ∞· · · k(ϕ 0 i,k1) −1_k ∞ = q X kK=1 pkKk(ϕ 0 i,kK) −1_k ∞ ! · · · q X k1=1 pk1k(ϕ 0 i,kK) −1_k ∞ ! . By Equation (3.5), we get P ~ kp~kk(ϕ 0 i,~k) −1_k ∞≤ αK. Hence, by Proposition 2.27 (b) A = 2 m X j=1 X ~_k p~_kVT_j,~_k(Dj),i f |J_T−1 j,~k| ◦ T −1 j,~k ≤ 2 m X j=1 X ~_k p~_k k(ϕ0_i,~_k)−1k∞ VDj,i(f |Dj) + 2Γ m X j=1 X ~ k p~_k Z Dj |f |dmn ≤ 2αK m X j=1 VDj,i(f |Dj) + 2Γkf k1 ≤ (f ) 2αKVIn_,if + 2Γkf k₁.

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Therefore, VIn_,iP_TKf ≤ A + 2 lkf k1 ≤ 2α K_V In_,if + 2(Γ + 1 l)kf k1 ≡ 2αK_V In_,if + γ_Kkf k₁ ≤ 2αK_V Inf + γ_Kkf k₁. Thus, VInP_TKf ≤ 2αKV_Inf + γ_Kkf k₁.

Remark 3.7. In Proposition 3.6, constants Γ depends on T~_k, and l depends on the

partition PK_{. Thus, γ}

K = 2Γ + 2_l is dependent on the fixed number K. All of these

constants are independent of any f in L1(In).

By Equation (3.6), we have the following lemma immediately.

Lemma 3.8. Let K ∈ N be a fixed constant such that 0 ≤ 2αK _{< 1. For each positive}

integer N , assume N = aK + b for a, b ∈ N and 0 ≤ b < K. Define β = 2K1α < 1.

There exists a constant γo = γo(K) which is independent of N such that we have the

following inequality

VInP_TNf ≤ 2βNV_Inf + γ_okf k₁. (3.8)

Note this inequality is useful when f ∈ BV . Proof. By Equation (3.6), for any constant N ∈ N,

VInP_TNf ≤ 2αNV_Inf + γ_Nkf k₁.

Let γM = max{γ1, γ2· · · , γK}. Then for all b = 1, 2, · · · , K − 1,

VInP_Tbf ≤ 2αbV_Inf + γ_Mkf k₁.

Therefore, for each N = aK + b, VInP_TNf = V_InP_TaK+bf ≤ 2αK_V In(P_T(a−1)K+bf ) + γ_Mkf k₁ .. . ≤ (2αK₎a_V In(P_Tbf ) + (2αK)a−1+ · · · + 2αK + 1 γ_Mkf k₁ ≤ (2αK₎a _2αb_V Inf + γ_Mkf k₁ + (2αK)a−1+ · · · + 1 γ_Mkf k₁ ≤ (2αK₎a_{· (2α}b_)V Inf + γM 1 − 2αKkf k1.

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Let γo= _1−2αγMK and β = 2 1 Kα. Then, βb = 2 b Kαb ≥ αb and βK = 2αK. Therefore, VInP_TNf ≤ (2αK)a· (2αb)V_Inf + γM 1 − 2αKkf k1 ≤ (βK)a· (2βb)VInf + γ_okf k₁ = 2βNVInf + γ_okf k₁.

Thus, both γo and β are independent of N . For big enough N , 2βN < 1 and

VInP_TN ≤ 2βNV_Inf + γ_okf k₁.

3.2 Existence of Absolutely Continuous

Invariant Measure

In this section, we present the main result on Theorem 3.12, which gives the existence of absolutely continuous invariant measure (ACIM). First, we state the mean ergodic theorem and a useful corollary in Dunford and Schwartz [14, VIII 5.1 and 5.3]. Theorem 3.9. [14] (mean ergodic theorem)

Suppose for every integer N ≥ 1, AN = _N1

PN −1

s=0 Ps, where AN is the average of

the iterates of the bounded operator P on the Banach space X. If the sequence of operators {AN} is uniformly bounded, then the set of those points x in X for which the

sequence {ANx}∞N =1 is convergent is a closed linear subspace consisting of all vectors

x for which the set {ANx} is weakly sequentially compact and lim N →∞

1 NP

N_{x = 0.}

Note that a set B is called fundamental, in [14], if the closure of the subspace spanned by B is equal to the whole space X.

Corollary 3.10. [14] Assume AN is defined as above. If the sequence {AN} is

uni-formly bounded, then it converges in the strong operator topology if and only if the sequence {_N1PN_{x} converges to zero for every x in a fundamental set and the sequence}

{ANx} is weakly sequentially compact for every x in a fundamental set.

We intend to apply the mean ergodic theorem and Corollary 3.10 to P = PT in

order to conclude convergence of the sequence {_N1 PN −1

s=0 PTsf }.

Proposition 3.11. Let T = {τk; pk}k=1,··· ,q be a random dynamical system satisfying

Equestion (3.5) which is constructed from piecewise C2 _{Jab lo´}_{nski transformations.}

Let PT be the Perron-Frobenius operator with respect to T . Then, for each f ∈ L1(In),

lim N →∞ 1 N N −1 X s=0 P_Tsf exists in L1_(In_).

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Proof. By the weak Lasota-Yorke inequality, in Lemma 3.8, there exist constants 0 < β < 1 and γo < ∞ such that for all N ∈ N and for all f ∈ BV (In),

VInP_TNf ≤ 2βNV_Inf + γ_okf k₁.

Let AN = _N1 N −1

P

s=0

P_Ts act on L1(In). Then for each f ∈ BV (In),

kANf k1 ≤ 1 N N −1 X s=0 kP_Tsf k1 ≤ kf k1 and VIn(A_Nf ) ≤ 1 N N −1 X s=0 (2βsVInf ) + γ_okf k₁ ≤ 2 N (1 − β)VInf + γokf k1. Therefore, there exist constants L, M < ∞ such that

{ANf } ⊂ {g ∈ L1(In) : kgk1 ≤ kf k1 = L, VIng ≤ M }

which is weakly sequentially compact by Lemma 2.26. We also have 1 NP N T f 1 = 1 NkP N T f k1 ≤ 1 Nkf k1 −→ 0.

Since BV ⊂ L1 _{is a dense linear subspace, BV is a fundamental subset in L}1_{. Thus,}

applying Corollary 3.10 to X = L1(In) and P = PT, we have that AN converges in

the strong operator topology on L1_(In_{), that is, for every f in L}1_(In_),

f∗ = lim

N →∞ANf

exists, where the limit is taken in L1_{- norm.}

Now we present the important theorem, which guarantees that PT has at least one

invariant density f∗. In other words, the result of the following theorem implies the existence of an absolutely continuous invariant measure, ACIM, (see Lasota and York [30]) for a random dynamical system T . Theorem 3.12 was established by Boyarsky and Lou [10], and it was proved under the condition of α < 1₃ in Equation (3.5), instead of α < 1. In this article we prove Theorem 3.12 without assuming α < 1₃ and prove it by using the weak Lasota-Yorke inequality, described in Proposition 3.6.

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Theorem 3.12. Let T = {τk; pk}k=1,··· ,q be a random dynamical system satisfying

Equestion (3.5) which is constructed from piecewise C2 _{Jab lo´}_{nski transformations.}

Then for each L1 function f ,

(a) there exists f∗ ∈ L1_{, lim} N →∞ 1 N N −1 P s=0 P_Tsf = f∗. (b) PTf∗ = f∗.

(c) If f also satisfies f ∈ BV (In_{), then f}∗ _{∈ BV (I}n_).

Proof. (a) Follows from Proposition 3.11.

(b) Given > 0, by (a), there exists No ∈ N such that for all N ≥ No,

1 N N −1 X s=0 P_Tsf − f∗ 1 < . kPTf∗− f∗k1 ≤ PTf∗− 1 N N −1 X s=0 P_Ts+1f 1 + 1 N N −1 X s=0 P_Ts+1f − f∗ 1 ≤ f∗− 1 N N −1 X s=0 P_Tsf 1 + 1 N N −1 X s=0 P_Tsf − f∗+P N T f − f N 1 ≤ 2 + 1 NkP N T f − f k1 ≤ 2 + 2 Nkf k1 −→ 0. That is, PTf∗ = f∗.

(c) By the calculation in the proof of Proposition 3.11, we know that if f ∈ BV (In), then there exist constants L, M < ∞ such that

{ANf } ⊂ {g ∈ L1(In) : kgk1 ≤ kf k1 = L, VIng ≤ M } ≡ D.

For D is a weakly sequentially compact subset of L1(In), and the L1- limit of ANf must be in D. So does f∗ in D ⊂ BV (In).

Ergodic theory of mulitidimensional random dynamical systems

Random Dynamical Systems

Li-Yu Shelley Hsieh

MASTER OF SCIENCE

Ergodic Theory of Multidimensional

Random Dynamical Systems

Li-Yu Shelley Hsieh

Supervisory Committee

Abstract

Table of Contents

List of Figures

Acknowledgements

Introduction

Preliminaries

2.1

Koopman and Perron-Frobenius Operators

2.2

Tonelli Variation on R

2.3

Generalized Bounded Variation on R

2.4

Piecewise C

Jab lo´

nski Transformation

Random Dynamical System

3.1

Weak Lasota-Yorke Inequality

3.2

Existence of Absolutely Continuous

Invariant Measure

_{Tonelli Variation on R}

_{Generalized Bounded Variation on R}