Renewal theory for operators in Banach spaces · the classical renewal theory and Banach spaces. A...

Renewal theory for operators in Banach spaces

Diplomarbeit im Fachbereich Mathematik

der Universität Bremen

vorgelegt von:

Johannes Kautzsch

im Februar 2011

Gutachter:

Prof. Dr. M. Keßeböhmer

Prof. Dr. B. O. Stratmann

CONTENTS CONTENTS

Contents

1 Introduction 1

2 The prime examples 32.1 The Farey and the Gauß system . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The α-Farey and the α-Lüroth system . . . . . . . . . . . . . . . . . . . . . 5

3 The toolbox 73.1 Basic facts about dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Jumping versus inducing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Return, passage and entry times . . . . . . . . . . . . . . . . . . . . 113.2.2 The basic concept of jumping and inducing . . . . . . . . . . . . . . 133.2.3 General induced and jump map . . . . . . . . . . . . . . . . . . . . . 183.2.4 Sum level sets revisited . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 The transfer operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.1 Defining the transfer operator . . . . . . . . . . . . . . . . . . . . . . 213.3.2 The Perron Frobenius operator . . . . . . . . . . . . . . . . . . . . . 223.3.3 The transfer operator and invariance . . . . . . . . . . . . . . . . . . 263.3.4 The Koopman operator . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.5 Explicit examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Renewal theory 304.1 Basic Relations of Renewal Theory . . . . . . . . . . . . . . . . . . . . . . . 30

4.1.1 The basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.2 A real world example . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1.3 Classical renewal theory in the α-Farey system . . . . . . . . . . . . 32

4.2 Renewal theory for the induced Farey system . . . . . . . . . . . . . . . . . 334.3 The delayed case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 Back to the α-Farey map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Further discussion 415.1 Delayed* sum level sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Back to ergodic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

References 49

I

“Lerne Denken und keine Gedanken.”

- Wolfgang Dietz

1 INTRODUCTION

1 Introduction

“Renewal theory arose from the study of ‘self renewing aggregates’, but more recentlyhas developed into the investigation of some general results in the theory of probabilityconnected with sums of independent non-negative random variables. These results areapplicable to quite a wide range of practical probability problems.” (Cox67, Preface)This application of the renewal theoretical idea is one facet of this thesis. The area of itis the notion of dynamical systems and ergodic theory. The name “ergodic” comes fromthe two greek words for work, “ergon” and way, “odos”. This theory began its developmentin the 1930’s with the first ergodic theorems by J. von Neumann and G.D. Birkhoff (cf.section 3.1). They state that the average over time is under certain circumstances equalto the average over space. The language for ergodic theory is given by measure theory.

In order to conceptualize a theory, it is necessary to have examples to elucidate thesituation. Through this thesis the leitmotif is built by two measure theoretical dynamicalsystems introduced in chapter 2, namely the Farey System and the α-Farey system with itsappropriate jump transformations, the Gauß and the α-Lüroth map. The definition of adynamical system is given in section 3.1. The term “jump transformation” is introduced insection 3.2. The problems discussed will be, whenever possible, held as general as possible.For a better understanding the theory will be exemplified with the help of these two, orrather four systems.

After introducing these two systems in chapter 2, the tools used in this thesis will bediscussed. This is done by first giving some useful facts and definitions from ergodic the-ory and the theory of dynamical systems. These basics are covered in section 3.1. In thissection the term “system” is put into a mathematical frame. Other tools for the later dis-cussion are the two concepts of “inducing” and “jumping”, established in section 3.2. Thesetwo concepts are kind of a time lapse to focus on the interesting parts during the long timebehavior of a system. With the introduction to these two concepts more terminology, like“sweep out sets” or “sum level sets” are established.One of the main tools used in this thesis is the transfer operator. A comprehensive ac-count of this operator is given in section 3.3. This transfer operator builds the link betweenthe classical renewal theory and Banach spaces. A Banach space is “a normed linear spacewhich is complete in its norm topology” (DS58, II.3.1, definition 2). The prime examples forinfinite dimensional Banach spaces are the spaces of integrable functions, Lp, 1 ≤ p ≤ ∞.This thesis will work with the two Banach spaces L1 and L∞ in particular. A standardreference for a more thorough explanation of the theory of linear operators on Banach orHilbert spaces is (DS58) and its sequels.

In fact, renewal theory is applicable to a wider range of problems than it was originallyintended. The problems addressed in this thesis come from the topics dynamical systemsand ergodic theory, but are solved with the help of renewal theory. One of the main ques-tions in this thesis is to determine the limiting behavior of the so-called sum level sets (cf.definition 14). Keßeböhmer, Munday and Stratmann proved a statement about this limit-ing behavior in the α-Farey case (cf. (KMS11)) with the help of Feller’s renewal theorem(cf. (EFP49) and (Fel68)). The relation valid in this case unfortunately does not hold inthe Farey case. This is the moment, when the title of the thesis is justified. The renewal

1

1 INTRODUCTION

theory will be lifted one level higher to operators acting on the Banach Spaces L1 andL∞, in which similar relations can be found. After all the needed tools are prepared, thisthesis gives a short introduction to the classical renewal theory before it gives an operatortheoretical renewal statement of Sarig (cf. (Sar02) and section 4.2). This statement isreproven in section 4.2. The next section 4.3 broadens this approach to something similarto the delayed recurrent event in the classical renewal theory.

For rounding off of this thesis, the last chapter, chapter 5, applies wider theory to obtainfurther results for the sum level sets. Finally, this thesis comes back to the infinite ergodictheory and has a closer look at a property of a dynamical system which is to possess a“uniformly returning set”. This property was first introduced by Keßeböhmer and Slassi in(KS07). Once more, the two example systems are consulted to illustrate the matter, andagain, a renewal theoretical argument will be the key to show that the α-Farey system hasthe uniformly returning property. In the end it will be shown, for the α-Farey system aswell as for the Farey system, that if one has the uniformly returning property on a certainsubset of the space, one can suggest uniformly returning for almost all of the space.

2

2 THE PRIME EXAMPLES

2 The prime examples

As mentioned in the introduction, it is always good to have examples in mind in orderto elucidate a theory. This thesis is referring to two main examples, each consisting oftwo somehow related systems. In this section only the maps are given, the term “system”implies more than just a map. The complete definition of a dynamical system will be givenin section 3.1. Though it will be spoken of “systems” here to be in accordance with thelanguage used later in this thesis.One example is represented by the two systems, given by the Farey and the Gauß map.The α-Farey and the α-Lüroth map constitute the second main example. The latter twosystems are kind of a linear correspondent to the first two systems. To understand whatis meant by that, the reader is referred the definitions in the two subsequent subsectionsand especially to figures 1 and 2. While the Gauß system has a longer known numbertheoretical pertinence, some other connections are, due to linearity, sometimes easier tofind in the α-Lüroth case.

2.1 The Farey and the Gauß system

The Farey map is given by

F (x) :=

x

1−xif x ∈ [0, 1

2 ];

1−x

xif x ∈ (1

2 , 1] := A1.(1)

The left branch of this map shall be denoted by F0(x) := x

1−xand the right one by F1(x) :=

1−x

x. Please note for later use, the inverse branches of F are given by F−1

0 (x) = x/(1 + x)and F−1

1 (x) = 1/(1 + x). The name A1 for the set (12 , 1] will be justified later. The Gauß

map is given by

G(x) :=

1x−

�1x

�if x ∈ (0, 1];

0 if x = 0.(2)

Here and throughout this thesis �x� denotes the integer part of x ∈ R, i.e. �x� := max{n ∈N ∪ {0}|n ≤ x}. Later it will be dwelled on the links between the Farey and the Gaußmap in greater detail. As it is well known, the Gauß system has a number theoreticalpertinence. A standard and basic reference for this is (Khi56). Every number x ∈ (0, 1]has a representation as a continued fraction. That means x can be described by

x =1

b1(x) + 1b2(x)+ 1

...

,

where all the bn’s are positive integers and for all n ∈ N given by

bn(x) =�

1Gn−1x

�.

Example. The continued fraction expansion of the golden ratios decimal fraction is [1, 1, 1, ...]G.This can be seen since

√5−12 is a fixed point of the Gauß map G.

3

2.1 The Farey and the Gauß system 2 THE PRIME EXAMPLES

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(a) The Farey map

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(b) The Gauß map

Figure 1: The Farey and the Gauß map

Note that if a number x ∈ (0, 1] is irrational, it has an infinite continued fraction ex-pansion. For all x ∈ I := [0, 1]\Q this continued fraction expansion is unique. That meansthere is a bijection from I to the space of infinite sequences over the infinite alphabet N,x can be written as πG(x) = [b1, b2, b3, ...]G.Finite sequences represent the rational numbers in (0, 1]. All numbers in (0, 1] having thesame k-first entries in their continued fraction expansion are a subinterval of the unit inter-val. Hence, as the rationals build a generator for the Borel σ-algebra, the finite sequencesdo as well for the σ-algebra in NN. This σ-algebra corresponds to the Borel one in (0, 1].In the space NN, the Gauß transformation corresponds to the left shift σ, given byσ([b1, b2, ...]G) = [b2, b3, ...]G. One can see that the following diagram commutes, wherethe commuting homeomorphism is πG : I −→ NN, with πG(x) = [b1, b2, ...]G. Here, itshall be noted that this resembles a situation which is introduced later and is called theconjugation of two dynamical systems, see definition 5.

IG ��

πG

��

I

πG

��

#

NNσ �� NN

Excluding the rationals of the unit interval is the same as excluding all the iteratedpreimages of 0 under the Farey map F . That this indeed is reasonable will be seen insection 3.2.

The introduction of the Farey and the Gauß system shall be concluded by defining thecylinder sets of the continued fraction expansion. First define the harmonic partition of(0, 1] by αH := (An;n ∈ N) with An := (1/(n + 1), 1/n]. As A1, the names An will get a

4

2.2 The α-Farey and the α-Lüroth system 2 THE PRIME EXAMPLES

deeper sense later.

Definition 1. Cylinder sets of the continued fraction expansion. The cylinder setsof the continued fraction expansion are defined by

CG(b1, b2, ..., bk) := {x ∈ (0, 1] : Gi−1(x) ∈ Abi , for all i = 1, ..., k}.

2.2 The α-Farey and the α-Lüroth system

The α-Lüroth and the α-Farey system somehow the linear correspondent of the previouslyintroduced systems, the Farey and the Gauß system. This is visualized in the two figures2 and 1. For another comprehensive account of those two systems the reader is referredto (KMS11). Let X = [0, 1] be equipped with the Borel σ-algebra B. Moreover α = {An :n ∈ N} shall be a countable partition of the unit interval. Here, the partition elementsare left open and right closed intervals and ordered from right to left, starting with A1.In addition the Ai’s only accumulate at the origin 0. The Lebesgue measure λ(Ai) of theAtom Ai ∈ α is denoted by ai. ti describes the Lebesgue measure of the i-th tail of α, i.e.ti =

�∞k=i

ak. Now the α-Farey map is given by

F(α)(x) =

0 if x = 0;

an−1(x−tn+1)an

+ tn if x ∈ An , n ≥ 2;

1−x

a1if x ∈ A1.

(3)

Analogous to the Farey system, the left branch of the α-Farey map is denoted by Fα,0(x) :=an−1(x − tn+1)/an + tn, x ∈ An, n ≥ 2. The right branch is called Fα,1 := (1 − x)/a1,x ∈ A1. For later purpose the inverse branches of the α-Farey map shall be mentioned.They are given by

F−1α,0(x) := an+1

an(x− tn+1) + tn+2 for x ∈ An, n ∈ N;

F−1α,1(x) := (1− a1x) for x ∈ [0, 1].

By convention, F−1α,0(0) := 0.

The α-Farey transformation maps each interval of the partition, but A1, linearly to itsright neighbor. A1 is mapped to all of X.The counterpart of the Gauß map is the α-Lüroth map given by

L(α)(x) =

(tn−x)an

if x ∈ An and n ∈ N;

0 if x = 0.

(4)

Similar to the continued fraction expansion of a number there exists a representation ofnumbers in [0, 1] relying on the α-Lüroth system. Every number x ∈ (0, 1] has a so-calledα-Lüroth expansion. It is given by πL(x) = [l1, l2, ...]α, with li ∈ N for i ∈ N. The li’s aredetermined by

li = k ⇔ Li−1(α) (x) ∈ Ak

The series (li) is finite if and only if Li−1(α) (x) = 0, for some i ∈ N. In this case x can be

represented in two different ways, namely x = [l1, l2, ..., (lk + 1)]α = [l1, l2, ..., lk, 1]α. If

5

2.2 The α-Farey and the α-Lüroth system 2 THE PRIME EXAMPLES

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(a) the α-Farey map

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(b) the α-Lüroth map

Figure 2: the α-Fary and the α-Lüroth map w.r.t. the harmonic partition.

(li)i∈N is an infinite sequence the α-Lüroth expansion is unique for x.If [l1, l2, ...]α is given one can determine the decimal notation of x by the formula (cf.(KMS11, p. 6))

x =∞�

n=1

(−1)n−1

��

i<n

ali

�= tl1 − al1tl2 + al1al2tl3 + ...

Example. If one considers the harmonic partition, i.e. An = ( 1n+1 , 1

n], then 2

3 = [1, 1, 1, ...]α.As in the Gauß system one can define cylinder sets for the α-Lüroth expansion.

Definition 2. Cylinder sets of the α-Lüroth expansion. The cylinder sets for theα-Lüroth expansion are given by:

C(α)(l1, l2, ..., lk) := {x ∈ (0, 1] : Li−1(α) (x) ∈ Ali , for all i = 1, ..., k}.

For later purpose λ(C(α)(l1, l2, ..., lk)) =k�

i=1ali should be mentioned here. For a deeper

introduction to these systems, similar number theoretical systems and the ergodic theorybehind it the reader is referred to (DK02).

6

3 THE TOOLBOX

3 The toolbox

For a better understanding of the later theory, this section gives an introduction to themain tools used in this thesis.

3.1 Basic facts about dynamics

First of all some nomenclature, definitions and useful facts from the field of dynamical sys-tems and ergodic theory shall be introduced. A standard reference for a deeper introductionto infinite ergodic theory is (Aar97). Also (Wal82) and (Den05) give an introduction todynamical systems and ergodic theory.

Definition 3. Dynamical system. Let X be a nonempty set and G be a semigroupwith a neutral element e. The tuple (X,G) is called a dynamical system if there is anassociative map ϕ such that

ϕ : G×X −→ X

ϕ(g, x) �−→ gx.

Note ϕ(e, x) = x. One says G acts on X via ϕ. Often a transformation T : X −→ X is givenand G = N. The action of G on X is then given by the iteration of T . ϕ(n, x) = Tn(x),with ϕ(0, x) = T 0(x) = x. In this case one writes (X,T ) for the dynamical system.Often X has some additional structure like a topology, T, or a σ-algebra, B. With thisstructure one can postulate continuity or measurability of T for example. (X,T,T) or(X,T,B) is then called a continuous respectively a measurable dynamical system. More-over a measurable dynamical system can be equipped with a measure µ. In this case thetuple (X,T,B, µ) is called a measure theoretical dynamical system. These measure theo-retical dynamical systems shall be of interest in this thesis.

Having defined dynamical systems one is lead to ask how two dynamical systems canbe compared. This leads to the next definition. (cf. (Den05), definition 65.)

Definition 4. Factor of dynamical systems. Let (Xi, Ti,Bi, µi), i = 1, 2 be two mea-sure theoretical dynamical systems. One calls (X1, T,B1, µ1) a factor of (X2, T2,B2, µ) ifthere exists a measurable surjection, i.e. an almost surely defined surjective measurablemap π : X1 −→ X2 such that π ◦ T1 = T2 ◦ π.

More than a factor is the conjugation of two dynamical systems.

Definition 5. Conjugation of dynamical systems. Two measure theoretical dynami-cal systems (Xi, Ti,Bi, µi), i = 1, 2 are said to be isomorphic, if there exists a measurablebijection, i.e. an almost sure defined bijective bimeasurable map π : X1 −→ X2 such thatµ1 ◦ π−1 = µ2 and π ◦ T1 = T2 ◦ π. That is to say the following diagram commutes.

(X1,B1, µ1)T1 ��

π

��

(X1,B1, µ1)

π

��

#

(X2,B2, µ2)T2

�� (X2,B2, µ2).

7

3.1 Basic facts about dynamics 3 THE TOOLBOX

When working with dynamical systems some classification of the systems is needed. Acommon starting point for this is to look at the properties of T .

Definition 6. Properties of T. Let (X,T,B, µ) be a measure theoretical dynamicalsystem. T is said to be

• absolutely continuous if and only if for all B ∈ B

µ(B) = 0 ⇒ µ(T−1B) = 0;

• non singular if and only if for all B ∈ B

µ(B) = 0 ⇔ µ(T−1B) = 0;

• measure preserving or µ is called T -invariant if and only if for all B ∈ B

µ(B) = µ(T−1B);

• ergodic if and only if

∀B ∈ B : T−1B◦= B µ− a.s. ⇒ µ(B)µ(X \ B) = 0.

Remark. For a measure µ, ◦= should here and throughout this thesis denote the almostsure equality of two sets. That means A

◦= B ⇔ µ ((A \ B) ∪ (B \ A)) = 0. If it is obviousto which measure one refers to, the µ-a.s. behind the ◦=-sign will be omitted.Example. The Farey map is measure preserving with respect to the measure µ = 1/x · λ.Whereas the α-Farey invariant measure, µα is determined by the density hα with respectto λ which is given by h(α) := dµα

dλ=

�∞n=1

tnan

· 1An . (cf. (KMS11), Lemma 2.6). Theα-Lüroth map preserves the Lebesgue measure and the invariant Lebesgue density for theGaußtransformation is g(x) = 1

lg 21

1+x.

Remark. lg(·) shall here and throughout this thesis denote the natural logarithm, i.e.lg(·) = loga(·)/ loga(e), for any a > 0.

Here only the proof for the Farey system is given. The calculation for the α-Farey andthe Gauß case is postponed here. It will be done later, with another neat way to showinvariance, see sections 3.3.3 and 3.3.5. The statement, that the α-Lüroth map is Lebesguemeasure preserving is easy to show an will be left to the reader.

Since the open intervals in [0, 1] are a generator for the Borel σ Algebra, it sufficesto show the statement for an arbitrary interval A = (a, b) , such that A ⊂ (0, 1]. WithF−1(A) = F−1

0 (A) � F−11 (A) = ( 1

1+b, 1

1+a) ∪ ( a

1+a, b

1+b) one calculates

µ(F−1(A)) =�

F−1(A)

1x

dλ

= −� 1

1+b

11+a

1x

dλ +� b

1+b

a1+a

1x

dλ

= −(− ln(1 + b) + ln(1 + a)) + ln(b)− ln(1 + b)− (ln(a)− ln(1 + a))= ln(b)− ln(a)

=�

A

1x

dλ = µ(A).

8


Looking at the long time behavior of sets in a dynamical systems one might ask whichsets return to themselves and which do not. This leads to the next definition.

Definition 7. Wandering sets. Let (X,T ) be a dynamical system. A set W is called awandering set, or wandering, if {T−nW : n ∈ N} consists of pairwise disjoint sets.

Remark. The union of two wandering sets is not necessary wandering. Look the union ofthe two wandering sets W and T−1W , for any wandering set W , for example.

Definition 8. Dissipative, conservative and Hopf decomposition. Let (X,T,B, µ)be a measure theoretical dynamical system. The dissipative part of the system (X,T,B, µ),D(T ), is the measurable union of the wandering sets. (D(T )c) = C(T ) is called the conser-vative part.The measurable union means, that D(T ) is a measurable set and every wandering set isalmost surely part of D(T ).X

◦= D(T )◦� C(T ) is called the Hopf decomposition of a Dynamical System. The sign

“A◦� B” denotes that the intersection of A and B is of measure zero.

A System, or T , is called dissipative repspectively conservative, if D(T ) ◦= X, respectivelyC(T ) ◦= X.

Taking a set E ⊂ C(T ) with µ(E) > 0 implies that almost every x ∈ E returns infinitelyoften to E under the iteration of T , Tn(x), n ∈ N.Maharam’s recurrence theorem gives an easy argument that the two main examples areboth conservative dynamical systems. It tells us (cf. (Aar97), Theorem 1.1.7).

Theorem 1. Maharam’s recurrence theorem Let (X,B, T, µ) be a measure theoreticaldynamical system. May µ be a σ-finite and T -invariant measure. T is conservative, ifthere exists an A ∈ B, with µ(A) < ∞, such that

X◦=∞�

n

T−nA.

For the proof see (Aar97).Remark. Those sets A are called sweep out sets, see definition 10.

With the Hopf decomposition it is now possible to analyze the conservative part and thedissipative part of a dynamical system separately. The behavior of conservative systemshas been studied a lot, whence for dissipative systems less questions are answered.In a conservative system one is interested in questions like how much time does one spentat a certain set A. How often is a set entered during a long, maybe infinite, period of time?Which ratio of the time, known as mean sojourn time, does the process spend at a set A?This leads to the definition of ergodic sums.

Definition 9. The ergodic sum. Let (X,T,B, µ) be a dynamical system with a σ-finitemeasure µ. For a measurable function f the ergodic sum Snf is given by

Snf :=n−1�

k=0

f ◦ T k.

9


For a set A ∈ {B ∈ B : 0 < µ(B) < ∞} =: B+ of finite positive measure one defines thesojourn time Sn(A) and writes

SnA := Sn1A :=n−1�

k=0

1A ◦ T k.

Remark. Note, that for a conservative ergodic measure preserving transformation T onehas for A ∈ B+ or f ∈ L1+

µ ,

limn→∞

SnA = ∞ resp. limn→∞

Snf = ∞ a.e. on X.

In the case one is working on a finite measure space X with µ(X) the famous Birkhoffergodic theorem gives a good description of the average behavior of Snf , resp. SnA. Here,the version used by Zweimüller (cf. (Zwe09), Theorem 1) shall be adopted.Theorem 2. Birkhoff ergodic theorem. Suppose (X,T,B, µ) is a measure theoreticaldynamical system with a measure preserving transformation T . Let f ∈ L1+

µ . If µ(X) < ∞,

limn→∞

1n

Snf =�

fdµ

µ(X)a.e. on X.

If µ(X) = ∞,

limn→∞

1n

Snf = 0 a.e. on X.

The proof is omitted here, see e.g. (Wal82) or, for the finite measure case, (Den05). Inthe infinite case one is also interested in the question how fast the ergodic average 1/n Snftends to zero. Is there maybe a sequence an which describes the behavior better than1/n? Unfortunately Snf is either underestimated or overestimated, as Aaronsons ergodictheorem states. ((Aar97, theorem 2.4.2)).Theorem 3. Aaronsons ergodic theorem. Suppose that T is a conservative ergodicmeasure preserving transformation of the σ-finite, infinite measure space (X,B, µ) and letan > 0, n ≥ 1. Then either

lim infn→∞

Snf

an

= 0 a.e. on X, ∀f ∈ L1+µ or

lim supn→∞

Snf

an

= ∞ a.e. on X, ∀f ∈ L1+µ .

A way out of this dilemma can be found via Hopfs ratio ergodic theorem. (cf.(Hop37))Here a version used by Zweimüller shall be cited (cf. (Zwe04)).Theorem 4. Hopfs ratio ergodic theorem. Let (X,T,B, µ) be a conservative measurepreserving dynamical system. Let µ be a σ-finite measure. Furthermore let f , g ∈ L1

µ, withg ≥ 0 and

�gdµ > 0. Then there exists a function Q(f, g) : X −→ R, such that

limn→∞

Snf

Sng= Q(f, g). a.e. on

�sup

n

Sng > 0�

.

If T is ergodic, one has Q(f, g) =R

X fdµRX gdµ

almost everywhere on X.

The proof of Hopfs ratio ergodic theorem is omitted here as well. There are differentways to proof this theorem, e.g. (Hop37) or (KK97). Another way to prove it was presentedby Zweimüller (Zwe04). The idea of this proof is very adroit. For his proof Zweimülleruses a technique which tries to view the infinite Space X through finite measure glasses.This technique is known as inducing. This is what the next subsection is about.

10

3.2 Jumping versus inducing 3 THE TOOLBOX

3.2 Jumping versus inducing

3.2.1 Return, passage and entry times

Let (X,T,A, µ) be a conservative, ergodic and measure preserving dynamical system. µshall be an infinite but σ-finite measure. As discovered in the previous subsection, in aconservative System one will almost surely return to a set E with µ(E) > 0 infinitely often.It is in deed a very interesting question, when and whether one will return to a certainset. Talking about renewal, this is what will be done later in this thesis, is linked to thatquestion. Before coming to the crucial definitions of the first return, the first entry and thefirst passage time, one should spent some time with thinking about which, in some sense“nice” or “good”, property a set E shall possess. Here the notion of a “sweep out set”, usedby R. Zweimüller (cf.(Zwe09)), shall be adopted.

Definition 10. Sweep out set. A measurable set A ∈ A is called a sweep out set, if�

n≥0

T−nA◦= X µ− a.s.

S. Isola (cf. (Iso09)) calls such sets “good sets”.

In this context a parameter is often used. It is known as the wandering rate of a set.

Definition 11. The wandering rate. Let (X,T,A, µ) denote a conservative, ergodicmeasure preserving dynamical system with the infinite, σ-finite measure µ. For a setA ∈ A, with 0 < µ(A) < ∞, the wandering wn(A) rate is given by

wn(A) = µ

�n�

k=0

T−kA

�.

Remark. The wandering rate of the α-Farey and the Farey system is slowly varying. Ameasurable function f : R+ → R+ is said to be slowly varying, if f(ξx)/f(x) −→ 1, forx −→∞ and ξ > 0. In particular, wn+1/wn −→ 1, for n −→∞.

Lemma 1. In a conservative and ergodic measure theoretical dynamical System, with anon singular Transformation T , any measurable set A, such that 0 < µ(A) < ∞, is a sweepout set.

Proof. The statement will be proven by contraposition. Let (X,T,B, µ) be a conservativeand ergodic measure theoretical dynamical system and let A ∈ B be such that 0 < µ(A) <∞. Now assume that A is not a sweep out set and define V := X \

�n≥0 T−nA. By the

assumption µ(V ) > 0. If one has a closer look at the construction of V , it is obvious thatT−1V = X \

�n≥1 T−nA ⊇ V . Induction yields this result for any n ∈ N. V ⊆ T−1V ⊆

11


T−2V ⊆ ... ⊆ T−nV . For further progress one has to construct a sequence of sets Wk.

W0 = V

W1 = T−1V \ T 0V

W2 = T−2V \1�

i=0

T−iV = T−2V \ T−1V

...

Wk = T−kV \k−1�

i=0

Wi = T−kV \k−1�

i=0

T−iV = T−kV \ T−(k−1)V.

Note T−1Wk−1 = Wk. Looking at the measure of the sets Wk only two options can befound, both leading into a contradiction. The first one is that there exists a k ∈ N suchthat µ(Wk) = µ(T−kV \ T−(k−1)V ) = 0. This assumption instantly yields µ almost surelyT−kV ⊆ T−k+1V , hence T−kV

◦= T−k+1V . Ergodicity then either implies µ(T−kV ) = 0,or µ((T−kV )c) = 0. Because V ⊂ T−kV , µ(T−kV ) = 0 obviously exposes V as a set ofmeasure zero, which is contrary to the assumption. But µ((T−kV )c) = 0 gives by an easyargument µ(A) = 0, a contradiction to the premise. Thus one has to conclude that for alln ∈ N, µ(Wn) > 0 holds. This, in connection with the fact that all the Wk’s are pairwisedisjoint, reveals W0 as a wandering set, eventually the final contradiction.

When looking at renewal questions in the dynamics of a system one is interested insomething like after how many steps is a point x mapped into a special set, what leads tothe next two definitions. In definition 12 and 13 (X,T,B, µ) is a conservative and ergodicdynamical system.

Definition 12. First return time. Let E ⊂ X ∈ B be a sweep out set. For a pointx ∈ E the first return time r : E −→ N ∪∞ is defined by

r(x) := min{n ≥ 1 : Tn(x) ∈ E},

where min ∅ is interpreted as ∞.

Remark. With this definition the set E can be decomposed into sets En fixed by theparticular first return times.

En := {x ∈ E : r(x) = n}.

The En’s are known as Level sets of the first return times. Since by assumption the systemwas conservative, one gets

E◦=

�

n∈NEn.

If one does not restrict himself to a starting point in E, a different description is needed.It would be nice to have a parameter which describes the first entry to the certain set. Thisleads to the next definition.

Definition 13. First entry time and first passage time. As before E ⊂ X shalldenote a sweep out set in the sigma algebra B. The first entry time into the set E isdefined by e : X −→ N0,

e(x) := min{k ≥ 0 : T k(x) ∈ E}.

12


Directly linked to the first entry time is the first passage time, p : X −→ N which is givenby

p(x) = 1 + e(x)

= 1 + min{k ≥ 0 : T kx ∈ E}

Example. If one recalls the Farey system and takes E = (1/2, 1], one immediately verifiesthat the first digit of the continued fraction expansion of a number x is exactly its firstpassage time, p(x).

Making use of the definitions above one is able to decompose X, similar to the decom-position of E, into the so-called level sets of the first passage time (cf. (Iso09), page 3).One defines An := {x ∈ X : p(x) = n}. Of course E = A1 and since the system was bypresumption conservative one gets

�

n∈NAn

◦= X µ-a.s.

Remark. The first passage time and the first return time are related via r = p ◦ T . Thisequation is verified by the relation T (En) = T|A1

(En) = An.Moreover for n ∈ N one has the relation F (An+1) = An. Regarding the Farey system

these sets can be determined explicitly. We have An = ( 1n+1 , 1

n]. So it obviously holds

F��

1n+2 , 1

n+1

��= ( 1

n+1 , 1n]. The notion used in subsection 2.1 for the harmonic partition

goes in accordance with this nomenclature for the sets.In the α-Farey case if one takes for E again the previously defined A1, the An’s of theα-Farey partition α are just the level sets of the first passage time. Calling the partitionsets (Ai)i∈N was thus coherent as well.In real life it sometimes does matter that a person is present at a certain place at a certaintime, whereupon it does not matter if the presence at this place is the first one. Thus inthis setting here one often is interested in those sets which arrive in the set E after k steps,no matter if it is the first arrival or not. Roughly speaking the presence in E before epochk is not of interest. With this idea in mind one can define the sum-level sets Ln.

Definition 14. Sum-level-sets The sum level sets for a sweep out set E are given by

Ln := {x ∈ X |Tnx ∈ E}. (5)

With the representation of the cylinder sets (cf. definition2) one can give another, oftenhandier, representation of the sum-level sets. In this representation the name sum-level-setis warranted. Before this thesis is heading towards this definition, the concept of the jumptransformation will be introduced as well as the concept of the induced transformation.These two concepts will be compared afterwards.

3.2.2 The basic concept of jumping and inducing

The concept of inducing relies on the idea that all the steps made in between two arrivalsto one certain set are squeezed together to one step. The jump transformation concepttakes the vantage point from the other side. One starts somewhere in the system, squeezesall steps up to the arrival in the certain set together and keeps then track where one isbrought to after the visit.

13


Definition 15. The jump transformation. For a sweep out set E ∈ B the jumptransformation T ∗ is defined as

T ∗E(x) := T ∗(x) := T p(x)(x).

If it is obvious which set one is talking about, the index E is omitted.

Since the kth level set is the set where p(x) = k, the jump transformation is on Ak

the k times iteration of the transformation itself, T ∗ = T k. The concept of jumping yieldsfurther interesting results. For example gives a straight forward calculation the followinglemma.

Lemma 2. The jump transformation F ∗ to the set E = A1 = (1/2, 1], of the Farey mapF , given by (1) coincides with the Gauß map G, (2).

Proof. Since on A1 the two maps F 1 = F and G are just the same, it will just be looked atthe case n > 1, hence an x ∈ An ⊂ [0, 1] for n ≥ 2 is chosen. In this case the first continuedfraction digit of x is equal to n, what is equivalent to saying G(x) = 1/x−b1(x) = 1/x−n.On the other hand, as mentioned above, the jump transformation is on Ak exact the ktimes iteration of T . Hence if one starts with an x ∈ An, it takes (n − 1) iterations of Fto reach the interval A1. That means one has to apply (n− 1) times F0 and eventually asingle time F1. Hence for x ∈ An,

F ∗(x) = Fn(x) = F1�Fn−1

0 (x)�

= F1

�Fn−2

0

�x

1− x

��

= F1

�Fn−3

0

�x

1− 2x

��

...

= F1

�x

1− (n− 1)x

�

=1− (n− 1)x

x− 1 =

1x− n.

In a similar manner it is shown in (KMS11, lemma 2.1), that the jump transformationto the set A1 of the α-Farey map coincides with the α-Lüroth map.In the inducing concept, contrary to the jump concept, one restricts himself to a certainsubset and takes the return time as iteration parameter. It is used for example by O. Sarig.(cf. (Sar02)) or R. Zweimüller (cf. (Zwe09)).

Definition 16. The induced transformation Let E be a sweep out set. The inducedtransformation T ind

E: E → E is given by

T ind

E (x) := T ind(x) := T (r(x))(x),

where r(x) is given by definition 12. In case the set E is clear from the context the indexis as well omitted.

14


Roughly speaking jumping describes the development of a point x taken somewhere inthe system. It brings the point into the certain set (here A1) and throws it back somewhereinto the system. One could call this procedure “in and out”. While inducing starts insidethe set, throws the point to a place somewhere in the system and waits for the return,in few words “out and in”. Intuitively, one is lead to suspect a link between these twoconcepts. And of course the question arises whether there is and where is that connectionbetween these two vantage points of a somehow similar concept.First one should have a closer look at how the jump transformation and the inducedtransformation look like in the Farey case, see also the two graphs of it (Figure 3). TakeA1 = (1/2, 1] as the sweep out set. As it has already been discovered the jump transfor-mation of the Farey map is the Gauß transformation. The induced transformation is givenby

F ind

A1=

F1(x) = 1x− 1 if x ∈ E1;

F 20 (F1(x)) = 1−x

2x−1 if x ∈ E2;

F 30 (F1(x)) = 1−x

3x−2 if x ∈ E3;

... .

In a nutshell F ind

A1= 1−x

nx−(n−1) , n ∈ N. Before it is time to have a look at the connection

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(a) The jump transformation

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(b) The induced transformation

Figure 3: The jump and the induced transformation of the Farey system w.r.t. (1/2, 1].

between these two concepts, a lemma should be given to determine the measure regardingthe induced system.

Lemma 3. Invariant measure for the induced system. Let (X,T,A, µ) be a measurepreserving, conservative, ergodic dynamical system with an infinite and σ-finite measureµ. Let E be a sweep out set, in particular E shall be of positive and finite measure. Then

15


the measure µE defined for A ∈ A ∩ E by µE(A) := µ|E(A)/µ(E) is a T ind

E-invariant

probability measure.

Remark. One should notice here and throughout this thesis the difference between thenotions µE and µ|E . Whereat µE is the measure in terms of the preceding lemma, µ|E isthe measure µ, but restricted to subsets of E.

Proof. The factor 1/µ(E) is for normalizing the measure to one. Let A ∈ A ∩ E begiven. T ind

E-invariance is now shown by a straight forward calculation using the disjoint

decomposition A = A ∩ (�

nEn).

Lemma 4. Let (X,T,B, µ) be a conservative ergodic measure preserving dynamical system.If A1 is a sweep out set with the property that T|A1

: A1 −→ X is bijective, then the inducedsystem (A1, T ind

A1,B ∩A1, µA1), and the jump system (X,T ∗,B, ν), where ν given by dν :=

dµ|A1◦ T−1

A1is the T ∗-invariant probability measure, are isomorphic. The commutating

homeomorphism is given by T|A1. That means the following diagram commutes almost

surely.

(A1,B ∩A1, µA1)T

ind��

T|A1

��

(A1,B ∩A1, µA1)

T|A1

��

#

(X,B ∩ I, ν)T∗

�� (X,B ∩ I, ν)

Remark. As one can easily see the condition, T|A1: A1 −→ X is bijective, is true for the

Farey system as well as for the α-Farey system, if one takes A1 = (1/2, 1].The sloppy “in and out” formulation above now sounds like “out-in and out again” for

the upper path and “out and then in-out” for the lower one. Of course this is no proof yet,but it gives the feeling of being on a correct route. Now the proof will be given.

Proof. First one has to show T|A1(T ind(x)) = T ∗(T|A1

(x)). Let x ∈ En ⊂ A1. ThenT ind = Tn−1

|X\A1

�T|A1

(x)�. But on the other hand one has as mentioned in a remark above

(page 13) T (En) = T|A1(En) = An. Furthermore on An, T ∗ equals Tn, where just the very

last iteration takes place in A1. That in turn implies T ∗(T|A1(x)) = T|A1

�Tn−1|X\A1

(x)�.

Now letting x = T|A1(x) proves the first assertion.

16


It is left to show that dµ|A1◦ T−1

|A1=: dν is T ∗-invariant. Let A ⊂ X ∈ B

ν(T ∗−1(A)) = µA1

�T−1|A1◦ T ∗−1(A)

�

= µA1

��T ∗ ◦ T|A1

�−1 (A)�

= µA1

��T|A1

◦ T ind

�−1(A)

�

= µA1

��T ind

�−1◦ T−1

|A1(A)

�

= µA1

�T−1|A1

(A)�

= ν(A)

Remark. The nomenclature jump transformation versus induced transformation is a littlespongily in the literature. Stefano Isola calls the transformation here defined as jump trans-formation the induced transformation, for example. This thesis stays with the definitions15 and 16.

Now it is time to look at the connection, stated in lemma 4 in the example of the Fareyand the Gauß system. First the two dynamical systems first shall be defined explicitly.Let A1 = A1 ∩ lim supn F−nA1. A1 is the set of all those points of A1 which returninfinitely often to A1. One should note here that λ(A1) = λ(A1), so, through the almostsurely glasses, everything is fine. As a σ-algebra for this system B ∩ A1 shall be takenand the measure is µA1 , in terms of lemma 3, µA1 = µ|A1

µ(A1) , where µ is the absolutelycontinuous measure with respect to the Lebesgue measure, defined via µ = 1

xλ. Notice,

µ(A1) = 1/ lg 2 The transformation is the induced transformation F ind : A1 −→ A1.The second system is given by the set I, defined in section 2.1, equipped with the Borelsigma algebra restricted to I. The invariant measure is determined by νG = 1

lg 21

1+xλ.

Finally the transformation here is the jump transformation F ∗, which coincides, as provenabove, with the Gauß map G. The factor 1

lg 2 is the normalizing factor, such that µA1

respectively νG are probability measures. Now the a corollary to lemma 4 can be stated.Corollary 1. The two systems (A1, F ind B ∩A1, µA1) and (I, G,B ∩ I, νG) are isomorphic.That means the following diagram commutes. The conjugating homeomorphism is givenby the right branch of the Farey map, F1 : A1 → I.

(A1,B ∩A1, µA1)F

r(x)��

F1

��

(A1,B ∩A1, µA1)

F1

��

#

(I,B ∩ I, νG)G

�� (I,B ∩ I, νG)

Proof. Of course the proof would follow directly from lemma 4, but in this case the proofcan be done by explicit calculation, which will be provided here. First one has to show

17


F1(F r(x)(x)) = G(F1(x)) for x ∈ A1. The thoughts here are exact the same as in theproof of lemma 4. Some basic considerations, mainly mentioned above, are the key. First,one should recall F (En) = An. Second, on An, it holds G(x) = F ∗(x) = Fn(x) =F1

�Fn−1

0 (x)�. Third on En, r(x) equals by definition n and since En ⊂ A1 one has for

n > 1 F r(x)(x) = Fn(x) = Fn−10 (F1(x)). The case n = 1 is trivial. To put it in a nutshell

F1(F r(x)) = F1(Fn−10 (F1(x))) = G(F1(x)).

The other point one has to show is that the two measures µA1 and ν are transferred intoone another by F1. Therefore one has to show

dµA1 ◦ F−11 =

1lg 2

1x + 1

dλ.

In the calculation the transformation formula for integration is used. Let f ∈ L∞µ . Onecalculates

�

X

fdµA1 ◦ F−11 =

1lg 2

�

A1

f ◦ F1dµ

=1

lg 2

�

A1

f ◦ F1(x)�

1x

+ (1− 1)�

dλ

=1

lg 2

�

A1

f ◦ F1(x) (F1(x) + 1) dλ

=1

lg 2

�

F1(A1)f(x)(x + 1)

��F−1�1 (x)

�� dλ

=1

lg 2

�

[0,1]f(x)(x + 1)

��

�1

x + 1

�� dλ

=1

lg 2

�

[0,1]f(x)

x + 1(x + 1)2

dλ.

=�

X

1lg 2

f(x)1

x + 1dλ.

3.2.3 General induced and jump map

In the previous section the case restricted to a bijective T : A1 −→ X was considered,quite a strong postulation. Now it will be broadened to more general subsets of X. Let(X,T,B, µ) still be the conservative, ergodic and measure preserving dynamical system.As well µ shall be an infinite but σ-finite measure. Here just one of the main examples isconsidered, the Farey system. One closely related generalization is to drop the injectivepart of the postulation. What happens if one induces, respectively jumps with respect to abigger subset of X than A1. As an easy example the set E := A1∪A2∪A3 = (1/4, 1] shouldbe taken. Not to get confused it will be adhered to the original names of the partition sets.These were the level sets of the first return time, respectively the first passage time withrespect to (1/2, 1]. The current level sets will be denoted with a “∼” on top. The level sets

18


of the first return time are

E1 = E1 ∪ E2 ∪ E3 ∪A2 ∪A3,

E2 = E4,

E3 = E5,

...

Whereas the level sets of the first passage time are given by A1 = E, A2 = A4, A3 = A5, etc.Some basic calculations yield the exact descriptions of the two maps. To get an impressionhow the maps look like, the reader is referred to figure 4. The induced transformationiterates the transformation until it returns to the starting set. By definition one hasF ind

|E : E −→ E, hence

F ind

E=

F1(x) = F (x) if x ∈ (1/4, 4/5];

F0(F1(x)) = 1−x

2x−1 if x ∈ (4/5, 5/6];

F 20 (F1(x)) = 1−x

3x−2 if x ∈ (5/6, 6/7];

F 30 (F1(x)) = 1−x

4x−3 if x ∈ (6/7, 7/8];

... .

The jump transformation starts somewhere in X, iterates until the starting point is mappedto the set of interest, here E, and performs one iteration more. Hence one gets

F ∗E

=

F1(x) = F (x) if x ∈ (1/4, 1];

F0(F1(x)) = x

1−2xif x ∈ (1/5, 1/4];

F 20 (F1(x)) = x

1−3xif x ∈ (1/6, 1/5];

F 30 (F1(x)) = x

1−4xif x ∈ (1/7, 1/6];

... .

Making the sets more complicated, the maps get different as well, but the root ideastays the same. After this first impression it is time to put it in a more general frame. Ifthe transformation is no longer an a.s. bijection between the set E and all of X, but isstill surjective, one has to weaken lemma 4, but one is still able to find a connection.

Lemma 5. Let (X,T,B, µ) be a conservative ergodic measure preserving dynamical system.Let E be a sweep out set, such that T (E) = X. Then the jump system (X,T ∗

E) is a factor

(cf. definition 4) of the induced system (E, T ind

E). The surjection is given by T|E : E −→ X.

That means T|E

�T ind

E

�= T ∗

E

�T|E

�.

19


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(a) The jump transformation

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(b) The induced transformation

Figure 4: The jump and the induced transformation of the Farey system w.r.t. (1/4, 1].

Proof. Basically the proof follows the same pattern as in lemma 4. On the set E1 one hasT ∗ = T ind = T and T : E1 −→ E1. So, obviously on E1 one has T|E

�T ind

�= T ∗(T|E).

Thus it is left to look at x ∈ En for n > 1. One has T (En) = An. Furthermore on An thejump transformation equals the n − 1 times iteration of T|Ec and one final application ofT|E . With the induced transformation it is the other way round. By definition of En onedoes not enter E for n − 1 steps, when starting in En. Hence one has to apply T|E onetime, then n− 1 times T|Ec and finally T|E concludes the induced transformation.

T ∗(T|E(x)) = T|E(Tn−1|Ec T|E(x)) = T|E1

(T ind(x)).

3.2.4 Sum level sets revisited

Before the powerful tool of the transfer operator is introduced, this section will be concludedwith a nice representation of the sum level sets for the sweep out set A1. Recall thedefinitions 1 and 2. Since Lα and G are the jump transformations of Fα and F respectively(cf. lemma 2), the Cylinder sets C(c1, c2, ..., ck) imply that every member of a certaincylinder set is in A1 at the c1’th period for the first time, the second time c2 periods laterand so on. That is to end up in A1 in the n-th period, the ci’s 1 ≤ i ≤ k have to sum upto n for a k. With this thoughts the expansion representation of the sum level sets can begiven.

Ln :=

�x ∈ C(c1, c2, ..., ck) :

k�

i=1

ci = n for some k ∈ N�

. (6)

By convention one defines L0 := (0, 1]. C(...) stands either for the Cylinder sets referringto the Farey map or the α − Farey map, or any other appropriate transformation. How

20

3.3 The transfer operator 3 THE TOOLBOX

the first sum-level sets for the Gauß map look like can be seen in figure 5. In terms of thecylinder sets the sum level sets can be written as follows (cf. (KMS11, page 14)).

L1 = C(1);L2 = C(2) ∪ C(1, 1);L3 = C(3) ∪ C(2, 1) ∪ C(1, 2) ∪ C(1, 1, 1);L4 = C(4) ∪ C(3, 1) ∪ C(2, 2) ∪ C(2, 1, 1) ∪ C(1, 3) ∪ C(1, 2, 1) ∪ C(1, 1, 2) ∪ C(1, 1, 1, 1);... ...

0.0 0.2 0.4 0.6 0.8 1.0

01

23

45

sum level sets

n

I I

II II II

II I

I II II I

I II II I I II II I

Figure 5: The first sum level sets of the Farey map.

3.3 The transfer operator

3.3.1 Defining the transfer operator

A very powerful tool when working with a dynamical system is the transfer operator T . Itwill also be a crucial tool in this thesis. The general setting may be as follows:Let (X,B, µ) be a measure space. T : X → X is a measurable and non singular transfor-mation. For now let f ∈ L1

µ and f ≥ 0. Now one is able to declare for all A ∈ B a finitemeasure via �

T−1A

fdµ.

Since the transformation T is non singular, this measure is absolutely continuous withrespect to µ. A corollary of the Radon-Nikodym theorem (cf. (LM95, Corollary 2.2.1))states that if one has a measure space (X,B, µ) and a finite measure ν on B which is

21


absolutely continuous with respect to µ, then there exists a unique element g of L1µ such

that for B ∈ B one has

ν(B) =�

B

gdµ.

Thus in this case one can find such a unique element in L1µ say T f , such that

ν(A) =�

T−1A

fdµ =�

A

T fdµ.

If one drops the condition f ≥ 0 and takes an f ∈ L1µ one has to decompose f in the usual

way via f+ := sup(0, f), f− := sup(0,−f) and finally f = f+ − f−. Using this procedureone can define the so-called transfer operator T for f ∈ L1

µ via T f := T f+ − T f−.

Definition 17. The transfer operator. Let (X,T,B, µ) be a measure theoretical dy-namical system. Let T be a measurable and non singular transformation. Then for A ∈ Band f ∈ L1

µ, the transfer operator T , acting on the Banach space L1µ, is defined by the

relation (cf. (KS08))

T : L1µ −→ L1

µ

f �→ T (f) :=d(ρf ◦ T−1)

dµ,

where ρf is the measure with density f with respect to µ. The transfer operator is char-acterized by the relation

�

A

T fdµ =�

T−1A

fdµ. (7)

Remark. The transfer operator deals with functions on the function space L1µ with respect

to a certain measure µ. Hence it always comes along with that measure. It describeshow probability densities behave under the action of the Transformation T . It is thus alsolinked with the transformation T itself.Remark. The transfer operator of the n times iteration of T is given by ˆ(Tn) = (T )n.

T−1(A) denotes the preimage of A under the action of T . Thus T does not need to beinjective. For an explicit calculation the right side of the latter equation sometimes has tobe broken down into a sum over pairwise disjoint Ai’s, whose union is A and on every Ai

the transformation is invertible. It will be done in 3.3.2.

3.3.2 The Perron Frobenius operator

In some special cases one can give an explicit representation of the transfer operator. Herethe case shall be considered, where one is dealing with the Lebesgue measure λ. MoreoverT may have some additional properties, specified in a second. An interval [a, b] ⊆ R shallbe the space X.A simple case to start with is an invertible and differentiable T with continuous derivative,on X. As well A shall be assumed to be an interval of the form [a, x], x ≤ b. The statementcan be generalized to measurable subsets of [a, b] by a standard argument. Later on thegeneralization to the case is done, where T is not invertible and differentiable on all of X

22


but on pairwise disjoint subsets A1, A2, ... ⊂ X whose union is all of X. In this case thedecomposition mentioned in the preceding paragraph, has to be done.

Equation (7) gives�

x

a

T f(s)dλ(s) =�

T−1[a,x]f(s)dλ(s).

Differentiating gives

T f(x) =d

dx

�

T−1[a,x]f(s)dλ(s)

= f(T−1(x))��

d

dxT−1(x)

��

= f(y)��

1T �(y)

�� , where: y = T−1(x).

Now it is time for the case, where T is not injective on all of X but on the pairwise disjointsubsets A1, A2, ..., whose union is A. Decomposing the integral into a sum of integralsover these subsets yields

T f(x) =�

y∈T−1(x)

��1

T �(y)

�� f(y),

what leads to the next definition.

Definition 18. The Perron Frobenius Operator. The transfer operator with respectto the Lebesgue measure λ is called Perron Frobenius operator, Lψ. It holds

Lψf =�

y∈T−1(x)

1|T �(y)|f(y). (8)

Often the Perron Frobenius operator is given in the more universal form

Lψf =�

y∈T−1(x)

e−ψ(y)f(y).

and with the evaluation function ψ(·) = lg(|T �(·)|) one comes to the representation (8).

Remark. The name of the transfer operator varies very much in literature. Sometimes thegeneral transfer operator is called Perron Frobenius operator, no matter which measureone refers to. The name Ruelle operator is as well common as combinations of these names.The transfer operator of a certain transformation and with respect to a certain measurehas from case to case his own name, as in the next example.Example. The Perron Frobenius operator of the Gauß Transformation, that means theTransfer operator with respect to the Lebesgue measure of the transformation G(x) givenby (2), is named after three mathematicians Gauß-Kuzmin-Wirsing operator. It is givenby

Gf(x) =�

k∈N

1(k + x)2

f

�1

x + k

�. (9)

23


Along with the transfer operator we get some nice properties and rules for workingwith it.

Lemma 6. For f ∈ L1λ

and g ∈ L∞λ

holds

�Lψfdλ =

�fdλ

and�

Lψ ((g ◦ T )f) dλ =�

gLψfdλ.

This lemma will be proven for the case, one is dealing with T to be the Farey map.For another adequate transformation it works just the same. In this case the System hasto be decomposed into the sets A1 and X \ A1 on both of them T is a bijective map toX = [0, 1].Remark. If one is dealing with another measure an equivalent statement, according to thecorresponding transfer operator, can be made.

Proof. First�Lψfdλ =

�fdλ is shown for f ∈ L1

λ. The other assertion then follows

directly from it.A straight forward calculation will lead to the destiny. The transformation formula forintegration is once again used within this calculation.

�

X

Lψfdλ(x) =�

X

�

y∈T−1(x)

e− lg(|T �(y)|) · f(y)dλ(x)

=�

[0,12 ]

1|T �(T−1

|[0,1/2](x))|· f

�T−1|[0,1/2](x)

�dλ(x)

+�

[ 12 ,1]

1|T �(T−1

|[1/2,1](x))|· f

�T−1|[1/2,1](x)

�dλ(x)

=�

T−1[0,12 ]

1|T �(x)| |T

�(x)|f(x)dλ(x)

+�

T−1[ 12 ,1]

1|T �(x)| |T

�(x)|f(x)dλ(x)

=�

X

f(x)dλ(x).

24


Now, if one inserts a (g ◦ T )f instead of f into the integral one gets�

Lψ((g ◦ T )f)dλ =�

(g ◦ T )fdλ

=� �

y∈T−1(x)

1|T �(y)| · (g ◦ T )(y)f(y)dλ

=�

[0,12 ]

1|T �(T−1

|[0,1/2](x))|· g

�T (T−1

|[0,1/2]|(x))�

f�T−1|[0,1/2](x)

�dλ

+�

[ 12 ,1]

1|T �(T−1

|[1/2,1](x))|· g

�T (T−1

|(1/2,1]|(x))�

f�T−1|(1/2,1](x)

�dλ

=�

[0,12 ]

g(x)1

|T �(T−1|[0,1/2](x))|

· f�T−1|[0,1/2](x)

�dλ

+�

[ 12 ,1]g(x)

1|T �(T−1

|[1/2,1](x))|· f

�T−1|(1/2,1](x)

�dλ

=�

g(x)�

y∈T−1(x)

1|T �(y)|f(y)dλ

=�

gLψfdλ.

Hence the assertions are proven.

Remark. The second statement resembles a statement which will be made later in thisthesis after introducing the notion of the Koopman operator (cf. lemma 9).

Working with transfer operators with respect to different measures one is lead to askfor connections between these operators. The next lemma will point out the connectionbetween a transfer operator with respect to a measure µ to the Perron Frobenius operator.The case shall be considered, where a measure µ, which has a density ϕ with respect tothe Lebesgue measure, is given.

Lemma 7. Let µ be a measure which is absolutely continuous with respect to the Lebesguemeasure λ. It may have the density ϕ with respect to λ. I.e. dµ = ϕdλ. Let T be thetransfer operator of T with respect to µ. Then the transfer operator with respect to µ canbe related to the Perron Frobenius operator. It holds

T f◦=Lψ(fϕ)

ϕµ-a.s. (10)

Proof. Let g ∈ L∞µ and f ∈ L1µ. A calculation now yields�

gT fdµ =�

(g ◦ T ) fdµ

=�

g ◦ T fϕdλ

=�

gLψ (fϕ) dλ

=�

gLψ (fϕ)1ϕ

dµ.

25


The first step in this calculation is justified in the proof of lemma 6 or with lemma 9 insubsection 3.3.4. Since g ∈ L∞µ and f ∈ L1

µ were arbitrary, the lemma is proven.

3.3.3 The transfer operator and invariance

As mentioned above, the transfer operator describes how densities behave under the actionof a transformation T . Thus T -invariance must somehow be reflected through the transferoperator. Of course it is, as stated in the following lemma.

Lemma 8. The transfer operator and T -invariance. Let (X,T,A, µ) be a non sin-gular dynamical system. Let T be the transfer operator of T with respect to µ. Moreoverρf denotes the measure which is absolutely continuous with respect to µ and has density f ,i.e. dρf = fdµ. ρf is T -invariant if and only if f is a fixed point of the transfer operatorT , i.e. T (f) = f .

Proof. Let A ∈ A. By the definition of the Transfer operator (7),�

A

T fdµ =�

T−1A

fdµ.

Let f be a fixed point of T , thus one has for any A ∈ A�

A

fdµ =�

A

T fdµ

(7)=

�

T−1A

fdµ.

Hence with T (f) = f follows the T -invariance. If one assumes T -invariance of ρf , this isequivalent to saying that for any A ∈ A,

�T−1A

fdµ =�A

fdµ, one gets�

A

fdµ =�

T−1A

fdµ

(7)=

�

A

T fdµ.

Thus the T -invariance implies that f is a fixed point of T , as well. This proofs theassertion.

Corollary 2. Use the conditions of lemma 8. µ is T -invariant, if and only if 1X is a fixedpoint of T .

Proof. Take f = 1X .

Corollary 3. The Perron Frobenius operator and T -invariance. Let (X,T,B, µf )be a non singular dynamical system where µf is absolutely continuous to the Lebesguemeasure λ. May µf have the density f with respect to λ. Then µf is T -invariant if andonly if f is a fixed point of the Perron Frobenius operator Lψ, i.e. Lψf = f .

Proof. Use Lψ instead of T and λ instead of µ and mimic the arguments of the proof oflemma 8.

26


Remark. In the same manner, a transformation preserves the Lebesgue measure, if 1X isan eigenfunction of the associated Perron Frobenius operator.Example. The Gauß measure. The density of the invariant measure of the Gauß transfor-mation is given by g(x) = 1

lg 21

1+x(cf. page 8). By corollary 3 this g must be a fixed point

of the Gauß-Kuzmin-Wirsing operator (9). In fact, it is, since

Gg(x) =�

k∈N

1(k + x)2

1lg 2

1

1 +�

1x+k

�

=�

k∈N

1(k + x)2

1lg 2

x + k

x + k + 1

=1

lg 2

�

k∈N

1(x + k)(x + k + 1)

=1

lg 2

�

k∈N

�1

x + k− 1

x + k + 1

�

=1

lg 21

x + 1= g(x).

3.3.4 The Koopman operator

Along with the transfer operator always comes another operator, the Koopman operator.Initially it was introduced by Koopman in 1931 in (Koo31). This operator is defined inthe following way (cf. (LM95, definition 3.3.1)).

Definition 19. The Koopman operator. Let (X,T,A, µ) be a measure theoreticaldynamical system with a non singular transformation T . For f ∈ L∞µ the Koopmanoperator with respect to T , UT : L∞µ −→ L∞µ is given by

UT f(x) = f(T (x))

Lemma 9. Properties of the Koopman operator. Let UT be as in definition 19. TheKoopman operator has the properties

• For all f, g ∈ L∞µ and a, b ∈ R one has UT (af + bg) = aUT f + bUT g.

• UT is adjoint to the transfer operator T . I.e. for every f ∈ L1µ and every g ∈ L∞µ

one has �gT fdµ =

�UT gfdµ.

• If T is µ-measure preserving, one has for f, g ∈ L∞µ�

fgdµ =�

UT fUT gdµ.

Proof. The first assertion follows from the vector space structure of L∞ itself and the proofof the second assertion is an analogous calculation as in the proof of lemma 6. For yetanother proof the reader is referred to (LM95).The third statement follows with

�fgdµ =

�fg1dµ =

�fgT (1)dµ =

�UT f UT gdµ. Here

one should remember that T -invariance is equivalent to T1 = 1.

27


Remark. Since one has�

fgdµ =�

UT fUT gdµ =�

f TUT gdµ for f, g ∈ L∞µ , it is easy tosee that g

◦= TUT g. Attention, the other way round, UT T g◦= g, is not true in general.

3.3.5 Explicit examples

This chapter shall be concluded with two explicit examples of the transfer operator. Thesetwo examples shall be F and Fα introduced in the beginning. The first one is the Farey map(cf.(KS08)). Recall (1) and the inverse branches of F . First the Perron Frobenius operatorwill be calculated to find the transfer operator with respect to the invariant measureµ = 1/xλ via formula (10). For this, note |F �

0(x)| = 1/(1 − x)2 and F−10 (x) = x/(1 + x),

hence |F �0(y)| = |F �

0(F−10 (x)) = (1 + x)2. Analogously one calculates with |F �

1(x)| = 1/x2

and F−11 (x) = 1/(1 + x) for the right branch |F �

1(y)| = |F �1(F

−11 (x)) = (1 + x)2. Now, by

equation (8) one has

Lψf =�

y∈F−1(x)

1|F �(y)|f(y)

=f

�x

1+x

�

(1 + x)2+

f�

11+x

�

(1 + x)2.

Consequently one gets for the transfer operator F with h(x) = 1/x,

F (f) =1hLψ (hf)

=1

h(x)

�h

�F−1

0 (x)�f

�F−1

0 (x)�

(1 + x)2+

h�F−1

1 (x)�f

�F−1

1 (x)�

(1 + x)2

�

= x

1 + x

x(1 + x)

f�

x

1+x

�

(1 + x)+

1 + x

1 + x

f�

11+x

�

(1 + x)

=1

(1 + x)f

�x

1 + x

�+

x

1 + xf

�1

1 + x

�.

In a similar manner one can determine the Transfer operator for the α-Farey map withrespect to the invariant measure, determined by its Lebesgue density hα =

�∞n=1

tnan1An .

Recalling the inverse branches of the α-Farey map, given on page 5, it is easy to determinethe Perron Frobenius operator of a map f ∈ L1

µα. With

��F �α,1

�F−1

α,1(x)�� = 1/a1 on the

one hand and on the other hand for x ∈ An,��F �

α,0

�F−1

α,0(x)�� = an/an+1 a straight forward

calculation gives

Lψ(f) =�

y∈F−1α (x)

1|F �

α(y)|f(y)

=1��F �

α,1

�F−1

α,1(x)��

f�F−1

α,1(x)�

+1��F �

α,0

�F−1

α,0(x)��

f�F−1

α,0(x)�

= a1f(1− a1x) +∞�

n=1

1An(x)an+1

an

f

�an+1

an

(x− tn+1) + tn+2

�.

28


Once more formula (10) needs to be applied to get to Tα(f). Please note that the facttn+1 + an =

�∞i=n+1 ai + an = tn is used here. As well on should notice the identity

hα

� ∞�

n=1

1An

an+1

an

(x− tn+1) + tn+2

�=

∞�

n=1

1An

tn+1

an+1.

Fα(f) =1

hα(x)

�

y∈F−1α x

hα(y)f(y)1

|F �α(y)|

=1

hα(x)

�a1hα(1− a1x)f(1− a1x)

+∞�

n=1

1An(x)an+1

an

hα

� ∞�

n=1

1An

an+1

an

(x− tn+1) + tn+2

�f

� ∞�

n=1

1An(x)an+1

an

(x− tn+1) + tn+2

��

=1

hα(x)

�a1

1a1

f(1− a1x)

+∞�

n=1

1An(x)an+1

an

tn+1

an+1f

� ∞�

n=1

1An(x)an+1

an

(x− tn+1) + tn+2

��

=1

hα(x)

�f(1− a1x) +

∞�

n=1

1An(x)tn+1

an

f

� ∞�

n=1

1An(x)an+1

an

(x− tn+1) + tn+2

��

=1

hα(x)f(1− a1x) +

hα(x)− 1hα(x)

f

� ∞�

n=1

1An(x)an+1

an

(x− tn+1) + tn+2

�.

During this calculation one finds also the hint why hα is indeed the density of the α-Fareyinvariant measure. A statement where the proof is still owed to the reader (cf. page 8).Because, if one uses the first part of the calculation and takes hα instead of f , one gets

Lψ(hα) = a1hα(1− a1x) +∞�

n=1

1An(x)an+1

an

hα

�an+1

an

(x− tn+1) + tn+2

�

= 1 +∞�

n=1

1An

tn+1

an

=∞�

n=1

1An

tnan

= hα.

Thus by corollary 3, hα is the Lebesgue density for the Fα-invariant measure.

29

4 RENEWAL THEORY

4 Renewal theory

4.1 Basic Relations of Renewal Theory

4.1.1 The basic theory

Renewal theory is a topic which originally comes from probability theory and randomwalks. But the idea itself finds its applications on a wider spectrum and the theory behindit often belongs to pure analysis. An application to real life and daily problems can befound in the replacement of broken light bulbs, for example. First some basic relations ofrenewal theory will be explained, but especially in this thesis one should always bear thetwo main examples the Farey and the α-Farey map in mind. The reader is referred to e.g.(Fel68), (Fel71) and (Kre05) for deeper relations, more theory and in particular for moreexamples, especially from the probabilistic point of view. Let rn, n ∈ N be a sequence ofreal numbers with the properties

rn ≥ 0 for n ∈ N, (11)∞�

n=1

rn = 1. (12)

Moreover

gcd{n ∈ N | rn > 0} = 1 (13)

shall be assumed. (13) is equivalent to the property that rn is not periodic. A sequencern has period p > 1 if rn = 0, if n �= kp, k, p ∈ N and p is the greatest integer with thatproperty. This periodic case has to be treated a little special but can easily be traced backto the non periodic case. Considering periodic rn’s would be redundant for the purposesof this thesis and is therefore left out.One can also find examples where (12) is weakened, e.g. in models for populations.By setting t0 = 1, another sequence tn can be defined recursively by the renewal equation

tn =n�

k=1

rktn−k. (14)

Recursively one immediately verifies that for all n ∈ N, 0 ≤ tn ≤ 1.

Definition 20. Renewal Pair. Two sequences tn and rn, n ∈ N with the properties(11)-(14) are called a renewal pair.

When working with infinite sequences, it is often worth to look at the generatingfunctions. The generating function of an infinite sequence an is for |s| < 1 given by

a(s) :=∞�

n=1

snan. (15)

With these generating functions one can state a lemma which will be of use later.

Lemma 10. The renewal equation (14) is equivalent to the following relation between thetwo generating functions of rn and tn.

t(s) =1

1− r(s).

30

4.1 Basic Relations of Renewal Theory 4 RENEWAL THEORY

For the proof in this context we refer the reader to (Fel68). Another proof will be givenlater in a little wider context. The reader is referred to page 35.A result due to Erdös, Feller and Pollard (EFP49) (cf. (Fel68) as well) describes thelimiting behavior of tn. Their discrete renewal theorem shall be quoted here without aproof.

Theorem 5. The renewal theorem. (cf.(EFP49)) Let rn and tn be a renewal pair. Letm :=

�∞k=1 krk Then

limn→∞

tn =1�∞

k=1 krk

=1m

, (16)

where 1/m := 0, if m = ∞.

4.1.2 A real world example

Before heading back to the main examples of this thesis it is worth having a look at a realworld example, to inject the formulas of the previous subsection with life. For instancea light bulb is checked after a certain period of time, e.g. every day at 7 am in themorning. If it is broken it is replaced by another identical, new light bulb. The timeused for the replacement will be, for the reason of simplification, ignored. Every lightbulb has a life experience that follows a certain distribution. All the life times of the lightbulbs are assumed to be identically and independently distributed. This implies that afterthe replacement of a light bulb, the procedure starts from scratch. One can define theprobabilities

tk := P(light bulb is broken in the morning of day k),rk := P(light bulb is broken for the first time in the morning of day k).

Now, if the question arises, whether on a certain day N , the light bulb has to be replaced,i.e. a renewal takes place, there are several options. It could either be the first replacement,that would be what rN would represent, or it could have been replaced the last time beforethe day N at day number k, k ≤ N . This would be represented by tk. After this lastreplacement it would have to be replaced the first time again at the day N . This “firsttime again” replacement is then represented by rN−k. Summing the probability of all thesemutually exclusive events gives tN , the probability that a replacement is necessary at daynumber N .

tN =N−1�

k=0

rN−ktk. (17)

In this example, the assumption was that all life times are identically distributed. Howeverone could think about this replacement example with a slight modification. The engineerwho is responsible for replacing these light bulbs may be appointed at a day, when thelight bulb is still in good order, but it was already shining for an unknown period of time.So the first life time has a distribution different to the subsequent ones. After the firstreplacement one finds himself again in the case considered before. The first replacementday may now have a distribution given by the sequence bn and the replacement, no matter

31

4.1 Basic Relations of Renewal Theory 4 RENEWAL THEORY

if it is the first one, at the day n may have distribution vn. This is what Feller calls adelayed recurrent event. (cf. (Fel68, XIII.5)). Equation (17) has now to be modified a bit.

vN =N−1�

k=0

bN−ktk (18)

= bN +N−1�

k=1

rkvN−k. (19)

The first equation can be interpreted as follows. The event that the light bulb is broken at7 am in the morning of the Nth day may have distribution vn. In (18) it is decomposed intothe sums of the probabilities that the first substitution was at day N − k, k = 0, ..., N − 1,bn−k and it has to be replaced k days later. This includes the case where the replacementat day N is the first one at all, this is the case in which k = 0. The vantage point of thesecond equality (19) is slightly different. Of course bN again is the first replacement on dayN . The sum represents the possibilities of a renewal on day N−k and the first replacementafter this change of the light bulb k days later. For later purposes this example should bekept in mind.Now this thesis will discuss some interpretations of the formulas above related to the twoprime example systems. For more and other basic introductory examples the reader mighthave a look at (Fel68).

4.1.3 Classical renewal theory in the α-Farey system

An important result for the α-Farey system due to Keßeböhmer, Munday and Stratmann(KMS11) is the following lemma (cf. (KMS11, Lemma 3.1)).

Lemma 11. Recall the α-Farey system as well as the meaning of an and L (α)n . The two

sequences an and λ(L (α)n ) build a renewal pair. That implies, that for n ∈ N one has

λ(L (α)n ) =

n�

k=1

akλ(L (α)n−k

). (20)

The lemma is proven by straight forward calculation, the reader might be referred to(KMS11).With this result and with the discrete renewal theorem of Erdös, Feller and Pollard, itis possible to make statements about the limiting behavior of λ(L (α)

n ). They proved thefollowing theorem (cf. (KMS11, theorem 1)).

Theorem 6. For the Lebesgue measure λ(L (α)n ) of the α sum level sets of a given partition

α of [0, 1] one has that�∞

n=1(L(α)n ) diverges, and that

limn→∞

λ(L (α)n ) =

0 if α is of infinite type;

(�∞

k=1 tk)−1 if α is of finite type.

Here, α is said to be of finite type, if for the tails tn of α one has that�∞

n=1 tn converges.If the sum diverges, α is said to be of infinite type.

Example. The harmonic partition is of infinite type, since�∞

n=1 tn =�∞

n=11n

= ∞.

32

4.2 Renewal theory for the induced Farey system 4 RENEWAL THEORY

4.2 Renewal theory for the induced Farey system

In the case of the Farey map the renewal equation as in the α-Farey example is unfortu-nately not valid. The relation (20) already fails for n = 2 in the Farey system. But if onetries to lift the theory a level higher, one is still able to find similar relations. This wasdone by Omri Sarig (cf. (Sar02)). His setting shall be modified to fit into the Farey settinghere.

Proposition 1. Consider the Farey system (I, F,B, µ). Let A1 := (1/2, 1] and En :=A1 ∩ {x| inf(k ≥ 1 : T kx ∈ [1/2, 1]) = n} and D be the unit disk in C, i.e. D := {z ∈ C :|z| < 1}. Let in addition T be the transfer operator of T with respect to µ. Now define thetwo operators

Tnf := 1A1 Tn(f1A1),

Rnf := 1A1 Tn(f1En).

Note R0 = 0 and T0 = I, where I denotes the identity. Then for all z ∈ D the renewalequation

T (z) = (I −R(z))−1 (21)

is valid. Here

T (z) :=∞�

n=0

znTn

and R(z) :=∞�

n=0

znRn

are the generating functions according to Tn and Rn respectively. Furthermore R(1) is thetransfer operator of T ind

A1with respect to µ.

Proof. The validity of the two equations

Tn =n�

k=1

RkTn−k (22)

=n−1�

k=0

TkRn−k, (23)

will be shown, because they imply the renewal equation (21) itself. (22) will be shownfirst.

33


Let g ∈ L∞µ and f ∈ L1µ.

�g

n�

k=1

RkTn−kfdµ =n�

k=1

�gRk

�1A1 T

n−k(f1A1)�

dµ

=n�

k=1

�g1A1 T

k

�1A1 T

n−k(f1A1)1Ek

�dµ

=n�

k=1

�(g1A1) ◦ T k · 1Ek Tn−k(f1A1)dµ

=n�

k=1

�(g1A1) ◦ Tn · 1Ek ◦ Tn−k · f · 1A1dµ

=n�

k=1

�g ◦ Tn ·

�1A1 ◦ Tn · 1Ek ◦ Tn−k · 1A1

�· fdµ.

Now it is time to have a closer look at the sets involved in this integral. First one shouldrecall the relation 1A1 ◦ T k = 1T−k(A1). Then one has to notice that the set A1 ∩ T−n(A1)equals the disjoint union over k = 1, ..., n of A1 ∩ T−n(A1) ∩ T−(n−k)(Ek).This result yields

�g

n�

k=1

RkTn−kfdµ =�

(1A1g) ◦ Tn1A1fdµ

=�

g1A1 Tn(1A1f)dµ

=�

gTnfdµ.

Since f ∈ L1µ and g ∈ L∞µ were arbitrary, assertion (22) is true. The proof of the second

equality follows a very similar pattern. Let again g be a L∞µ function and f an element ofL1

µ.

�g

n−1�

k=0

TkRn−kfdµ =n−1�

k=0

�(g1A1)T

k

�1A1 T

n−k(f1En−k)�

dµ

=n−1�

k=0

�g ◦ Tn · 1A1 ◦ Tn · 1A1 ◦ Tn−k · f · 1En−kdµ

=n�

k=1

�g ◦ Tn · 1A1 ◦ Tn · 1A1 ◦ T k · f · 1Ekdµ.

This time the manner is a little different, but the result is the same. First one notices thatthe sets Ek are part of the sets T−kA1 and second, that the set T−nA1 ∩A1 is a subset of

34


the disjoint union�

n

k=1 Ek ∩A1 =�

n

k=1 Ek. Hence the calculation can be completed via

�g

n−1�

k=0

TkRn−kfdµ =�

(g1A1) ◦ Tnf1A1dµ

=�

g1A1 Tn(f1A1)dµ

=�

gTnfdµ.

(23) is valid as well.As stated above the validity of these equations implies the renewal equation. For this

calculation one should bear R0 = 0, T0 = I and z ∈ D in mind. As

R(z)T (z) =∞�

k=0

zkRk

∞�

k=0

zkTk

=∞�

k=1

zkRk

∞�

k=0

zkTk

=∞�

k=1

zk

�k�

l=1

RlTk−l

�

=∞�

k=1

zkTk

=∞�

k=0

zkTk − T0

= T (z)− I.

And the other way round,

T (z)− I =∞�

k=1

zkTk

=∞�

k=1

zk

k−1�

l=0

TlRk−l

=∞�

k=0

zkTk

∞�

k=0

zkRk = T (z)R(z).

Hence R(z)T (z) = T (z)− I = T (z)R(z), what is equivalent to I = (I −R(z)) T (z) =T (z) (I −R(z)). That proves the assertion (21), hence the first part of the proposition.Now another similar calculation yields that R(1) is the transfer operator of T ind withrespect to µ. Let g and f be as above.

�g∞�

k=1

Rkfdµ =� ∞�

k=1

g1A1 Tk(f1Ek)dµ

35

4.3 The delayed case 4 RENEWAL THEORY

=� ∞�

k=1

g ◦ T k1A1 ◦ T k1Ekfdµ

=� ∞�

k=1

g ◦ T k1T−k(A1)1Ekfdµ

=�1A1g ◦ T indfdµ

=�1A1gT ind(f)dµ.

One simplification arises from the above mentioned fact Ek ⊂ T−k(A1). The other simplyrepresents that by definition on Ek, r(x) = k, hence the k times iteration on T is just thetransformation T ind. Summing over all k finishes the proof, since A1 =

�∞k=1 Ek.

4.3 The delayed case

In the previous section the assertions were restricted to the interval A1. Loosely speakingone just considered the renewal case in which the beginning is precisely a start from scratch.But what happens if one finds himself in an initial position outside the interval A1? Thisresembles the situation which Feller calls a delayed recurrent event (cf. (Fel68, XIII.5) orequations (18) and (19) in this thesis).

Proposition 2. Adopt the setting of proposition 1. Define the four operators Tn, Rn, Vn

and Bn given by

Tnf = 1A1 Tn(f1A1), (24)

Rnf = 1A1 Tn(f1En), (25)

Vnf = 1A1 Tn−1(f), (26)

Bnf = Tn−1(f1An). (27)

These four operators fulfill the delayed renewal equations given by

Vn = Bn +n−1�

k=1

Tn−kBk =n�

k=1

Tn−kBk (28)

= Bn +n−1�

k=1

RkVn−k. (29)

The operator Bn plays the part of the first entrance to A1 at time period n. WhereasRn is, as in the restricted case, the first return to A1 after the nth iteration of T . Vn

represents the appearance of the event “Entering A1” at time n and finally, Tn is the returnto A1 after n steps.

36

4.3 The delayed case 4 RENEWAL THEORY

Remark. Note the difference between the two phrases “after n iterations” and “at time n”.The first equation can be interpreted as follows. To end up in A1 in the nth period,

it could either be the first arrival, this is what Bn stands for, or the first arrival in A1

could have taken place at any time period k, k ≤ n. This would imply that one has tocome back to A1 in n− k steps. The latter is consequently represented by Tn−kBk. Sincethose possibilities are mutually exclusive for different k, one has to sum over all thosepossibilities.The second equality has to be seen a little different. Of course Bn again stands for the firstpresence at A1 in the nth period. If this presence is not the first one, one has to have beenthere before and this is the first return after the last attendance. Hence one has gotten toA1 at some point n − k, this represented by Vn−k and returned to it the first time afterk more steps. This first return is given by Rk. For the calculation one should bear therelation 1A1 ◦ T k−1 = 1Lk

in mind.

Proof. First (28) is shown. One has

�gBnfdµ +

�g

n−1�

k=1

Tn−kBkfdµ

=�

gTn−1(f1An) +n−1�

k=1

�g1A1 T

n−k

�1A1 T

k−1(f1Ak)�

dµ

=�

g ◦ Tn−1f1Andµ +n−1�

k=1

�(g1A1) ◦ Tn−k

�1A1 T

k−1(f1Ak)�

dµ

=�

g ◦ Tn−1f1Andµ +n−1�

k=1

�g ◦ Tn−11A1 ◦ Tn−11A1 ◦ T k−1f1Akdµ.

On the one hand one has Ai ⊂ Li for all i ∈ N and on the other one should notice thatthe sum level set Ln is part of the disjoint union of the Ak’s itself, Ln ⊂

�n

k=1 Ak. Hencethe calculation can be completed by

�gBnfdµ +

�g

n−1�

k=1

Tn−kBkfdµ

=�

g ◦ Tn−1f1Ln1Andµ +n−1�

k=1

�g ◦ Tn−11Lnf1Akdµ

=n�

k=1

�g ◦ Tn−11Lnf1Akdµ

=�

g ◦ Tn−11Lnfdµ

=�

(g1A1) ◦ Tn−1fdµ

=�

g1A1 Tn−1fdµ

=�

gVnfdµ.

37

4.4 Back to the α-Farey map 4 RENEWAL THEORY

After verifying the first assertion, the second is checked in a similar manner.

�gBnfdµ +

�g

n−1�

k=1

RkVn−kfdµ

=�

gTn−1(1Anf)dµ +n−1�

k=1

�g1A1 T

k

�1Ek1A1 T

n−k−1f�

dµ

=�

g ◦ Tn−11Anfdµ +n−1�

k=1

�g ◦ Tn−11A1 ◦ Tn−11Ek ◦ Tn−k−1fdµ

=�

g ◦ Tn−11Ln1Anfdµ +n−1�

l=1

�g ◦ Tn−11Ln1En−l ◦ T l−1fdµ.

Noticing that the sets T−(l−1)En−l are disjoint for different l’s and their union over l =1, ..., n− 1 is Ln, finishes the calculation.

�gBnfdµ +

�g

n−1�

k=1

RkVn−kfdµ

=�

g ◦ Tn−11A1 ◦ Tn−1fdµ

=�

(g1A1) ◦ T (n−1)fdµ

=�

g1A1 Tn−1fdµ

=�

gVnfdµ.

Since f ∈ L1µ and g ∈ L∞µ were chosen to be arbitrary, the proposition is proven.

4.4 Back to the α-Farey map

Since one has now found a renewal equation for operators in the Farey system, the questionsuggests itself, whether these operator equations go in accordance with the renewal resultsof the α-Farey system (20). That means one has to show that the equation

λ(L (α)n ) =

n�

k=1

akλ(L (α)n−k

)

follows from the formulas (24)-(27), if one chooses T = F(α) and T to be the associatedTransfer operator. Whenever one is working with transfer operators one needs of coursea measure for the system to refer to. In the α-Farey system this measure, as mentionedbefore, has a density which is given by

h(α) :=dµα

dλ=

∞�

n=1

tnan

· 1An .

38


Since

λ(Ln) =�1A1 ◦ Fn−1

(α) dλ

=�1A1 ◦ Fn−1

(α)

1h(α)

dµα

=�1A1 T

n−1 1h(α)

dµα

=�

Vn

�1

h(α)

�dµα,

λ(Ln) is clearly represented by the operator Vn (26). Taking the equality (28) into accountone gets

λ(Ln) =�

Vn

1h(α)

dµα

(28)=

�Tn−1

�1An

1h(α)

�dµα +

� n−1�

k=1

1A1 Tn−k

�1A1 T

k−1(1Ak

1h(α)

)�

dµα

=�1An

1h(α)

dµα +n−1�

k=1

�1A1 ◦ Tn−11A1 ◦ T k−11Ak

1h(α)

dµα

=�1Andλ +

n−1�

k=1

�1A1 ◦ Tn−11Ak

1h(α)

dµα

= λ(An) +n−1�

k=1

�1A1 ◦ Tn−k−1T k(1Ak

1h(α)

)dµα

= λ(An) +n−1�

k=1

�1Ln−k

T k(1Ak

1h(α)

)dµα.

Now one should have a closer look at the transfer operator itself. If one now makes useof the relation (10) T (·) = 1

h(α)Lψ(·h(α)) and of the exact representation of Lψ (8), the

calculation can be finished. For the transfer operator T , one gets

T k(1Akh−1(α)) =

1h(α)

Lk

ψ(1Ak

h(α)

h(α))

=1

h(α)

�

y=T−kx

1|T k�(y)|

1Ak(y).

The fact that the jump transformation coincides with the α-Farey map and in additionthe fact that this map is linear, reduces the derivative of T k to T k

� = ∆T k(x)/∆x = 1/ak.

39


Thus

λ(Ln) = λ(An) +n−1�

k=1

�1Ln−k

ak

1hα

dµα

= λ(An)λ(L0) +n−1�

k=1

ak

�1Ln−k

dλ

=n�

k=1

akλ(Ln−k).

Hence, the question from the beginning of this section, page 38 can be answered affirma-tive.

40

5 FURTHER DISCUSSION

5 Further discussion

5.1 Delayed* sum level sets

Looking another time at the two main examples some more general theory shall be appliedto obtain further results. An interesting question is for example how do the sum levelsets behave if one starts looking at the process at a later time. That is to say, how doesthe Limit of Ln behave if one misses the first presence in A1. In the α-Lüroth case thesituation occurs to be rather simple, but not so in the Farey case.Remark. The word “delayed” in the context of this section stands for missing the wholefirst epoch. It is thus a difference to the “delayed” used in subsection 4.3. For not gettingconfused, here the sum level sets are called delayed* sum level sets.

First of all these delayed* sum level sets shall be described. The sum level sets in thecontinued fraction expansion representation or the α-Lüroth expansion respectively aregiven by (6).

Ln =

�[a1, ..., ak]|

k�

i=1

ai = n, for some k ∈ N�

.

Now, since one does not want to bother about the first presence in A1, the first expansionentry may be arbitrary. So the delayed* sum level set denoted by L ∗

n then is defined by

L ∗n :=

�[a1, ..., ak]|

k�

i=2

ai = n, a1 ∈ N, k ∈ N, k ≥ 2

�.

Hence one can say L ∗n is just the preimage of Ln under the jump transformation.

L ∗n = F ∗−1Ln.

So, after this first short approach, the attention should first be drawn towards the α-Fareysystem, respectively the α-Lüroth system. Since the α-Lüroth transformation is λ invariantone gets the same limiting behavior as in the non delayed* case. One has for k ∈ N,

λ(L ∗n ) = λ(L−1

α Ln) = λ(Ln).

For this limiting behavior the reader is referred to subsection 4.3 and to (KMS11). Thequestion would loose a lot of its appeal if it would be that easy in the Farey case as well.Of course it is not, since the Gauß transformation is not measure preserving with respectto λ. But with the help of two results obtained by Keßeböhmer and Stratmann (KS10) asimilar behavior for the delayed sum level set can be encountered.First another result by Keßeböhmer and Stratmann should be mentioned. They showedthat (cf. (KS))

λ(Ln) ∼ 1log2 n

.

For two sequences bn and cn, n ∈ N, “∼” means limn→∞bncn

= 1.Now a proposition from (KS10) shall be cited, without the proof. (cf. Proposition 1.3. in(KS10)). The notation is slightly translated to fit into this context. May “∗ − lim” denotethe weak convergence of measures.

41

5.2 Back to ergodic theory 5 FURTHER DISCUSSION

Proposition 3. For each interval [β, γ] ⊂ (0, 1], one has that

∗ − limn→∞

�lg n

lg(γ/β)λ|F−n(β,γ)

�= λ|(0,1).

For the delayed sum level set of the Farey transformation one calculates

λ(L ∗n ) = λ(G−1Ln)

=�1Ln ◦Gdλ

=�1LnG(1)dλ

=�1

F−(n−1)(( 12 ,1])G(1)dλ,

where G was defined in (9). With γ/β = 2 one gets by the use of Proposition 3,

lg(n− 1)lg 2

λ(L ∗n ) =

lg(n− 1)lg 2

�1

F−(n−1)(( 12 ,1])G(1)dλ

∗−limn→∞−→�

G(1)dλ.

With the exact representation of the Gauß-Kuzmin-Wirsing operator the latter integralcan be determined explicitly.

�G(1)dλ =

� �

k∈N

1(k + x)2

· 1dλ

=�

k∈N− 1

k + x

��1

0

=�

k∈N− 1

k + 1+

1k

= 1.

Therefore one obtains λ(L ∗n ) ∼ 1

log2(n−1) . Since the logarithm is slowly varying one has1

log2 n∼ 1

log2(n−1) . Consequently the limiting behavior of Ln and L ∗n is the same.

5.2 Back to ergodic theory

In subsection 3.1 a short introduction to the theory of dynamical systems and ergodictheory was given. Now, further results from ergodic theory will be discussed. In thissubsection (X,T,A, µ) is a conservative, ergodic and measure preserving dynamical system,with the infinite and σ-finite measure µ. First the notion of uniformly returning sets shall begiven. This nomenclature was primarily introduced by Keßeböhmer and Slassi in (KS07).

Definition 21. Uniformly returning (cf. (KS07, definition 2.3)). Let (X,T,A, µ) bea measure theoretical dynamical system. A set A ∈ A with 0 < µ(A) < ∞ is calleduniformly returning for f ∈

�L1

µ

�+, if there exists a positive increasing sequence bn � ∞such that

bnTnf −→ µ(f) µ-a.s. uniformly on A. (30)

42


The set A is called uniformly returning if it is uniformly returning for any f ∈�L1

µ

�+.Here, µ-a.s. uniform convergence on A means uniform convergence in L∞µ |A∩A.

Lemma 12. In the Farey system (1/2, 1] is uniformly returning. In the α-Farey systemA1 is uniformly returning, if α is of infinite type.

Remark. Recall that the harmonic partition is of infinite type (cf. example page 32).

Proof. For the Farey system this statement is proven for a class of functions in (KS08).For the α-Farey system this statement will be proven here by an explicit calculation. Forthe preliminary calculations the reader is referred to subsection 3.3.5 of this thesis. Asbefore, hα denotes the density of the Fα-invariant measure with respect to the Lebesguemeasure. To get an impression of these iterations the first two are done by hand, the proofwill follow after it. One should look at the function f = 1/hα. It obviously is in L1

µ, since�[0,1] 1/hαdµ =

�[0,1] dλ = 1.

Fα

�1hα

�=

1hα

Lψ(1)

=1hα

�a1 +

∞�

n=1

1An(x)an+1

an

�.

Looking only at A1, one gets

1A1

1hα

Lψ(1) = 1A1

1hα

�a1 +

a2

a1

�

= 1A1

1hα a1

(a1a1 + a2)

= 1A1

1hα a1

λ(L2).

The second iteration of Fα is

Fα

�Fα

�1hα

��= Fα

�1hα

Lψ(1)�

=1hα

Lψ

�hα

1hLψ(1)

�(31)

=1hα

�Lψ

�a1 +

∞�

n=1

1An(x)an+1

an

��

=1hα

(a1(Lψ(1)) + Lψ

� ∞�

n=1

1An

an+1

an

�

=1hα

a1

�a1 +

∞�

n=1

1An(x)an+1

an

�+ a1

a2

a1+

∞�

n=1

1An(F−1α,0(x))

an+1

an

∞�

k=1

1Ak

ak+1

ak

=1hα

a1

�a1 +

∞�

n=1

1An(x)an+1

an

�+ a1

a2

a1+

∞�

n=1

1An(x)an+1

an

an+2

an+1.

43


Focussing on A1 yields,

1A1

1hα

L2ψ(1) = 1A1

1hα

a1

�a1 +

∞�

n=1

1An(x)an+1

an

�+ a1

a2

a1+

∞�

n=1

1An(x)an+1

an

an+2

an+1

=1

hα a1

�a3

1 + a1a2 + a2 + a1 + a3�

(32)

=1

hα a1λ(L3).

The summands in the second last line (32) of the calculation are the Lebesgue measure ofthose cylinder sets, whose union is L3. Now one can make a guess where this is boilingdown to.

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4

!-sum level sets

n

IIII

I II

II

II

I I

Figure 6: The first α-sum level sets of the α-Farey map, w.r.t. the harmonic partition,with the the point 3/4 and its preimages.

For further iterations and to make a statement about the limiting behavior of Fnα another

calculation will be performed. Inductivly one notices (cf. (31)) Fnα (1/h) = 1/hLn

ψ(1).

With the exact formula for the Perron Frobenius operator, one gets

1hα

1A1Ln

ψ(1) =

1hα

1A1

�

y∈F−nα (x)

1|Fn

α

�(y)|1(y).

For completing in this argumentation three things are worth to mention (cf. definition2). First, in every cylinder set, C(l1, l2, ..., lk) of the α-Farey map, whose α-Farey expan-sion entries add up to n+1,

�k

i=1 li = n+1, for some k ∈ N, is one and only one preimage

44


of x under Fnα (cf. figure 6).

Second, on each of these cylinder sets the derivative of Fnα is constant and its absolute

value equal to λ(A1)/λ(C(l1, l2, ..., lk)) = a1/(al1 · ... · alk).

Third, the disjoint union of these cylinder sets is the n+1st sum level set,�P

li=n+1 C(l1, l2, ..., lk) =Ln+1.

Hence,

1hα

1A1Ln

ψ(1) =

1hα

1A1

�

y∈F−nα (x)

1|Fn

α

�(y)|

=1hα

1a1

�

C(l1,...lk)Pki=1 li=n+1,k∈N

λ(C(l1, ...lk))

=1hα

1a1

λ(Ln+1).

Since α was assumed to be of infinite type, λ(Ln) tends to zero, by theorem 6. Moreoverfor all iterations the result is a constant function on A1. The convergence is thus uniformlyas well, and the assertion is proven.

With the definition of uniformly returning it is an interesting question under what kindof additional requirements to the system one can suggest the almost sure convergence ofthe series in (30) on further subsets of X or even on almost all of X. Here the mattershall be discussed for further subsets in the α-Farey system and the Farey System. Thesesystems satisfy some nice properties which shall be mentioned once again.([0, 1], Fα,B, µα), respectively ([0, 1], F,B, µ), is a conservative, ergodic, measure preservingdynamical system with the infinite, σ-finite measure µα = hαλ, respectively µ = hλ. Thepartition sets (An)n∈N are the level sets of the first passage time and satisfy

T : An −→ An−1, µ-a.s. bijective, for n > 1.

and T : A1 −→ X is µ-almost surely bijective, in particular Fα|A1and F|A1

consist of onlyone branch and Fα as well as F , of two full branches. That implies that every x ∈ A1 hastwo preimages. One of it is in A1, the other in X \A1. With this preliminary observationsthe two final theorems can be stated.

Theorem 7. Uniformly returning for the α-Farey system. Let ([0, 1], Fα,B, µα) bethe α-Farey system with respect to the partition (An)n∈N, given by the level sets of the firstpassage time. If A1 is uniformly returning, then so is every A which is bounded away from0.

Proof. If one can show the almost sure convergence of the series (30) in a set B ∈ A oneof course has the almost sure convergence on any subset of B. And since A3 is to A2

what A2 is to A1, namely the level set of the second passage time, it is sufficient to showthe statement for almost every x ∈ A2. Since X

◦=�∞

n=1 An, the rest will follow by aninductive argument. The crucial fact one has to show is that if A1 is uniformly returning,A2 is as well.

45


Let x0 ∈ A2 and x1 denote the two preimages of Fα(x0). With f ∈ L1µ one has,

hα(x0)F �

α(x0)Fn

α f(x0) =hα(x0)F �

α(x0)Fn

α f(x0) +hα(x1)F �

α(x1)Fn

α f(x1)−hα(x1)F �

α(x1)Fn

α f(x1)

=�

i∈{0,1}

hα(xi)F �

α(xi)Fn

α f(xi)−hα(x1)F �

α(x1)Fn

α f(x1)

=�

i∈{0,1}

hα(xi)hα(xi)

1F �

α(xi)Ln

ψ(hαf)(xi)−

hα(x1)F �

α(x1)Fn

α f(x1).

This is the time to note once more that xi ∈ F−1α (Fα(x)), i = 0, 1. With the exact

representation of the Perron Frobenius operator one gets,

hα(x0)F �

α(x0)Fn

α f(x0) =�

y∈F−1α (Fα(x))

1F �

α(y)Ln

ψ(hαf)(y)− hα(x1)

F �α(x1)

Fn

α f(x1)

= Ln+1ψ

(hαf)(Fα(x0))−hα(x1)F �

α(x1)Fn

α f(x1)

= hα(Fα(x0))Fn+1α (f)(Fα(x0))−

hα(x1)F �

α(x1)Fn

α f(x1).

Hence,

Fn

α f(x0) =F �

α(x0)hα(x0)

�hα(Fα(x0))Fn+1

α (f)(Fα(x0))−hα(x1)F �

α(x1)Fn

α f(x1)�

.

Now one is in the position to proceed with the final step. One knows that x1 as well asFα(x0) is in A1, hence the two summands converge for a suitable normalizing sequence.From (KS07, proposition 3.1) it is known that (bn) is slowly varying if and only if thewandering rate (cf. definition 11) is slowly varying, which it is (cf. remark page 11). Onehas bn ∼ wn(A1). Since wn is slowly varying, one has for n −→ ∞, wn/wn+1 −→ 1.Therefore,

limn→∞

wnFn

α f(x0) = limn→∞

wn

�F �

α(x0)hα(x0)

�hα(Fα(x0))Fn+1

α (f)(Fα(x0))−hα(x1)F �

α(x1)Fn

α f(x1)��

=F �

α(x0)hα(x0)

hα(Fα(x0)) limn→∞

wn

wn+1wn+1F

n+1α (f)(Fα(x0))

− F �α(x0)

hα(x0)hα(x1)F �

α(x1)lim

n→∞wnFn

α f(x1)

−→ F �α(x0)

hα(x0)hα(Fα(x0)) µ(f)− F �

α(x0)hα(x0)

hα(x1)F �

α(x1)µ(f) (33)

= µ(f)F �

α(x0)hα(x0)

�hα(Fα(x0))−

hα(x1)F �

α(x1)

�

Note that the convergence in (33) is µα-a.s. uniformly on A1. Now it is time to have a

46


closer look at the constants. Since x0 ∈ A2 and x1 ∈ A1, one has

F �α(x0)

hα(x0)

�hα(Fα(x0))−

hα(x1)F �

α(x1)

�=

a1

a2

a2

t2

�t1a1− t1

a1a1

�

=a1

t2

�t1a1

(1− a1)�

= t1

= 1.

Note that 1− a1 = t2.Hence for x ∈ A2, limn→∞wnFn

α f(x0) = µ(f). That is, the set A2 is uniformly returningas well.The inductive argument, mentioned in the beginning of this proof, gives the conclusion.Assume An is uniformly returning. If x0 ∈ An+1, Fα(x0) ∈ An and F−1

α (Fα(x0)) ={x0, x1}, x1 ∈ A1. With the same calculation one gets

limn→∞

wnFn

α f(x0) = µ(f)F �

α(x0)hα(x0)

�hα(Fα(x0))−

hα(x1)F �

α(x1)

�.

Finally, it is left to show, that the latter constant is equal to one as well, if x0 ∈ An+1. Itis, since

F �α(x0)

hα(x0)

�hα(Fα(x0))−

hα(x1)F �

α(x1)

�=

an+1

tn+1

an

an+1

�tnan

− t1a1

a1

�

=an

tn+1

1an

(tn − an)

=tn+1

tn+1

= 1.

This thesis shall be concluded with an analogous statement about the Farey system.

Theorem 8. Uniformly returning for the Farey system. Let ([0, 1], F,B, µ) be theFarey system. Let (An)n∈N denote the level sets of the first passage time. Recall thatµ = hλ, with h(x) = 1/x. Every A which is bounded away from 0 is uniformly returning.

Proof. Since A1 is uniformly returning (cf. (KS08)), the same pattern as in the proofof the previous lemma can be applied. It holds F−1F (x0) = {x0, x1}, whereat x0 ∈A2, respectively, for the inductive argument, x0 ∈ An+1. Moreover x1 ∈ A1. With anequivalent calculation as in the proof of theorem 7 one has,

limn→∞

wnFnf(x0) = µ(f)F �(x0)h(x0)

�h(F (x0))−

h(x1)F �(x1)

�,

for x0 ∈ A2, An+1 respectively. All that is left to show is, that the latter constant is equalto one. For this one should note that x1 = 1− x0, because the Farey map is symmetric tox = 1/2. Furthermore,

F �(x0) =1

(1− x0)2,

47


h(F (x0)) =1− x0

x0,

h(x1) = h(1− x0) =1

1− x0,

F �(x1) = F �(1− x0) =1

(1− x0)2.

Hence,

F �(x0)h(x0)

�h(F (x0))−

h(x1)F �(x1)

�=

x0

(1− x0)2

�1− x0

x0− (1− x0)2

1− x0

�

=1

1− x0− x0

1− x0

= 1,

what proves the assertion.

48

REFERENCES REFERENCES

References

[Aar97] J. Aaronson. An introduction to infinite ergodic theory. Mathematical surveysand monographs; 50. American Mathematical Society, Providence, RI, USA,1997.

[Cox67] D.R. Cox. Renewal theory. Science paperbacks. 58. Methuen, London, 1967.

[Den05] M. Denker. Einführung in die Analysis dynamischer Systeme. Springer, Berlin,Germany [et.al.], 2005.

[DK02] K. Dajani and C. Kraaikamp. Ergodic theory of numbers. The MathematicalAssociation of America, Washington, DC, USA, 2002.

[DS58] N. Dunford and J. T. Schwartz. Linear operators, General theory, volume oneof Pure and applied mathematics : A Series of Texts and Monographs ; 7,1.Interscience Publishers, Inc., New York [u.a.], 1958.

[EFP49] P. Erdős, W. Feller, and H. Pollard. A property of power series with positivecoefficients. Bulletin of the American Mathematical Society, 55(2), 1949.

[Fel68] W. Feller. An Introduction to Probability and its Applications, volume 1. JohnWiley Sons Inc., New York, USA, third edition, 1968.

[Fel71] W. Feller. An Introduction to Probability and its Applications, volume 2. JohnWiley Sons Inc., New York, USA, second edition, 1971.

[Hop37] E. Hopf. Ergodentheorie. Springer, Berlin, Germany, 1937.

[Iso09] S. Isola. From infinite ergodic theory to number theory (and possibly back).https://unicam.it/∼stefano.isola/ergoinfinita.pdf, 2009.

[Khi56] A. Khinchine. Kettenbrüche. D.G. Teubner Verlagsgesellschaft, Leipzig, Ger-many, second edition, 1956.

[KK97] T. Kamae and M. Keane. A simple proof of the ratio ergodic theorem. Osaka J.Math, 34(3):653–657, 1997.

[KMS11] M. Keßeböhmer, S. Munday, and B. O. Stratmann. Strong renewal theoremsand Lyapunov spectra for α-Farey and α-Lüroth systems. to appear in ErgodicTheory and Dynamical Systems, 2011.

[Koo31] B. O. Koopman. Hamiltonian Systems and Transformations in Hilbert Space.Proceedings of the National Academy of Sciences, 17(5):315–318, may 1931.

[Kre05] U. Krengel. Einführung in die Wahrscheinlichkeitstheorie und Statistik. ViewegStudium, Wiesbaden, Germany, 8th edition, 2005.

[KS] M. Keßeböhmer and B. O. Stratmann. On the Lebesgue measure of sum-levelsets for continued fractions. to appear in Discrete and Continuous DynamicalSystems.

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REFERENCES REFERENCES

[KS07] M. Keßeböhmer and M. Slassi. Limit laws for distorted critical return timeprocesses in infinite ergodic theory. Stochastics and Dynamics, 7(1):103–121,2007.

[KS08] M. Keßeböhmer and M. Slassi. A distributional limit law for the continuedfraction digit sum. Mathematische Nachrichten, 281(9):1294–1306, 2008.

[KS10] M. Keßeböhmer and B. O. Stratmann. A dichotomy between uniform distribu-tions of the Stern-Brocot and the Farey sequence. preprint: arXiv: 1009.1823,2010.

[LM95] A. Lasota and M. C. Mackey. Chaos, Fractals, and Noise Stochastic Aspects ofDynamics. Springer, New York, USA [et.al.], second edition, 1995.

[Sar02] O. Sarig. Subexponential decay of correlations. Inventiones mathematicae,150:629–653, 2002.

[Wal82] P. Walters. An introduction to Ergodig Theory. Springer, New York, USA [et.al.], 1982.

[Zwe04] R. Zweimüller. Hopf’s ratio ergodic theorem by inducing. Colloquium Mathe-maticum, 101(2):289–292, 2004.

[Zwe09] R. Zweimüller. Surrey notes on infinite ergodic theory. 2009.

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Danksagung. Bedanken möchte ich mich bei meinen Eltern, sowie deren Lebensgefährten,die mich während meines Studiums nicht nur finanziell immer unterstützt haben.Ebenso geht ein herzlicher Dank an Tanja Schindler, Stephan Gessner und meinen VaterKarl Kautzsch, die diese Arbeit Korrektur gelesen haben.Bernd Otto Stratman gebührt ein herzlicher Dank, da er sich als Zweitgutachter zur Ver-fügung gestellt hat.Ganz besonders bedanke ich mich bei meinem Professor Marc Keßeböhmer; für die interes-sante Aufgabenstellung, die hervorragende Betreuung, die immer offene Tür und die vielenintensiven Gespräche, die diese Arbeit in dieser Form möglich gemacht haben.Vielen herzlichen Dank!

Bremen im Februar 2011,Johannes Kautzsch

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