A Z2-CONSTRUCTION OF A K-AUTOMORPHISM THAT COMMUTES WITH A RANK-1TRANSFORMATION

By

NICHOLAS SHARPE

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2016

c⃝ 2016 Nicholas Sharpe

I dedicate this to everyone.

ACKNOWLEDGMENTS

Thanks for all the help I have received in studying and writing this dissertation, from

my advisor Jonathan King, and others in the Ergodic Theory community: Doctors Benji

Weiss, Michael Keane, Aimee Johnson, and Steve Kalikow. Thanks to the faculty and

staff at the University of Florida Mathematics Department including Rick Smith and Jean

Larson, in addition to the continued Teaching Assistantship appointments that made

this work possible. Finally, I say thank you to the internet, for providing around-the-clock

LaTeX help, and all those who have contributed to the trove of online information.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

CHAPTER

1 DEFINITIONS AND THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1 Introduction to Measurable Dynamical Systems . . . . . . . . . . . . . . . 131.1.1 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.2 Isomorphism In Ergodic Theory . . . . . . . . . . . . . . . . . . . . 151.1.3 Partitions, Part 1/3: Basics . . . . . . . . . . . . . . . . . . . . . . . 161.1.4 Symbolic Shift Space . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.1 Entropy H of a Partition . . . . . . . . . . . . . . . . . . . . . . . . 211.2.2 Example: Calculating Entropy (of a Partition) . . . . . . . . . . . . 211.2.3 Entropy h(T ,P) of a Transformation w/r/t/ a Partition . . . . . . . . 231.2.4 Entropy h of a Transformation . . . . . . . . . . . . . . . . . . . . . 23

1.3 Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.1 Stationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.2 Partitions, Part 2/3: P-names . . . . . . . . . . . . . . . . . . . . . 25

1.4 Words and Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.4.1 Words Over The Alphabet A = [1..z ] . . . . . . . . . . . . . . . . . 271.4.2 Rank-1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.5 Cutting and Stacking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.5.1 Chacon’s Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.5.2 Alternative Picture of Chacon’s Map . . . . . . . . . . . . . . . . . 321.5.3 Chacon’s Map Revisited as a Symbolic Shift System . . . . . . . . 37

1.6 The K Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.6.1 Partitions Part 3/3: Independence and Autonomy . . . . . . . . . . 371.6.2 K -Automorphism Definition . . . . . . . . . . . . . . . . . . . . . . 391.6.3 Example of a K-Automorphism . . . . . . . . . . . . . . . . . . . . 39

1.7 Projective Limits and Ergodic Properties . . . . . . . . . . . . . . . . . . . 401.8 Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2 THE CONSTRUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.1 Setup and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2 The Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.1 Stage 1: Horizontal Staircase Concatenation . . . . . . . . . . . . 43

5

2.2.2 Stage 2: Vertical Concatenation with Shifts . . . . . . . . . . . . . 442.2.3 Stage 3: Concatenating the Pages . . . . . . . . . . . . . . . . . . 47

2.3 Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.4 Addressing Added Spacer . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.5 Defining R(n) and U(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 PROVING RANK-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1 Band One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Band b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 PROVING THE K PROPERTY . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1 Band One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Band b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 APPLYING THE WEAK-CLOSURE THEOREM . . . . . . . . . . . . . . . . . . 63

APPENDIX

A LEMMATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

B LEBESGUE SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6

LIST OF FIGURES

Figure page

1-1 Stage One of Chacon’s Transformation: Cutting. . . . . . . . . . . . . . . . . . 28

1-2 Stage One of Chacon’s Transformation: Stacking. . . . . . . . . . . . . . . . . 29

1-3 Stage Two of Chacon’s Transformation: Cutting and Stacking. . . . . . . . . . . 31

1-4 Stage n of Chacon’s Transformation. . . . . . . . . . . . . . . . . . . . . . . . . 31

1-5 Alternative Chacon: Stage 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1-6 Alternative Chacon: Stage 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1-7 Alternative Chacon: Stage 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1-8 Alternative Chacon: Stage 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1-9 Alternative Chacon: Stage 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1-10 Alternative Chacon: Stage ∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1-11 A Third View of Chacon’s Transformation . . . . . . . . . . . . . . . . . . . . . 36

2-1 Cutting and Stacking Legend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2-2 The 0-block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2-3 Horizontal Staircase Concatenation. . . . . . . . . . . . . . . . . . . . . . . . . 43

2-4 An Infinite Staircase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2-5 Two Staircases, Stacked with a Shift. . . . . . . . . . . . . . . . . . . . . . . . . 45

2-6 Vertical Concatenation, with Shifts. . . . . . . . . . . . . . . . . . . . . . . . . . 45

2-7 Cutting Out a Page. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2-8 A Page, with Added Shift Spacer. . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2-9 The (n + 1)-block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2-10 An n-Page Inside the (n + 1)-block. . . . . . . . . . . . . . . . . . . . . . . . . . 49

5-1 Shifting Left by W(n) units Equals one Shift Downward. . . . . . . . . . . . . . 64

5-2 Shifting Left by W(n+1) units, and Right by W(n) units. . . . . . . . . . . . . . . 65

7

LIST OF SYMBOLS, NOMENCLATURE, OR ABBREVIATIONS

[1..∞) = Z+ the positive integers {1, 2, ...}.

Z2 = Z× Z the 2D plane of integer coordinates, the coordinate systemof time values for Z2 symbolic shift space.

[t1..t2] an integer interval, the time values from t1 to t2, inclusive:{t ∈ Z|tt 6 t 6 t2}. Here t1, t2 ∈ Z, with t1 6 t2.

[t1..t2) a left-closed, right-open integer interval, the time valuesfrom t1 to t2: {t ∈ Z|tt 6 t < t2}. Here t1, t2 ∈ Z, with t1 < t2.

z ∈ [2..∞) the number of letters in the alphabet, the letter z alwaysstands for a positive integer, 2 or larger.

A = [1..z ] the alphabet, over which words and symbolic shift space aredefined.

ω = (ω1,ω2, ...,ωℓ) a word over the alphabet A. Here each ωi∈ A. Word ω is afinite sequence of length ℓ comprised of members of A.

|ω| the length of the word ω, the non-negative integer ℓ shownin the line above.

ω n ζ the left-to-right concatenation of the word ω with the word ζ.For example, (3, 1, 4)n(1, 5)=(3, 1, 4, 1, 5).

X a set where a sigma-field, measure, and transformation areor will be defined. For example X = [1..z ]Z

d

, where d ∈ [1..2].

� a (complete) sigma-field (or sigma-algebra), or simply fieldof subsets of X .

µ a non-atomic probability measure defined on �.

X the triple (X , �,µ), a standard Lebesgue probability space,called a space.

(T : X) the quadruple (X , �,µ,T ), a space X with a measure-preserving transformation T defined on X , called a systemor simply a transformation.

µ|E the measure µ conditioned on the (positive-mass) set E∈�,given by µ|E(B) := µ(B∩E)

µ(E).

8

⊗Z P([1..z ]) the sigma-field generated by cylinder sets in the space

X = [1..z ]Z, where P([1..z ]) means the set of all subsets of [1..z ].

P,Q both denote partitions of X , usually written P=(α1, ...,αz). Theα1, ...,αz are referred to as the atoms of P.

x [0..∞) the forward P-name of x∈X , a singly-infinite sequence of thecoordinates of x∈X w/r/t P: (x0, x1, ...) where xt=j if P=(α1,...,αz)and x ∈ T−t(αj).

x(−∞..∞) the (full) P-name of x∈X , a bi-infinite sequence of coordinates ofx∈X w/r/t P: (...x−1, x0, x1, ...) where xt=j if x∈T−t(αj .)

x [t1..t2] the time t1 to time t2 P-name of x∈X , a word, w , of length 1+t2−t1over the alphabet A, consisting of (consecutive) coordinates of xw/r/t P. So w=(xt1, .., xt2) , where xt=j if x∈T−t(αj .)

x [t1..t2) the left-closed, right-open time t1 to time t2 P-name of x∈X ,a word, w , of length t2−t1 over the alphabet A, consisting of(consecutive) coordinates of x w/r/t P. So w=(xt1, .., xt2−1) , wherext=j if x∈T−t(αj .)

x [0] the time-zero coordinate value of x∈X w/r/t P: x [0]=j ifP=(α1,...,αz) and x∈αj .

x [t] the time t coordinate value of x∈X : x [t]=j if P=(α1,...,αz) andx∈T−t(αj .).

P 4 Q indicates that the partition P is refined by the partition Q.

P ⊥ Q indicates that partition P is independent of partition Q.

P ⊥εQ indicates that partition P is ε-independent over the partition Q. Hereε ∈ (0, 1).

P fεQ indicates that partitions P and Q are ε-autonomous. Here ε ∈ (0, 1).

P |E the partition P conditioned on the set E .

H(P) the entropy of the partition P.

h(T ,P) the entropy of the transformation T w/r/t the partition P.

h(T ) the entropy of the transformation T .

9

n ∈ Z+ represents a stage during a cutting and stackingconstruction.

B(n) the n-block, a finite matrix over A, constructed at stage n

during a cutting and stacking construction.

H(n) ∈ Z+ the height of the unique n-block B(n) in the Z2 construction.

W (n) ∈ Z+ the width of the unique n-block B(n) in the Z2 construction.

U(n) ∈ Z+ the up number, for the n-block B(n) in the Z2 construction,influences the growth of H(n).

R(n) ∈ Z+ the repeat number or “right number ”, for the n-block B(n)in the Z2 construction, influences the growth of W (n).

� the set of all “random walk” words π during stage n of the Z2

construction, uniquely identifying each Page.

Xb :=([0..1]b

)Z width-b shift space, viewed horizontally in Ch3, andvertically in Ch4.

�b :=⊗

Z P([0..1]b

)a width-b factor field of �.

µb := µ|�bthe measure µ conditioned on the width-b factor field.

Xb the triple (Xb, �b,µb).

a gray rectangle, stands for the symbol ‘0’, used for imagesduring cutting and stacking constructions.

a white rectangle, stands for the symbol ‘1’, used forimages during cutting and stacking constructions.

a light blue Rectangle, also stands for the symbol ‘1’. Thedifferent color is used to distinguish the added spacerduring stage n of a cutting and stacking construction fromall spacer which was added at a previous stage.

a green rectangle, stands for the symbol ‘2’, used tolabel the n-block B(n) during stage n of a cutting andstacking construction.

10

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

A Z2-CONSTRUCTION OF A K-AUTOMORPHISM THAT COMMUTES WITH A RANK-1TRANSFORMATION

By

Nicholas Sharpe

May 2016

Chair: Douglas CenzerMajor: Mathematics

In the field of Mathematics known as Dynamical Systems, where transformations

of ’spaces’ are studied, the sub-field of Ergodic Theory is concerned with the study

of measure-preserving maps of (non-atomic) Lebesgue Probability spaces. Some

Descriptive Set Theory results by Matt Foreman and others indicate that these maps

and many of their sub-classes of maps will not be classified(5), yet some special

families of maps have been classified(7). While the K -Automorphisms have proved

difficult to classify(8), this dissertation develops the relationship between a finite entropy

K -Automorphism and a Rank-1 transformation, which are shown to have a surprising

connection with one another- the K -Automorphism arises as a limit of a sequence of

powers of the Rank-1 transformation.

K -Automorphisms are transformations with random behavior, while Rank-1

transformations have a deterministic asymptotic behavior. This is made precise with

the notion of entropy, a function which assigns a non-negative, extended real number

to each transformation. While every K -Automorphism must have positive entropy(2),

all Rank-1 transformations must have zero entropy. Entropy is a ’weak invariant’ for

isomorphism, meaning that if two transformations or ’systems’ are isomorphic, then

it follows that they must have the same entropy. Thus, as one might expect, it is not

possible for a K -Automorphism to be isomorphic to a Rank-1 transformation. There is,

11

however, some common ground among these two classes of transformations, and this

dissertation constructs two maps which highlight this relationship.

In Chapter 2, two transformations S and T are constructed, each as a sub-action

of a larger Z2 system that acts on {0, 1, 2}Z2. S and T themselves are Z actions, or

one-dimensional symbolic shift spaces. As a natural consequence of S and T being the

horizontal and vertical sub-actions of the larger system, they commute with each other.

In Chapter 3, it is proven that the horizontal action S is a Rank-1 transformation, and in

Chapter 4 it is proven that the vertical action is a K -Automorphism. Formally:

For any Lebesgue space X, there exists measure-preserving transformations

S ,T : X → X such that

• S is a Rank-1 transformation,

• T is a K -Automorphism, and

• S ◦ T = T ◦ S .

12

CHAPTER 1DEFINITIONS AND THEORY

1.1 Introduction to Measurable Dynamical Systems

Measure-preserving transformations on Lebesgue probability spaces are the focus

of Ergodic Theory. Here are the basic definitions along with a brief discussion of these

concepts. Throughout the entire document, z is always a positive integer, 2 or larger. In

addition, the symbol A will always stand for the set [1..z ]. Refer to the Notation Section

for a quick reference of definitions of the symbols used in this document.

DEFINITION A (Lebesgue) probability space, hereafter referred to as a space,

is a probability space X := (X , �,µ) with the additional assumption that X is a

Lebesgue space (see Appendix B). Here X is a set, � a (complete) sigma-field (or

“sigma-algebra”), and µ is a non-atomic probability measure defined on �. non-

atomic means

∀x ∈ X : µ({x}) = 0. (1–1)

Probability measure on � means that µ is a measure on � with

µ(X ) = 1. (1–2)

Lebesgue is complicated (2).

DEFINITION A measure-preserving transformation, or simply a transformation,

of a space X, written (T : X), also called a measure-preserving function, is an

invertible function

T : X → X , (1–3)

satisfying the measure-preserving property

∀E ∈ � : µ(E) = µ(T−1(E)).

13

The pair (T : X), or equivalently the quadruple (X , �,µ,T ), consists of the space X

together with the transformation T and is referred to as the system or even simply the

transformation when the space (X , �,µ) is understood.

1.1.1 Ergodicity

DEFINITION A system (X , �,µ,T ) is ergodic if

∀E ∈ � : T−1(E) ⊂ E ⇒ µ(E) ∈ {0, 1}. (1–4)

If T is an ergodic transformation of a probability space, then X has no sets with

measure strictly between 0 and 1 which are invariant under T . Said differently, If X

admits an invariant set for T , that set must either be of full or null measure.

Ergodic transformations are the basic objects of study in Ergodic Theory: As it

turns out, on Lebesgue spaces, a general (non-Ergodic) transformation T decomposes

(11) into a collection of Ergodic components, and so by studying an arbitrary ergodic

transformation, we are studying the building blocks of all measure-preserving

transformations. This is somewhat analogous to primes in Number Theory, or simple

groups in Group Theory.

The Bernoulli shifts form a fundamental class of examples of symbolic Ergodic

systems. They have positive entropy and are “completely non-deterministic”, in a

more extreme sense than K -Automorphisms. The Bernoulli shifts admit a generating

“time-zero partition” P (see §1.1.4, p.19) which is moved completely independently:

Under their time-zero symbol partition (defined below), conditioning on a set of

coordinate positions for points in the space (conditioning on a cylinder set) yields no

change in the probabilistic distribution (§1.1.3, p.16) of other coordinate positions

(cylinder sets) in the space.

14

The K -Automorphisms also form a class of Ergodic transformations, which (a more

general notion than the Bernoulli shifts) only require that asymptotically no information is

retained, after the coordinates are assuredly far enough apart (§1.6.2, p.39).

1.1.2 Isomorphism In Ergodic Theory

DEFINITION Transformation (T2 : X2) is a factor of (T1 : X1) if there exists a

measurable, measure-preserving ϕ: X1 → X2 such that there exists a measurable subset

E ⊂ X1 with µ1(E) = 1 and ∀x ∈ E :

ϕ ◦ T1 = T2 ◦ ϕ, (1–5)

or equivalently, the following diagram is commutative:

T1

X1 −→ X1

ϕ ↓ ↓ ϕ

X2 −→ X2

T2 .

The map ϕ is called a factor map.

DEFINITION Transformation T2 is isomorphic to T1 if T2 is a factor of T1 with the

additional conditions:

• the factor map ϕ is one-to-one up to a null set,

• ϕ−1 is a measure-preserving function.

DEFINITION Fix a non-atomic Lebesgue probability space X:=(X , �,µ). Define the

set � to be the collection of all invertible transformations on X:

� := {T : X → X∣∣T is invertible and T ,T−1 both preserve µ} . (1–6)

15

DEFINITION An invariant for isomorphism in � is a set R together with a function

ι : � → R such that

∀T1,T2 ∈ �, T1∼= T2 ⇒ ι(T1) = ι(T2). (1–7)

DEFINITION A complete invariant for isomorphism in � is a set R together with a

function ι : � → R such that

∀T1,T2 ∈ �, T1∼= T2 ⇐⇒ ι(T1) = ι(T2). (1–8)

While entropy does not serve as a complete invariant for the larger class of

transformations known as the K -Automorphisms, entropy remains an important

consideration while analyzing relationships among transformations of any kind.

1.1.3 Partitions, Part 1/3: Basics

DEFINITION Let X be a space. A (finite) partition P of X is an ordered, finite

collection P = (α1, ...,αz) of measurable subsets of X such that

• X =∪z

j=1 αj ,

• ∀α, β ∈ P : α ̸= β ⇒ µ(α ∩ β) = 0.

The elements α ∈ P are called the atoms of the partition P.

DEFINITION The distribution of P is the probability vector

d(P) := (µ(α1),µ(α2), ...µ(αz)) ∈ [0, 1]z . (1–9)

DEFINITION Let X be a space, and consider a partition P of X. For a set E ∈ �, we

form the partition P conditioned on the set E , written P |E , given by

P |E := {α ∩ E | α ∈ P} . (1–10)

Note that P |E is not a partition of the set X , but a partition of the set E .

16

DEFINITION Consider two partitions P,Q of X. Say that P is refined by Q, written

P 4 Q, (1–11)

if ∀α ∈ P, ∃β1, ...,βk ∈ Q such that

αa.e.=

k∪i=1

βi . (1–12)

When considering two partitions of the same space together, they are each refined

by their join.

DEFINITION The join of two partitions P and Q of the same space X is a third

partition:

P ∨Q := {α ∩ β |α ∈ P, β ∈ Q} . (1–13)

Of course the empty set often appears in the above set, we tacitly remove it. The join

of two partitions is the minimal common refinement of the two partitions involved in the

join.

DEFINITION Let (T : X) be a system, and consider P = (α1, ...,αz), a partition of

X. For any k ∈ Z+, the partition P is pushed k times by T to produce another partition

T−k(P) of X:

T−k(P) :=(T−k(α1),T

−k(α2), ...,T−k(αz)

). (1–14)

Consider a transformation (T : X). As T is measure-preserving, P and T−k(P)

have identical distributions, ∀k ∈ Z. These translated versions of P are joined together to

produce common refinements:

k2∨i=k1

T−i(P) = T−k1(P) ∨ T 1−k1(P) ∨ T 2−k1(P) ∨ ... ∨ T−k2(P). (1–15)

Note that if [k1..k2] ⊂ [t1..t2] then

k2∨i=k1

T−i(P) 4t2∨i=t1

T−i(P). (1–16)

17

With this collection of refining joins of translations by T , the collection of partitions∨k2i=k1

T−i(P) has a limiting smallest sigma-field which contains each of these “finitely

joined sigma fields”.

DEFINITION Let P be a partition of a system (T : X). Say that P generates �

under T , or simply “P generates” if � is the smallest complete sigma-field which

contains all of the finite sigma-fields{k∨

i=−k

T−i(P)

∣∣∣∣∣ k ∈ Z+

}. (1–17)

A sigma-field generated as above has the property of being T -invariant:

DEFINITION Let (T : X) be a system and suppose �0 is a sigma-field on X. Say

that �0 is T -invariant if

∀E ∈ �0 : T−1(E) ∈ �0. (1–18)

1.1.4 Symbolic Shift Space

First we define a Bernoulli shift, and then move on to consider more general shifts,

which arise naturally from a partition P and transformation (T : X).

DEFINITION A probability vector is a finite tuple p⃗ := (a1, a2, ..., az) such that

• ∀j ∈ [1..z ] : aj > 0, and

•∑z

j=1 aj = 1.

DEFINITION Given a probability vector p⃗, the Bernoulli shift associated to p⃗ is the

measure space (X , �,µ), with

Set X := [1..z ]Z, and field � :=⊗Z

P([1..z ]), with measure µ :=⊗Z

(µ0), (1–19)

where P([1..z ]) denotes the set of all subsets of [1..z ], and µ0 is the probability measure

on the finite set [1..z ] given by p⃗:

∀j ∈ [1..z ] : µ0(j) = aj . (1–20)

18

More general than a Bernoulli shift but with the same measure space (X, �), we

consider for any z ∈ [2..∞), a general shift-invariant non-atomic Lebesgue probability

measure. These arise naturally as processes given any partition of a system (T : X)

(§1.3, p.24).

DEFINITION For z ∈ [2..∞), consider the measure space (X , �,µ), with

X := [1..z ]Z, and field � :=⊗Z

P([1..z ]), with measure µ, (1–21)

such that µ is a shift-invariant non-atomic Lebesgue probability measure on �, and

T : X → X the left shift. Such a system (T : X) is called a symbolic shift space.

Elements of X are typically denoted as

X = [1..z ]Z =

(

· · · x−1 x0 x1 · · ·) ∣∣∣∣∣∣∣

For each t ∈ Z,

value xt is in [1..z ],

. (1–22)

NOTATION Let X := [1..z ]Z, and consider a partition P of X. For all x ∈ X, the

following is used to denote various subsets of coordinates of x w/r/t P. ∀t ∈ Z and

∀t1 < t2 ∈ Z,

x [t] := xt , (1–23)

x [t1..t2] := (xt1, xt1+1, ..., xt2) , (1–24)

x [t1..t2) := (xt1, xt1+1, ..., xt2−1) , (1–25)

x [t1..∞) := (xt1, xt1+1, ...) , and (1–26)

x(−∞..t2] := (..., xt2−1, xt2) . (1–27)

The above notions of considering subsets of coordinates of the points x ∈ X extend

to an abstract space (T : X) where we consider a partition P of X which gives rise to

P-names of points in the (T ,P) process. These P names are precisely the lists of

coordinates seen above. This is discussed further in the P-names section, §1.3.2 (p.25).

19

DEFINITION Let X := [1..z ]Z. Consider a finite set of distinct integers

J = {j1, ..., jm} ⊂ Z, and a word C = (C1, ...,Cm) over A. The cylinder set corre-

sponding to J at C is

{x ∈ X | ∀k ∈ [1..m], x [jk ] = Ck}. (1–28)

As the cylinder sets generate the (usual, Borel) sigma-field � on X , any measure

defined on � is determined by its values on the cylinder sets. This important fact allows

the (T ,P) process to be formally viewed as a symbolic shift space (see §1.3, p.24).

DEFINITION A commonly used partition of a symbolic shift space is the time-zero

symbol partition P, defined as follows:

P := (α1,α2, ...,αz) , where αj := {x ∈ X | x [0] = j}. (1–29)

1.2 Entropy

The word entropy is used in Mathematics for several related notions. With a system

and a transformation in mind, the definition and discussion of the transformation’s

entropy must be preceded by the entropy of the transformation with respect to a partition

P of X , which in turn must be preceded by the entropy of a partition itself.

To give an idea of how entropy is defined through familiar notions, the process

of defining entropy is analogous with the process of defining the Lebesgue Integral:

Partitions play a similar role as the simple functions- while the entropy of a partition

is similar to the integral of a simple function. The entropy of a transformation w/r/t

a partition is like the Lebesgue Integral of a general function taken over a proper

sigma-field of the whole measure space. The entropy of a transformation is like the

Lebesgue Integral of a general (measurable) function across the entire measure space.

20

1.2.1 Entropy H of a Partition

DEFINITION Let P be a partition of the probability space X. The entropy of the

partition P is the non-negative real number

H(P) :=∑α∈P

µ(α) · log(

1

µ(α)

), (1–30)

where, by convention, we extend by continuity to define

0 · log(1

0

)= 0. (1–31)

The quantity H(P) is also referred to by some Ergodic Theorists as “distribution

entropy”, or the distropy of the partition P, in order to avoid confusion with the quantities

h(T ,P) and h(T ) which are separately defined in the next definitions.

This H(P) has no reference to a particular transformation T , just the space X itself

and P. To discuss the properties of the H function, note that as P = (α1, ...,αz) is the

variable for the function H, we explore various kinds of partitions P. Restricting H to only

partitions with exactly z atoms, H takes on its maximum value of log(z) only when P

consists of atoms which all have the same measure of 1z. On the other hand, if one or

few of the atoms in the partition P is/are much larger than the others (‘much larger’ in

a relative sense), then the quantity H(P) will be small. An example of this is calculated

below.

1.2.2 Example: Calculating Entropy (of a Partition)

EXAMPLE Let X be a space. For each K ∈ [1..∞), put

P(K) = (α1, ...,αz), with µ(α1) =z · K + 1− z

z · Kand (1–32)

for j = 2, 3, ..., z , define µ(αi) =1

z · K. (1–33)

21

We do not specify what X is: Any space will do. In a non-atomic Lebesgue space,

such a partition always exists (2). Further, we do not bother with how each atom αi is

defined, just that their measures are as given above. The way the partition actually sits

in the space is irrelevant1 : Only d(P) = (µ(α1), ...,µ(αz)) , the distribution of P, (§1.1.3,

p.16) affects the value of the function H when applied to P.

Note in (1–32) and (1–33) that for each K > 1, the atom α1 has a lot more mass

than all of the other equally-weighted atoms. We take K larger and larger to find that the

mass of the first atom α1 goes to 1 while the mass of each other atom is headed to 0.

We calculate H (P(k)) to be

H (P(k)) =z · K + 1− z

z · Klog

(z · K

z · K + 1− z

)+

z − 1

z · Klog (z · K) . (1–34)

The first term corresponds to α1 while the second term is the sum of the terms

corresponding to the other z − 1 atoms. Note that z is fixed here while we vary K , and

both of the terms in the above sum evidently approach 0 as K → ∞.

Function H quantifies how evenly distributed the mass of the atoms of P is across

the measure space, so that H(P) near zero is synonymous with P having a small

number of large-mass atoms, while H(P) near its maximum value of log(z) corresponds

to P consisting of atoms with roughly evenly-distributed mass. Note that as z increases,

(and correspondingly, P becomes a partition which has more atoms) H(P) has the

potential to grow as large as log(z), and thus in any space X, H can become unbounded

when we consider partitions with (finite, but) arbitrarily many atoms.

This example shows how one thinks about and calculates the entropy of a partition

inside a measure space. But this is merely the entropy of the partition P that was

1 When we later introduce a transformation T , the way that P sits in X becomesimportant.

22

calculated, not the entropy of an actual transformation T . For this one must consider

how T moves the atoms of the partition about the space.

1.2.3 Entropy h(T ,P) of a Transformation w/r/t/ a Partition

With H defined we may now define the entropy with respect to a partition. This is

not yet the entropy of T , but it is working toward that definition which follows afterward.

DEFINITION The entropy of a transformation w/r/t a Partition P is the limit

h(T ,P) := limℓ→∞

1

ℓH

∨j∈[1..ℓ]

T j−1(P)

, (1–35)

which always exists. (2; 11; 9)

Note that the join in the above equation has z ℓ many atoms, so that ℓ log(z) is an

upper bound for H, therefore h(T ,P) 6 log(z).

The limit (1–35) allows for upper and lower bounds on the measure of P-names of

large length: In the following equation, let h denote h(T ,P). Given ε > 0, ∃ t ∈ Z+ such

that ∀ℓ > t:

ℓ [h− ε] < H

(ℓ∨1

T j−1 (P)

)< ℓ [h+ ε] . (1–36)

This bound reveals useful information about how the masses of P-names behave as we

consider longer and longer P-names.

1.2.4 Entropy h of a Transformation

DEFINITION The entropy of transformation T is the value

h(T ) := sup {h(T ,P) | P a partition of X} . (1–37)

If two different systems are known to be isomorphic, then they must have the same

entropy . While the converse is false, it becomes true when we restrict the class of

23

transformations to Bernoulli shifts: Ornstein showed (7) in the 1970s that two Bernoulli

shifts which have the same entropy are isomorphic .

1.3 Processes

DEFINITION A random variable over the alphabet A is (11) a probability space

(Y , �, ρ), together with a measurable function ψ : Y → A. Often the measure ρ is not

mentioned explicitly, instead one refers to such expressions as

Prob {ψ(y) = a} instead of ρ(ψ−1(a)). (1–38)

1.3.1 Stationary Processes

DEFINITION A (two-sided, stationary) process is (11) a bi-infinite sequence

of random variables (ψk)k∈Z with each of the functions defined on the same measure

space (Y , �, ρ), such that the stationarity condition is met: ∀C1, ...,Cm ∈ [1..z ] :

∀t ∈ Z : Prob {ψt+k(y) = Ck | k ∈ [1..m]}is constant in

the translation amount t.(1–39)

Given a process, one constructs the system S : (X , �,µ), where S is the left-shift

and

X := [1..z ]Z, with field � :=⊗Z

P([1..z ]), (1–40)

The measure µ is defined on cylinder sets (§1.1.4, p.20) as follows.

Consider J = {j1, ..., jm} ⊂ Z, and C1, ...,Cm ∈ [1..z ], and

E := {x ∈ X | xj1 = C1, ..., xjk = Ck}. Then (1–41)

µ(E) := Prob{ψj1 = C1, ...,ψjk = Ck}. (1–42)

24

Under the above identification of processes with symbolic shift spaces, processes

are the same as symbolic shift spaces as previously defined (§1.1.4, p.19).

On the other hand, given a transformation, one constructs many processes; one for

every partition. Fix a non-trivial partition P = (α1, ...,αz) of system (X , �,µ,T ). This

partition encodes a factor of the system as a process in the following way:

(Y , �, ρ) := (X , �,µ), and ∀k ∈ Z : ψk(y) := y [k ]. (1–43)

This is known as the (T,P)-process.

The pair (T ,P) is a lot more than just an input for the entropy function h. A process

can be identified as a (complete) shift-invariant measure on a finite alphabet symbolic

shift space, say a symbolic shift system “(S : Y)”. This (S : Y) is a factor of the original

(T : X). If P generates (§1.1.3, p.18), then in fact T and S are isomorphic systems.

Indeed: If h(T ) < log(z), and T is ergodic, then the Krieger Generator Theorem (§1.3.2,

p.26) guarantees that a partition P exists with no more than z atoms, such that system

(T : X) and system (S : Y) ∼= (T ,P) are isomorphic under the factor map that takes

points to P-names. This is formalized below.

1.3.2 Partitions, Part 2/3: P-names

DEFINITION Let (T : X) be a system, and P = (α1, ...,αz) a partition of X. Define

π : X → [1..z ]Z by

π(x) := (..., x−1, x0, x1, ...) , where xt = j if x ∈ T−t(αj), (1–44)

called the P-name of the point x ∈ X . This π(x) is also denoted by x(−∞..∞). In

addition, the following notational shorthands are used, which extend those put in place

on p.19 to an arbitrary system, along with a partition of that system. ∀t ∈ Z and

∀t1 < t2 ∈ Z:

x [t] := xt , (1–45)

25

x [t1..t2] := (xt1, xt1+1, ..., xt2) , (1–46)

x [t1..t2) := (xt1, xt1+1, ..., xt2−1) , (1–47)

x [t1..∞) := (xt1, xt1+1, ...) , and (1–48)

x(−∞..t2] := (..., xt2−1, xt2) . (1–49)

When there are two partitions in consideration, say P and Q, then the notation

xP [0..∞) and xQ [0..∞) distinguishes between P-names from Q-names. Analogously,

xP(−∞..∞) and xQ(−∞..∞)are used for the infinite P-name of x and the bi-infinite

P-name of x respectively.

The P-name mapping π is a measure-preserving Factor Map which carries (T : X)

to the (T ,P) process. This (T ,P) process is represented as a symbolic shift space

(S : Y). On (S : Y), the action of T (as well as the sigma-field and measure) has

been conjugated to the shift map with measures on cylinder sets provided by the

corresponding measure of the points x ∈ X moved under T according to those cylinder

sets.

KRIEGER GENERATOR THEOREM Let (T : X) be an ergodic system with h(T ) <∞.

If z ∈ [2..∞) is such that h(T ) < log(z), then there exists a generating partition P of

(T : X) such that, with S = (Y , �, ν) denoting the symbolic shift space associated to the

(T ,P) process, then,

(T : X) ∼= (S : Y) , (1–50)

under the Factor Map π that sends points to P-names (§1.3.2, p.25).

By virtue of Krieger’s Generator Theorem, so long as h(T ) < ∞ and T is ergodic,

the system (T : X) is isomorphic to a shift map S on the space [1..z ]Z induced by

the partition P. Various partitions produce factor processes of T , i.e. factor shift

transformations. Under the assumption that T is ergodic and that P is finite and also a

generating partition, the original system T is isomorphic to the shift map S under the

map that sends points to P-names.

26

1.4 Words and Names

1.4.1 Words Over The Alphabet A = [1..z ]

DEFINITION A word ω over the set A is a finite sequence of elements from A:

ω = (ω1,ω2, ..,ωℓ) , where each ωi ∈ A. (1–51)

The non-negative integer ℓ is the length of ω, often written as ℓ = |ω|.

DEFINITION Consider two words η = (η1, ..., ηk) and ζ = (ζ1, ..., ζℓ) over a common

alphabet A. The concatenation of η and ζ, written η n ζ, is

η n ζ := (η1, η2, ..., ηk , ζ1, ζ2, ..., ζℓ) . (1–52)

DEFINITION Let ω and ζ be two words over the set A such that |ζ| > |ω|. Let

ε ∈ (0, 1). Say that ζ is (1-ε)-packable by ω if ∃J ∈ [1..∞) and (possibly empty) words

s(0), s(1), ..., s(J) over A such that

ζ = s(0)n ω n s(1)n ...ω n s(J), with

∑J

i=0 |s(i)||ζ|

< ε. (1–53)

In other words, ζ is (1-ε)-packable by ω if more than (1-ε) of the coordinates of ζ are

occupied by disjoint copies of ω, each separated by a (possibly empty) string.

1.4.2 Rank-1 Definition

DEFINITION A system (X , �,µ,T ) is Rank-1 if ∃P, a generating partition of X,

such that ∀ε > 0, ∃t ∈ Z+, a repetition word ω, and a set G ∈ � with µ(G) > 1− ε, such

that:

∀x ∈ G ,∀ℓ > t : The length-ℓ P-name of x is (1− ε)-packable by ω. (1–54)

27

Reference (4) has other equivalent forms of Rank-1; the one presented here is used for

the proofs in chapter 3. The length-ℓ P-name of x is x [0..ℓ), defined in (1–47).

1.5 Cutting and Stacking

In this section, we construct Chacon’s map (2) as an explicit self-map of the unit

interval.

Chacon’s transformation is built by cutting and stacking (11; 1; 8; 10). Constructions

by cutting and stacking are iterative: At each stage a new piece of the real line is

adjoined to what is considered to be the whole space at that stage. In addition, a

consistent rule for how points are moved into the newly added “spacer” and how they

move out of the spacer. Each assigned rule for “moving points” will be a measure-

preserving operation by construction. The limit map will be a self-map defined on a finite

sub-interval of the real line which preserves Lebesgue measure.

The notion of the spacer is important for building a measurable partition of the

measure space which will bring out desirable properties in the transformation T . By

labeling part of the measure space as spacer, we are implicitly creating a partition that

consists of two atoms– one with those points who are designated inside a spacer set,

and then its complement. The resulting partition of the measure space (in the limit)

generates the sigma-field of the measure space under the action of T .

1.5.1 Chacon’s Map

Throughout this subsection, let T denote Chacon’s map.

Figure 1-1. Stage One of Chacon’s Transformation: Cutting.

28

Begin with an interval of the real line: [0, a) for some a ∈ R+. Divide this interval into

three equal sub-intervals: [0, a3

), [ a

3, 2a

3

), and [2a

3, a). Next, introduce a new sub-interval,

[a, b1) for some b1 > a such that the length of [a, b1) is a3

(the same length as each

of the three subintervals). This newly-added sub-interval is called the added spacer

during this stage of the construction. Notice that the added spacer occupies(14

)1 of the

measure of the current whole space [0, b1). The light blue color indicates that the spacer

is newly-added at this stage in the construction, and will be white after this step and in

the limit.

Now we define a rule for T on part of the measure space that we have so far.

The limit map has an identical description on this part of the space, the difference

between the rule we will now give and the limit map is that we will leave part of the rule

undefined for some points in the space, for now. During continued stages of the iterative

construction, more of the space and map are defined. In the limit, a rule or map T is

defined for all points, excepting a nullset, in the limit measure space.

T is defined by visually stacking the sub-intervals atop one another to indicate

that T maps each interval to the one above it, and hence each point to the one directly

above it. As a consequence of this, each subinterval has the interval beneath it as it’s

T -pre-image.

Figure 1-2. Stage One of Chacon’s Transformation: Stacking.

Note that the action of cutting intervals and stacking them to define a mapping

only partially defines a rule for the thus-far-constructed space: We have not defined

where the top-interval points are mapped. The rule for these points will be defined

later, and in fact the top sub-interval will have points that travel to all far corners of the

measure space. Also note that the assigned mapping is one-to-one and hence invertible

29

everywhere with the exception of the bottom row of the stack (because no points have

yet been assigned to target this subinterval).

This completes stage one of the construction of Chacon’s transformation. Ornstein

called the object which has been constructed a “Gadget” (8) or, more specifically, a

“1-Gadget”, and consists of the following data:

DEFINITION For n ∈ Z+, an n−gadget is

• a subinterval In of R,

• a (partial2 )self-mapping Tn : In → In,

• a (finite) sigma-field �n with In ∈ �,

• Lebesgue measure on �n, and

• A Lebesgue-measurable partition Pn of In.

After one has defined a sequence of Gadgets, provided that the limiting sub-interval

limn→∞(In) is bounded, a finite T -invariant measure µ is produced for which the partition

P = limn→∞(Pn) is measurable (11); we normalize it to make it a probability measure.

Chacon Stage n + 1

The spacer is placed above the second of the three evenly-divided columns of the

n-block, as shown above. The three columns are then stacked together to produce a

taller (n + 1)-block, according to the rule

“Block1, Block2, Spacer, Block3 ‘’ as in

Block3

Spacer

Block2

Block1

. (1–55)

2 The domain of Tn is a proper subset of In, but is most of In; all, but the top level.

30

Let n ∈ Z+, and suppose that the n-Gadget has been constructed. Even though

the images below explicitly show the creation of the 2-Gadget from the 1-Gadget, the

reader will note at the end of the construction that the images provide an abstract stage

of the construction, with the 1-Gadget in the image taking the role of the n-Gadget and

the 2-Gadget representing the (n + 1)-Gadget.

Figure 1-3. Stage Two of Chacon’s Transformation: Cutting and Stacking.

Figure 1-4. Stage n of Chacon’s Transformation.

31

This process of cutting and stacking is continued indefinitely, and in the limit, a map

T is constructed that preserves Lebesgue measure on the unit interval

X :=

[0,

3a

2

)= [0, 1) , (1–56)

setting a = 23

to normalize. Let α0 := {x ∈ X |x < a} and α1 := {x ∈ X |x > a}. This

partition P = (α0,α1) generates the sigma-field on [0, 1) under the action of T . In fact,

‘α0’ and ‘α1’ stand for 0 and 1 here. This is made precise by applying the P-name map

π : X → Y , where Y is a symbolic shift space determined by the partition P. In the

construction, subintervals were labeled as gray or white, which creates the P-names

of points in the measure space w/r/t T . All points in α0 were in the original piece of the

measure space [0, a), whereas all points that lie in α1 were at some point adjoined to the

measure space as added spacer.

1.5.2 Alternative Picture of Chacon’s Map

There is another sequence of pictures of Chacon’s transformation, another

perspective, that allows one to see the whole constructed measure space at each

stage, along with the entire space in a limit photo. Unlike the previous construction

where the space [0, 1) was not consistently displayed in a fixed landscape, in this case

we can see the number line and the Lebesgue measure in action. This is an important

perspective for symbolic cutting and stacking, so that the measure is understood to be

equivalent to one on the real line yet abstracted away from any specific intervals on the

line. As before, we fix a ∈ R+ and begin by cutting the interval [0, a) into thirds:

Figure 1-5. Alternative Chacon: Stage 0.

32

We add a spacer interval atop the second interval, but without any other stacking:

The stacking is replaced with the description that each interval either maps to the one

above it, or if there is not an interval above it, then it maps to the bottom interval in the

adjacent column on the right side. Therefore, in Figure (1-6) and the figures that follow,

the interval below the newly-added spacer is mapped to the newly-added spacer, as

in the stacking that was done in the first discussion of Chacon’s map in the previous

section. The last interval on the right remains undefined for T at this stage, and the

first interval on the left has no pre-image for T at this stage. The green coloring is

used to denote where the map has been so-far defined on the measure space, and the

red where the map has yet to be defined. The temporary orange and black arrows in

the pictures indicate the newly defined mapping at this stage.This rule accomplishes

the same usual stacking as previously discussed in the last section, but allows for an

alternative perspective of the measure space.

Figure 1-6. Alternative Chacon: Stage 1.

Next, we divide the remaining red intervals into even thirds, and add a spacer

above the middle-third on the right-hand red zone as shown below. The mapping is now

extended on 23’s of the red portions from the previous stage: On the right-hand side,

the left third maps to the middle third on the left hand-side (on the opposite ‘end’ of the

measure space) . The middle third on the right-hand side maps to a small added spacer

above it, which is mapped to the right third on the left-hand side of the measure space.

Again, the orange and black arrows indicate this mapping.

33

Figure 1-7. Alternative Chacon: Stage 2.

The above arrows will be removed in subsequent images, with the understanding

that the mapping remains the same. Continuing, we “cut” again, and designate a new

mapping again on 23

of the remaining red section of the measure space exactly as in the

previous step3 .

Figure 1-8. Alternative Chacon: Stage 3.

The same pattern is continued inductively, here is one last depiction of the fourth

gadget.

In the limit, we now have a probability space with a sigma-field, measure,

transformation, and generating partition, all viewable in the following picture:

There is a way in which the two pictures can be united, which showcases the added

spacer becoming negligible while also offering another perspective of the measure

space and transformation. First, choose a large value of n ∈ Z+ and consider the

3 The difference though is that the red section has decreased in size by a factor of athird, which continues to grows exponentially small as we continue.

34

Figure 1-9. Alternative Chacon: Stage 4.

Figure 1-10. Alternative Chacon: Stage ∞.

n−Gadget as first described in the previous section. This Gadget consists of a single

tall stack of symbols (of zeros and ones). Inside this large “n”-stack one can find

(n + 1)-blocks and (n − 2)-blocks and all (n − k)-blocks for k 6 n. Even though it

would look quite tall and skinny, we stretch this so that it visually appears to be a large

rectangle as pictured in Figure 1-11.

This rectangle represents a large part of the measure space (n was taken large)

and we may choose a point in the measure space by selecting any location in this block

which corresponds to either a ‘0’ or a ‘1’ in a particular location along with neighboring

zeros and ones all the way up and down the block until the end of the n-block is reached.

These are partial P-names for points in the limiting measure space. Since we assume

n is large, we imagine the top and bottom of this “n” stack of symbols as being most of

the measure space in the precise sense that reaching the end (the top or bottom) of the

n-block is a low-probability event.

35

Of course, the measure space is more than just the n-block, there are all the

(n + k)-blocks to consider for k ∈ Z+. These are incorporated into the picture by slicing

the large rectangle into thirds vertically and adding a spacer interval4 atop the middle

strip as pictured below. The mapping goes (on the left-third of the n-block) upward to

the top of the stack and then from the left-third top to the middle-third bottom, all the

way up to the middle-third top and up into this new piece of spacer and then to the

bottom of the right-third. This extended mapping (extended beyond the n-block to the

top-left, top-middle, and to the newly added spacer) comprises the (n + 1)-Gadget or

“(n + 1)-block”.

Figure 1-11. A Third View of Chacon’s Transformation

This process is continued to produce the (n + 2)-block by merely adding another

spacer interval, of 13

the length of the previous atop the middle-third of the right-third of

the n-block, as pictured. The later (n + k)-block (k ∈ Z+) are each added by drawing a

shorter newly added spacer interval above the right side of the n-block to indicate the

mapping, as pictured in Figure 1-11.

4 Spacer interval may be thought of as either an actual sub-interval of R or as a singlesymbol ‘1’ depending on intended perspective.

36

1.5.3 Chacon’s Map Revisited as a Symbolic Shift System

Chacon’s transformation can also be defined by symbolic cutting and stacking, as

follows:

(X , �,µ) is taken to be the measure space generated in the following way using

cutting and stacking: A sequence of blocks B0,B1, ... is defined as follows. Let B0 be the

string of symbols that is simply ‘0’. Continuing inductively:

∀n ∈ Z+ : Bn := Bn−1 n Bn−1 n 1n Bn−1. (1–57)

The set X thus becomes

{x ∈ [0..1]Z|∀k ∈ Z,∀n ∈ Z+,∃j ∈ Z s.t. x [j ..j + |B(n)|) = Bn and k ∈ [j ..j + |B(n)|)

}.

(1–58)

1.6 The K Property

1.6.1 Partitions Part 3/3: Independence and Autonomy

The future and the past of most of the points in the space must be considered

while defining and proving the condition for a K -Automorphism. In a rough sense, if a

sub-collection of coordinates is considered in the future and another sub-collection in the

past of the measure space w/r/t (T : X) (i.e. cylinder sets are produced, one depending

on some positive coordinate values, and another on negative values) then, so long as

the gap between the positive and negative coordinates chosen is sufficiently large, we

have that the past and future coordinate values are nearly independent of one another.

Nearly independent is made rigorous in the form of ϵ-independence, defined below.

When talking about points that belong to a symbolic shift space (T : X) we refer to

the past and future of a point, or of a set of points. These are notions that refer to the

specific coordinate values of points with respect to a given partition P under the action

37

of T . With (T ,P), one considers the P-names of all points in the space as a new space

which is measurably conjugate to the original X by the P-name map (see the Words

and Names section §1.4 on page 27 for further information). The 0th-coordinate of the

P-name of a point x is referred to as the time-zero value of the point x . The negative

coordinate values in the P-name of a point x , written x(−∞..− 1], constitute the future of

the point x w/r/t (T : X), while the positive values, written x [1..∞), constitute the past of

the point x w/r/t (T : X).

Thus we have the past and the future of the entire measure space symbolized as∨∞k=1T

k(P) and∨−∞

k=−1Tk(P), respectively.

For the remainder of this section, fix a space X.

DEFINITION Partition P is said to be ε-independent over the partition Q, written

P ⊥εQ, (1–59)

if ∃ a sub-collection C of atoms of Q such that both

∑β∈C

µ(β) > [1− ε], and (1–60)

∀β ∈ C :∑α∈P

∣∣∣∣µ(α ∩ β)µ(β)

− µ(α)

∣∣∣∣ 6 ε. (1–61)

A related notion to ε-Independence of two partitions is their Lack-of-independence,

or “Lack’.

DEFINITION Partitions P and Q are said to be ε-independent if

⟨P,Q⟩ :=∑

α∈P,β∈Q

|µ(α ∩ β)− µ(α)µ(β)| 6 ε. (1–62)

A third notion of independence of partitions is the Autonomy Number and the

concept of ε-Autonomy:

DEFINITION Partitions P and Q are called ε-autonomous, written

P fεQ, (1–63)

38

if

[H(P) +H(Q)]−H(P ∨Q) 6 ε. (1–64)

The quantity [H(P) +H(Q)]−H(P ∨Q) is called the autonomy number of P and Q.

1.6.2 K -Automorphism Definition

DEFINITION A system (T : X) is a (finite-entropy) K -Automorphism if ∃ a partition

P of X such that

• P generates �, and

• ∀s ∈ Z+,∀ε > 0,∃ g ∈ Z+ such that ∀ℓ ∈ Z+:

ℓ∨i=1

T−i(P) ⊥ε

g+s∨i=g+1

T−i(P). (1–65)

If T is a K -Automorphism then ∀ε > 0, a fixed length of the future is ε-independent

of the infinite past, if separated by a large enough gap which may depend on ε. This

version of the definition can be found in (8).

1.6.3 Example of a K-Automorphism

Bernoulli shifts (§1.1.4, p.19) are K -Automorphisms. This is because, for a Bernoulli

shift, the measure µ is a product measure, so the formula µ(α ∩ β) = µ(α)µ(β)

holds whenever α and β are cylinder sets (§1.1.4, p.20) which are defined over disjoint

coordinate indices. Thus, the quantity µ(α∩ β)µ(α)

− µ(β) = 0, so the sum over all such atoms

is also 0.

Therefore, ∀ℓ ∈ Z+,∀N ∈ Z+, the partition P of X consisting of P-names over

time values [1..ℓ] is independent of the partition Q of X made of up of P-names

over time values [−N..0]. Thus any Bernoulli shift (T : µ) satisfies the definition of

K -Automorphism with a gap of g = 1, for every ε > 0.

39

1.7 Projective Limits and Ergodic Properties

THEOREM 1 Suppose (T : X) is a measurable system, and that

F1 ⊂ F2 ⊂ F3 ⊂ ... (1–66)

is an increasing sequence of T -invariant sub-fields of � such that � is the smallest

complete sigma-field that contains {Fj |j ∈ [1..∞)} . If ∀j ∈ Z+, the restriction T �Fjis a

Rank-1 transformation, then (T : X) is a Rank-1 transformation. (4)

THEOREM 2 Suppose (T : X) is a measurable system, and that

F1 ⊂ F2 ⊂ F3 ⊂ ... (1–67)

is an increasing sequence of T -invariant sub-fields of � such that � is the smallest

complete sigma-field that contains {Fj |j ∈ [1..∞)} . If ∀j ∈ Z+, the restriction T �Fjis a

K -Automorphism, then (T : X) is a K -Automorphism.

Theorem K1 follows from a more general result ((13), Prop 4.6, p.213).

1.8 Statement of Results

All (non-atomic) Lebesgue spaces are isomorphic as measure spaces, and so

the construction in this dissertation applies not just to symbolic shift spaces, but

also constructs a Rank-1 and a K -Automorphism on any general Lebesgue space.

The construction takes place in chapter 2, while proofs of obtaining a Rank-1 and a

K -Automorphism are contained in chapters 3 and 4, respectively. To summarize the

results:

For any Lebesgue space X, there exists measure-preserving transformations

S ,T : X → X such that

• S is a Rank-1 transformation,

• T is a K -Automorphism, and

40

• S ◦ T = T ◦ S .

See chapter 5 for further discussion.

41

CHAPTER 2THE CONSTRUCTION

2.1 Setup and Notation

The letters H and W will be reserved for the height and width of the Z2 n-block.

Throughout the construction, we use two functions (which specify dimensions of

sub-objects):

R,U : Z+ → Z+.

These functions will each approach +∞ as n → +∞, and we let the further rules for

definition of U,R to remain unspecified until the end of the construction.

DEFINITION A block is a finite matrix with entries from the alphabet A := [0..2] =

{0, 1, 2}. The symbols 0 and 1 are represented visually as gray( ) and white( )

respectively, while black is used for drawing boundaries among symbols. When new

spacer is added during the construction, it will be colored light blue ( ) to distinguish

it from previously added spacer, as was done in chapter 1 with the construction of

Chacon’s map(1.5.1). The symbol 2 is reserved as a“labeler” symbol and is represented

visually as green( ). For example:

Figure 2-1. Cutting and Stacking Legend.

2.2 The Construction

An initial block is defined, and then by induction a sequence of blocks B(0),B(1), ...

is defined. Each successive block is created through three stages, outlined here:

• Horizontal Staircase Concatenation

42

• Vertical Concatenation with Shifts

• Concatenating the Pages

These steps are divided into 3 subsections below.

Stage 0: The 0-block

Begin with the ‘0-block’, which we define to be the rectangular block shown below,

using the symbols 0 and 1.

Figure 2-2. The 0-block.

Now assuming that the n-block has been constructed, we describe below how the

‘(n + 1)-block’ is inductively defined.

2.2.1 Stage 1: Horizontal Staircase Concatenation

Figure 2-3. Horizontal Staircase Concatenation.

Concatenate two copies of the n-block with one another to produce a wider

non-rectangular shape, by shifting the right-hand copy down one unit, as pictured.

The bold borders are used for visual aid to identify copies of the original block.

Concatenate another copy of the n-block on the right-side of the already-concatenated

two blocks, again shifting down one unit. Continue this countably many times. This

creates an infinite staircase. In the next step these staircases will be stacked atop one

another, with shifts.

43

Figure 2-4. An Infinite Staircase.

2.2.2 Stage 2: Vertical Concatenation with Shifts

Here an unbounded region of symbols is constructed, and a finite rectangle is cut

out which captures the essence of what takes place in the unbounded region. This cut

produces one Pagen+1 (π). This process is then repeated for each π to produce many

Pages.

Consider a finite enumeration (“shift-sequence”)

π :=(π1,π2, ...,πU(n)

), (2–1)

a word of length U(n) over the alphabet A(n) := [0..W (n)).

Let � be the set of all such π. Now each π ∈ � will give rise to a Pagen+1 (π), inside

of which the π dictates how the staircases will be shifted when stacked: ∀j ∈ [1..|π|],

the number πj decides how far to shift the ‘next’ staircase atop the previous within the

Pagen+1 (π). Each successive shifted staircase is stacked on top of the already-stacked

bunch of staircases, culminating in what will become one Page, as follows:

Fix a particular π ∈ �. To construct the Pagen+1 (π) corresponding to π, begin with

one Infinite Staircase, and add the second on top with a shift to the right by π1 many

units.

Stack another Infinite Staircase, atop the previous, again using a shift, this time of

π2 many units. Continue stacking with shifting according to π to create an unbounded

44

Figure 2-5. Two Staircases, Stacked with a Shift.

region of symbols by finitely-many copies of the Infinite Staircases on one other, each

successive stacking done with a shift provided by the corresponding value in π.

Figure 2-6. Vertical Concatenation, with Shifts.

Now cut out a rectangular block, like a cookie-cutter, from this infinite landscape of

symbols, to produce Pagen+1 (π), as follows:

Impose an (x , y)-axis with the origin at the bottom left corner of any one of the

n-blocks in the bottom Infinite Staircase we began with before stacking the rest,

so that each individual cell is assigned to a unique point in Z × Z. We next “cut”

45

a large rectangle, which contains all of the symbols which lie inside the rectangle

[0..R(n)W (n)) × [0..H(n)U(n)). This height (controlled by the factor U(n)) ensures that

we see every shift provided by the shift-sequence π(n) in this rectangular Page that

is cut out. This width (controlled by the multiplicative factor R(n)) ensures that we run

enough copies of the n-block horizontally so as to see many copies of the nth repetition

word.

Figure 2-7. Cutting Out a Page.

There are some empty places where there is neither a zero or one, in these places

the symbol 1 is added. All added spacer is displayed in the light blue color.

For every π ∈ �, Pagen+1 (π) is created, producing many Pages.

The partial copies of the n-block occurring around the border of the Page are all

declared to be spacer, and the resulting figure is thus a rectangular array of symbols.

46

Figure 2-8. A Page, with Added Shift Spacer.

2.2.3 Stage 3: Concatenating the Pages

We are now ready to form B(n+1), the (n+1)-block. First, horizontally concatenate

all of the Pages:

~B(n + 1) = nπ∈�

Pagen+1(π). (2–2)

This puts all the pages into one single rectangle. Lastly, adjoin n columns with labeler

symbol ‘2’ (pictured in green) on the left-hand-side of this large concatenation, and then

n rows with ‘2’ on the bottom. The result is B(n + 1). This is the Z2 (n + 1)-block.

47

Figure 2-9. The (n + 1)-block.

2.3 Formulae

This completes the construction of the Z2 system. The set X of all infinite lattices

over the set [0..2] which contain a nested n-block structure are where the probability

measure µ is supported:

X =

x ∈ [0..2]Z2

∣∣∣∣∣∣∣∣∣∣∀s, t ∈ Z,∀n ∈ Z+,∃k1, k2 ∈ Z s.t.

s ∈ [k1..k1 + H(n)), t ∈ [k2..k2 +W (n)), and

(xi ,j)i∈[k1..k1+H(n)),j∈[k2..k2+W (n)) = B(n)

. (2–3)

• The height of an (n + 1)-Page is H(n)U(n), and has width R(n)W (n).

• For each n, there are W (n)U(n) many (n + 1)-Pages which make up the (n +1)-block.

• The Width of the (n + 1)-block is

W (n + 1) = n +W (n)R(n) ·W (n)U(n). (2–4)

48

• The Height of the (n + 1)-block is

H(n + 1) = n + H(n)U(n). (2–5)

Figure 2-10. An n-Page Inside the (n + 1)-block.

2.4 Addressing Added Spacer

Next we count how many symbols of spacer there are in each new block that is

formed in the Z2 construction.

SPACER LEMMA ∀n ∈ Z+: In the (n + 1)-block, the ratio of added spacer to the size

of the whole n-block is less than εn, where (εn)∞n=1 is a sequence of epsilons such that∑∞

n=1 εn <∞.

Proof: While building the (n + 1)-block, in creating the Pages, each Page contains

an amount of shift spacer, which was colored light blue in the pictures. In order to count

this shift spacer, we first we calculate how many copies of the n-block could possibly fit

into each Page. We divide the number of symbols in a Page by the number of symbols

49

in the n-block to get

[# of n-block copies in a (n + 1)− Page ] 6 R(n)U(n)

H(n)W (n). (2–6)

Now, for each copy of the n-block in an (n+1)−Page, for the purpose of counting we

associate to it the height 1, width ℓ stretch of added spacer that follows on its right-side

in the (n + 1)−Page, where ℓ ∈ [0..W (n)). This ℓ depends on how far the next level

of Staircases above were shifted when stacked, and must be a number that is no more

than W (n). Since we know an upper bound for the maximum number of n-block copies

in a (n + 1)−Page, we multiply that bound by W (n) to obtain

[# added shift spacer symbols in an (n + 1)− Page ] 6 R(n)U(n)

H(n). (2–7)

We now multiply over all (n + 1)-Pages to obtain a bound on the total number of

added spacer symbols in the (n + 1)-block:

[# added spacer in (n + 1)-block ] 6 W (n)U(n)[R(n)U(n)

H(n)

]+ n · [W (n + 1) + H(n + 1)] .

(2–8)

The right hand term above comes from the labeler symbols (the 2’s) that are added

at the bottom and left ends of the (n + 1)-block.

Finally we are now ready to compare this to the the total number of symbols in the

(n + 1)-block, which is

H(n + 1)W (n + 1) =[n +W (n)R(n)W (n)U(n)

][n + H(n)U(n)] . (2–9)

Forming the desired ratio and simplifying, the following quantity is obtained:

added spacer in (n + 1)-block

# of total symbols in (n + 1)-block. (2–10)

This is less than or equal to

W (n)U(n)[R(n)U(n)H(n)

]+ n ·

[n +W (n)R(n)W (n)U(n) + n + H(n)U(n)

][n +W (n)R(n)W (n)U(n)] [n + H(n)U(n)]

, (2–11)

50

which is less than

W (n)U(n)[R(n)U(n)H(n)

]+ n ·

[n +W (n)R(n)W (n)U(n) + n + H(n)U(n)

][W (n)R(n)W (n)U(n)] [H(n)U(n)]

. (2–12)

This is further manipulated into

1

H(n)2U(n)+

n

H(n)U(n)+

2n2

H(n)U(n)W (n)R(n)W (n)U(n)+

n

W (n)R(n)W (n)U(n). (2–13)

This completes the proof, as we merely require that the above quantity is summable,

which can certainly be done, by choosing appropriately large values for the functions

R(n) and U(n) (which are chosen in the next section).

2.5 Defining R(n) and U(n)

DEFINITION Select R : Z+ → Z+ such that

• limn→∞ R(n) = ∞,

• limn→∞H(n)R(n)

= 0,.

• R satisfies equation (2-14) below.

DEFINITION Select U : Z+ → Z+ such that

• limn→∞ U(n) = ∞.

• U satisfies equation (2-14) below.

SPACER CONDITION We require that the functions R and U satisfy

∞∑n=0

an converges, (2–14)

where

an :=1

H(n)2U(n)+

n

H(n)U(n)+

2n2

H(n)U(n)W (n)R(n)W (n)U(n)+

n

W (n)R(n)W (n)U(n).

(2–15)

51

CHAPTER 3PROVING RANK-1

In this chapter we consider the horizontal-factors of the Z2 system obtained by

restricting the Z2 measure space onto a horizontal width-b band, b ∈ Z+. For each such

b, we consider the space

Xb := (Xb, �b,µb) , (3–1)

where

Xb :=([0..2]b

)Z, with field �b := �

ZP([0..2]b

), and measure µb := µ|�b

. (3–2)

The space [0..2]b is viewed here as a column vector space:

[0..2]b =

x1

x2

...

xb

∣∣∣∣∣∣∣∣∣∣∣∣∣For each i ∈ [1..b],

value xi is in [0..2]

. (3–3)

So that the space([0..2]b

)Z is viewed as the symbolic shift space

Xb =

· · · x−1,1 x0,1 x1,1 · · ·

· · · x−1,2 x0,2 x1,2 · · ·

· · · ......

... · · ·

· · · x−1,b x0,b x1,b · · ·

∣∣∣∣∣∣∣∣∣∣∣∣∣For each i ∈ [1..b] and j ∈ Z,

value xi ,j is in [0..2],

. (3–4)

The goal is to show, for fixed b, that this factor-space of sequences, together with its

shift map, is a Rank-1 transformation. When a < b, then the b-band sequences contain

the a-band sequences as factors. The full horizontal-shift group element T that acts on

the constructed Z2 system may be found by taking a projective limit as b → ∞ of the

horizontal shift maps acting on the width-b factor spaces. Therefore, since the Rank-1

52

property is sealed under projective limit (see the Rank-1 section of chapter 1) we need

only obtain a result about a width-b factor Xb having the Rank-1 property.

In this section, to prove the Rank-1 property, we use the convergence of the

following quantity:

limn→∞

[H(n)

R(n)

]= 0. (3–5)

3.1 Band One

DEFINITION ∀n ∈ Z+, we define the nth repetition word ω = ω(n) to be the

left-to-right, bottom-to-top (”raster”) scan of the Z2 n-block. Thus for each n ∈ Z+, the

word ω(n) has length W (n)H(n). Below is an image of the raster scan of a block, the

word ω =(ω1, ...,ωW (n)H(n)

). The scan begins at the bottom-left corner of the block, the

bottom row is scanned first, and then concatenated with the 2nd row up, and et cetera

until the top row of the block is finally read off.

Let P = (α0,α1,α2) be the partition of X1 defined by

For a ∈ [0..2], αa := {x ∈ X1| x [0] = a} . (3–6)

The repetition word ω(n) recurs in P-names of points of the measure space, so that

in a long sample sequence of bandwidth 1 from the band-1 horizontal factor, most of the

symbols that occur are in fact among distinct copies of the word ω(n) being repeated.

53

THEOREM 1.1 The 1-band is Rank-1. ∀ε > 0, ∃n ∈ Z+ and ∃G ∈ �b with

µ(G) > 1− ε, such that ∀Z > W (n + 1),

∀x ∈ G : x [−Z ..Z ] is (1− ε)-packable by ω(n). (3–7)

Proof:

Let ε > 0. Begin by choosing a δ > 0 small enough that

(1− δ)2 > 1− ε. (3–8)

Now choose p ∈ Z+ large enough that

r1, r2 ∈ (1− δp, 1) ⇒ r1 · r2 > 1− δ. (3–9)

For a fixed value of n, a row of the Z2 (n + 1)-block that does not consist of any

partial copies of the n-block that were converted to spacer in the construction will be

referred to as a good row.

Let Gn be the set of points in the measure space whose time-zero coordinate is

contained in a Good Row of the (n + 1)-block.

Now choose n > 1 such that:

• µ(Gn) > 1− δ,

• the Z2 (n + 1)-block occupies more than (1-δ) of the Z2 space,

• H(n)H(n)+1

> 1− δp and 2H(n)+2R(n)

< δp.

Below, we count the number of coordinates present in an arbitrary row of the Z2

(n + 1)-block, and compare this to the number of coordinates in any Good Row of this

(n + 1)-block that are occupied by a distinct copy of the repetition word ω(n).

Claim: There are at least

R(n)W (n)− 2 [|ω|+W (n)]

|ω|+W (n)(3–10)

54

many occurrences of ω in each Page, and thus at least

|ω| ·[W (n)U(n)

]· R(n)W (n)− 2 [|ω|+W (n)]

|ω|+W (n)(3–11)

many coordinates in a single Good Row of the Z2 (n + 1)-block that contain distinct

readings of the the word ω(n).

To prove the first part of the claim, first note that there is a width of R(n)W (n)

many coordinates in a Page. By the staircase-like construction, after an initial entrance

on the left-hand side of a Page of a Good Row of the (n + 1)-block of at most length

|ω| + W (n), we find perfect copies of the word ω separated by little random gaps of

spacer. Each such gap of spacer is no more than W (n) — the width of the Z2 n-block.

At the right-hand end of a Good Row of the (n + 1)-block, a copy of ω perhaps does not

finish displaying, but the number of coordinates there with a (perhaps) partial copy of ω

is not any more than |ω| +W (n). Therefore, by dividing the R(n)W (n)− 2 [|ω|+W (n)]

many coordinates by |ω| +W (n), we count how many times ω, along with a largest gap

of W (n) units of added spacer, must at least occur.

The second part of the claim follows by applying this count to the W (n)U(n) many

Pages that make up the entire Z2 (n + 1)-block.

Next, we compare this lower bound to the total number of coordinates that are

present in each horizontal row of the Z2 (n + 1)-block, which is:

R(n)W (n) ·W (n)U(n).

Therefore the fraction of coordinates in any one of the good rows of the Z2 (n +

1)-block that are occupied by distinct copies of ω = ω(n) is at least

|ω| ·[W (n)U(n)

]· R(n)W (n)−2[|ω|+W (n)]

|ω|+W (n)

R(n)W (n) ·W (n)U(n). (3–12)

55

Using |ω| = W (n)H(n) and simplifying, this becomes

H(n)

H(n) + 1

(1− 2H(n) + 2

R(n)

)> 1− δ. (3–13)

Therefore with probability greater than 1 − δ, a point x ∈ G passes through a

Good Row of the (n + 1)-block and thus any portion of it’s P-name that includes an

(n + 1)-block is (1-δ)-packable by distinct copies of ω. Further, given any positive integer

Z > W (n + 1), the width-1 partial P-name of such a point x from time coordinates -Z to

Z , written x [−Z ..Z ], is (1 − δ)-packable with copies of the (n + 1)-block. Since each of

these copies of the (n + 1)-block are (1− δ)-packable with copies of ω, we conclude that

the partial P-name x [−Z ..Z ] is (1− ε)-packable by copies of the nth repetition word ω.

3.2 Band b

Fix b ∈ [2..∞). In this section, we show that the width-b horizontal factor is a Rank-1

transformation. We define ω = ω(n) in this case to be the left-to-right, bottom-to-top

width-b-raster scan of the n-block (the n-block from the Z2-Rank-1). Thus ω begins by

reading the bottom width-b band of the Z2 n-block, and transitions to the next width-b

band that is shifted up by one coordinate, and continuing until the top of the Z2 n-block

is reached. Note that ω(n) has length W (n)[H(n) − (b − 1)], and that each letter in the

word ω(n) is a b-tuple (which we think of as stacked vertically, as in a column vector) of

values from [0..2].

THEOREM 1.2 The b-band is Rank-1. ∀ε > 0, ∃n ∈ Z+ ∃ G ∈ �b with µ(G) > 1− ε

such that ∀Z > W (n + 1), ∀x ∈ G :

x [−Z ..Z ] is (1− ε)-packable by ω(n). (3–14)

Proof:

With b ∈ Z+, let ε > 0, and let δ < ε be such that (1− δ)2 > 1− ε.

56

A width-b band of the Z2 (n + 1)-block that does not consist of any partial copies

of the n-block that were converted to spacer in the construction will be referred to as a

good band.

Let Gn be the set of points in the measure space whose time-zero, width-b

coordinate is contained in a Good Band of the (n + 1)-block.

Choose n ∈ Z+ such that:

• µ(Gn) > 1− δ,

• H(n) > b,

• the Z2 n-block occupies more than (1-δ) of the Z2 space,

•[1− 2[H(n)−(b−1)+R(n)]

R(n)L(n)

] [H(n)−(b−1)

H(n)−(b−1)+R(n)

]> 1− δ

We count the number of occurrences of the repetition word ω inside the

Z2 (n + 1)-block as we pass through from left-to-right with a band of width-b:

We find copies of the repetition word ω(n) inside each Page. As when b was 1 in

the previous section, we see a horizontal width of R(n)W (n) many coordinates in this

Page. By the staircase construction, the word ω appears in each good band of the Page.

Now the bandwidth is b and the vertical shifts between the adjacent n-block copies are

all only 1 unit, and the width-b-raster scan stops once the top line of the block is seen for

the first time. So after we finish scanning a copy of ω, we have to shift through b many

stretches of W (n) coordinates before we find another copy of ω (while we scan away the

rest of the top of the Page along with increasingly more of the next Page). This gives us

an upper bound on the number of coordinates we must shift through in order to find the

next ω: b ·W (n). In the beginning we may need to wait as long as |ω| + b ·W (n) many

coordinates to ensure we begin a fresh copy of ω, and we also may not have a whole

copy on the right-hand end, so we arrive at a lower bound for the number of copies of ω

that are seen:

57

R(n)W (n)− 2 (|ω|+ bW (n))

|ω|+ bW (n)(3–15)

Inside a Page, then, we find in shifting through a good band from left-to-right with a

bandwidth of b, the fraction of coordinates covering the path with exact copies of ω to be

at least

|ω|R(n)W (n) ·W (n)U(n)

·W (n)U(n) · R(n)W (n)− 2 (|ω|+ bW (n))

|ω|+ bW (n), (3–16)

which manipulates into

R(n)W (n)− 2 (|ω|+ bW (n))

R(n)W (n)· |ω||ω|+ bW (n)

. (3–17)

Now |ω| = W (n)[H(n) − (b − 1)], so the fraction of coordinates covering the path

with exact copies of ω is at least

[1− 2[H(n)− (b − 1) + R(n)]

R(n)

]· H(n)− (b − 1)

H(n)− (b − 1) + R(n)> 1− δ. (3–18)

Now ∀Z > W (n + 1), the time-zero width-b coordinate vector x [−Z ..Z ] is (1 −

δ)-packable by copies of the Z2 n-block. In these blocks we have distinct copies of our

repetition word ω occurring more than 1 − δ of the time as we shift from left-to-right

through the Z2 (n + 1)-block inside of a good window of bandwidth b. Thus the partial P

name x [−Z ..Z ] is (1− ε)-packable by distinct copies of ω(n).

58

CHAPTER 4PROVING THE K PROPERTY

First the width-1 vertical factor of the system is shown to be a K -Automorphism.

Then the necessary modifications for the general argument for a width-b factor is given

in the last section of the chapter.

4.1 Band One

THEOREM 2.1 The width-1 down-shift is a K -Automorphism of (X ,µ). More

precisely,

∀ε > 0,∀k ∈ Z : ∃G ∈ Z+ such that ∀g,∀s ∈ Z+:

−g∨i=−(g+s)

T i(P) ⊥ε

k∨i=1

T i(P).

Proof: Let ε > 0 and choose δ > 0 such that√δ < ε

6. Let k ∈ Z+.

Fix M ∈ Z+ such that the section of the M-block obtained by removing its top and

bottom H(M) many rows, more than 1 − δ of the measure space remains. (This can be

done, since U(M) → ∞.)

Fix G ∈ Z+ such that G > H(M), the height of the M-block.

Now fix g > G , and also s ∈ Z+. Finally, choose N > M such that the probability that

vertical time-value range [−(g + s)..k ] lies in an (N + 1) page is greater than 1− δ. Such

a column is referred to as a good column.

With probability greater than 1 − δ, an element of∨−g

i=−(g+s)Ti(P) is an equivalence

class of points in the measure space which have specified coordinates in some

rectangular subset of an (N + 1)-page, and an element of∨k

i=1Ti(P) is an equivalence

class of points whose coordinates are specified in a different rectangular subset of a

different (N + 1)-page. We condition to the set E where all columns are Good Columns,

59

and µ(E) > 1 − δ (see (1.1.3). Now an element of∨k

i=1Ti(P|E) is made up entirely of

the interior of the (M + 1)-block, so that elements of∨k

i=1Ti(P|E) therefore have the

form

[partial Column][bit][Column][bit]...[bit][Column][bit][partial Column], (4–1)

where [bit] either stands for a single spacer or nothing at all, and [Column] stands for a

column of the M-block of width 1.

Now we compare the µ distribution on∨k

i=1Ti(P|E) to the conditional distribution

on∨k

i=1Ti(P|E) conditioned on

∨−gi=−(g+s)T

i(P|E).

An element of∨−g

i=−(g+s)Ti(P|E), by viewing the corresponding coordinates as

sitting inside of the (N + 1)-block, consists of a history of passing through N-blocks with

random (single) spacers in between. Looking forward from this “recorded history” by a

gap of g coordinates in the vertical direction, the partition of∨k

i=1Ti(P|E) is considered

in relation to the other partition∨−g

i=−(g+s)Ti(P|E). Since the gap g that separates their

coordinates is larger than H(M), it is equally likely that an M-block should begin at

any particular location in more than g units above the highest coordinate position of an

element in∨−g

i=−(g+s)Ti(P|E). There have been more than H(M) units of added spacer

in that gap length g > G to randomize where an M-block is to be expected.

With c ∈ R from the Independence/Autonomy Lemma in Appendix A, the

distributional difference is small:

−g∨i=−(g+s)

T i(P|E) fcδ2

k∨i=1

T i(P|E), (4–2)

so that the independence is achieved:

−g∨i=−(g+s)

T i(P|E) ⊥√δ

k∨i=1

T i(P|E).

60

With the set E as described above, we conditioned on E in order to achieve

independence. But, by the ε-Independence Lemma 3 in the Appendix, we still have that

−g∨i=−(g+s)

T i(P) ⊥6√δ

k∨i=1

T i(P).

Since 6√δ < ε, the proof is complete.

4.2 Band b

THEOREM 2.2 The width-b down-shift is a K -Automorphism of (X ,µ). More

precisely,

∀ε > 0,∀k ∈ Z : ∃G ∈ Z+ such that ∀g,∀s ∈ Z+:

−g∨i=−(g+s)

T i(P) ⊥ε

k∨i=1

T i(P),

where P is the width-b time-zero partition of the measure space Xb.1

Proof: Let ε > 0 and choose δ > 0 such that√δ < ε

6. Let k ∈ Z+.

Fix M ∈ Z+ such that the section of the M-block obtained by removing its top and

bottom H(M) many rows, as well as the left and right b many columns, more than 1 − δ

of the measure space remains. (This can be done as U(M) → ∞ and b is fixed.)

Fix G ∈ Z+ such that G > H(M), the height of the M-block.

Now fix g > G , and also s ∈ Z+. Finally, choose N > M such that the probability that

vertical time-value range −(g + s) to k , of width b lies in an (N + 1) page is greater than

1− δ, as was done in the previous subsection.

1 Xb was defined in chapter 3 as a horizontal symbolic shift space, but can also beviewed as a vertical symbolic shift space, as is done here.

61

With probability greater than 1 − δ, an element of∨−g

i=−(g+s)Ti(P) is an equivalence

class of points in the measure space which have specified coordinates in some

rectangular subset of an (N + 1)-page, and an element of∨k

i=1Ti(P) is an equivalence

class of points whose coordinates are specified in a different rectangular subset of a

different (N + 1)-page. So, condition on the set E where the probability that a vertical

time-value range −(g + s) to k , of width b, lies in an (N + 1) page is greater than 1 − δ,

and µ(E) > 1 − δ. An element of∨k

i=1Ti(P|E) is made up entirely of the interior of the

(M+1)-block, so that elements of∨k

i=1Ti(P|E) therefore have the form C1nC2n ...nCb

where each Ci has the form

[partial Column][bit][Column][bit]...[bit][Column][bit][partial Column], (4–3)

oriented in the vertical direction, where [bit] refers to a width-b combination of spacer

and no spacer between columns, and [Column] refers to a width-b column of the

M-block. Now as in the previous subsection, we compare the µ distribution on∨k

i=1Ti(P|E) to the conditional distribution on

∨k

i=1Ti(P|E) conditioned on∨−g

i=−(g+s)Ti(P|E), and apply the ε-Independence Lemma 3 in the Appendix to complete

the proof.

62

CHAPTER 5APPLYING THE WEAK-CLOSURE THEOREM

King’s Weak-closure Theorem (3) says that the commutant of a Rank-1 transformation

is the weak-closure (see (5–3) below) of the powers of the Rank-1. Therefore, for any

Rank-1 transformation S which is known to commute with another transformation T , it

necessarily follows that T is a limit of powers of S under the weak operator topology (6).

Consider the class of transformations � as discussed in 1.1.2 (p.15). The set � of all

invertible transformations on X is given the weak topology by declaring that a sequence

(Sm) ⊂ � converges to a limit S ∈ � if

∀B ∈ � : limm→∞

[µ(S−1m (B)△ S−1(B)

)+ µ (Sm(B)△ S(B))

]= 0. (5–1)

A metric which realizes the weak topology is defined as follows. First, choose a

countable generating sub-algebra (E1,E2, ...) and put

d(S ,T ) :=

∞∑j=1

µ(T−1(Ej)△ S−1(Ej)

)2j

. (5–2)

DEFINITION Consider a system (S : X). The weak-closure of S is the d-closure of

the set {Sk∣∣ k ∈ Z} ⊂ �. (5–3)

Applying the Weak-closure Theorem (3), there exist sequences (nj)j∈Z+ such

that ∀ε > 0,∃N ∈ Z such that ∀j > N, distance d(Snj ,T ) < ε. Each member in

this sequence is a Rank-1 transformation, as any non-zero power of a Rank-1 is still

Rank-1 (4). Thus we have a sequence of zero-entropy, deterministic transformations

defined on a common measure space which converge to the completely positive entropy

K -Automorphism, which has no zero-entropy factors.

An actual sequence (nj)j∈Z+ is in fact evident from the construction. Note that inside

an (n + 1)-Page, shifting a sub-region of coordinates (the purple region in Figure (5-1) to

the left by W (n) many units (to the red region in Figure (5-1)) has the effect of vertically

63

shifting that region down by 1 unit. Therefore, we have the explicit formula

limn→∞

SW (n) = T . (5–4)

Figure 5-1. Shifting Left by W(n) units Equals one Shift Downward.

In fact, because T /∈ {Sk |k ∈ Z}, it follows1 that another curious sequence exists

for S , a rigidity sequence.

DEFINITION Consider a system (S : X). A sequence of integers r1, r2, ... is called a

rigidity sequence for S if

limj→∞

S rj = Id, (5–5)

the identity transformation on X.

A rigidity sequence for S can be found by applying the previous idea of shifting

horizontally by W (n) to achieve a vertical shift of 1 unit, with a twist.

1 See (3) for details.

64

Figure 5-2. Shifting Left by W(n+1) units, and Right by W(n) units.

Here, we first shift to the left by W (n + 1) units (indicated by the green arrow in

Figure (5-2)), which has the effect of shifting vertically down by one unit. Next, we simply

shift to the right by W (n) many units (indicated by the red arrow in Figure (5-2)), which

shifts one unit up. The composition of these two shifts is the identity shift, on most2

sub-regions of symbols in the (n + 1)-block. This leads to the following formula:

limn→∞

SW (n+1)−W (n) = Id. (5–6)

2 Near the edge of an (n+1)-Page, the Identity is not always achieved, but this is asmall portion of the space.

65

APPENDIX ALEMMATA

The following lemmata are tools for proving equivalences of various notions of

partial independence.

INDEPENDENCE/AUTONOMY LEMMA There is a positive constant c such that

∀ε > 0, if P and Q are two partitions of X such that if P fcε2

Q, then P and Q are√ε-independent. ((11), Lemma IV.2.3, p.224),((12), Lemma 6.1, p.26).

ε-INDEPENDENCE LEMMA 1 Consider P and Q partitions of X , a set E ⊂ X , and

an ε ∈ (0, 1) such that µ(E) > 1− ε2. If conditioned partitions1 P|E and Q|E satisfy that

P|E ⊥ Q|E , then P is (3ε)-independent over Q ((12), Lemma 6.1, p.26).

ε−INDEPENDENCE LEMMA 2 Consider P and Q partitions of X . If P is

ε-independent over Q, then

∑α,β

|µ(α ∩ β)− µ(α)µ(β)| 6 3ε. (A–1)

((12), Lemma 6.1, p.26).

ε-INDEPENDENCE LEMMA 3 Consider P and Q partitions of X , a set E ⊂ X , and

an ε ∈ (0, 1) such that µ(E) > 1 − ε2. If conditioned partitions P|E and Q|E satisfy that

P|E is ε-independent over Q|E , then P is (6ε)-independent over Q.

Proof: Set

C := {β ∈ Q | µ(β ∩ E) > [1− ε] · µ(β)} . (A–2)

Certainly∑

β∈C µ(β) > [1− ε]. Using Independence Lemma 2 from above,

∑α∈P,β∈Q

∣∣∣∣µ(α ∩ E ∩ β ∩ E)

µ(E)− µ(α ∩ E)

µ(E)· µ(β ∩ E)

µ(E)

∣∣∣∣ 6 3ε. (A–3)

1 See (1.1.3) for definition of P|E .

66

Now fix a β ∈ ξ. From the above inequality, there are δα ∈ R such that∑

α∈P δα 6 3ε

and

µ(α ∩ β ∩ E)

µ(E)=

µ(α ∩ E ∩ β ∩ E)

µ(E)=

µ(α ∩ E) · µ(β ∩ E)

µ(E)µ(E)+ δα. (A–4)

Now to show the (6ε)-independence, we compute that

∑α∈P

∣∣∣∣µ(α ∩ β)µ(β)

− µ(α)

∣∣∣∣ (A–5)

6∑α∈P

∣∣∣∣µ(α ∩ β ∩ E)

µ(β)− µ(α ∩ E)

∣∣∣∣ +∑α∈P

∣∣∣∣µ(α ∩ β ∩ E c)

µ(β)− µ(α ∩ E c)

∣∣∣∣ . (A–6)

We first show that the left-hand sum of (A–6) is less than 4ε, and then that the

right-hand sum is less than 2ε. First note that

µ(E)− µ(β ∩ E)

µ(β)< µ(E) + ε− 1 6 ε, (A–7)

andµ(β ∩ E)

µ(β)− µ(E) 6 µ(β ∩ E)

µ(β)+ ε2 − 1 6 ε2 < ε, (A–8)

and therefore ∣∣∣∣µ(β ∩ E)

µ(β)− µ(E)

∣∣∣∣ < ε. (A–9)

Next, using (A–4), the left hand sum of (A–6) is equal to

∑α∈P

∣∣∣∣µ(α ∩ E) · µ(β ∩ E)

µ(E)µ(β)+ δαµ(E)− µ(α ∩ E)

∣∣∣∣ (A–10)

=∑α∈P

µ(α ∩ E)

∣∣∣∣ µ(β ∩ E)

µ(E)µ(β)− 1 +

δα · µ(E)µ(α ∩ E)

∣∣∣∣ (A–11)

6∑α∈P

µ(α ∩ E) ·∣∣∣∣ µ(β ∩ E)

µ(E)µ(β)− 1

∣∣∣∣+∑α∈P

µ(α ∩ E) · |δα| · µ(E)µ(α ∩ E)

(A–12)

6 µ(E) ·∣∣∣∣ µ(β ∩ E)

µ(E)µ(β)− 1

∣∣∣∣+ 3ε · µ(E) 6 4ε. (A–13)

67

For the right hand sum of (A–6), first note that since

µ(β ∩ E) + µ(β ∩ E c) = µ(β), (A–14)

we therefore haveµ(β ∩ E)

µ(β)< ε. (A–15)

Now we return to the right-hand sum of (A–6) and compute

∑α∈P

∣∣∣∣µ(α ∩ β ∩ E c)

µ(β)− µ(α ∩ E c)

∣∣∣∣ (A–16)

6∑α∈P

µ(α ∩ β ∩ E c)

µ(β)+∑α∈P

µ(α ∩ E c) (A–17)

µ(β ∩ E c)

µ(β)+ µ(E c) (A–18)

< ε+ ε2 < 2ε. (A–19)

The proof is now complete.

68

APPENDIX BLEBESGUE SPACES

Lebesgue spaces play an important role in Ergodic Theory. One nice way to view

a Lebesgue space is via the notion of isomorphism of measure spaces, or “measure

algebras”. Glasner (2) takes this approach, defining a homomorphism and also an

isomorphism among measure algebras. The representation, or equivalent definition of a

Lebesgue spaces becomes

DEFINITION A non-atomic Lebesgue space is a measure space (X, �) that is

isomorphic (2) to the unit interval [0, 1] with Lebesgue measure.

In this dissertation, all spaces are assumed Lebesgue.

69

REFERENCES

[1] N. Friedman, Replication and stacking in ergodic theory, The AmericanMathematical Monthly 99 (1992), 31–41.

[2] E. Glasner, Ergodic theory via joinings, American Mathematical Society, USA,2003.

[3] J. King, The commutant is the weak closure of the powers, for rank-1 transforma-tions, Ergodic Theory & Dynamical Systems 6 (1986), 363–384.

[4] J. King, Joining-rank and the structure of finite rank mixing transformations, JournalD’Analyse Mathematique 51 (1988), 182–227.

[5] B. Weiss M. Foreman, D. Rudolph, The conjugacy problem in ergodic theory, TheAnnals of Mathematics, Princeton, 2011.

[6] J. King N. Friedman, P. Gabriel, An invariant for rigid rank-1 transformations,Ergodic Theory & Dynamical Systems 8 (1988), 53–72.

[7] D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances InMathematics 4 (1970), 337–352.

[8] D. Ornstein and P. Shields, An uncountable family of k-automorphisms, (1973).

[9] K. Petersen, Ergodic theory, Cambridge University Press, Cambridge, 1983.

[10] D. Rudolph, Fundamentals of measurable dynamics, Clarendon Press, Oxford,1990.

[11] P. Shields, The ergodic theory of discrete sample paths, American MathematicalSociety, USA, 1961.

[12] P. Shields, The theory of bernoulli shifts, University of Chicago Press, Chicago,1973.

[13] J. Thouvenot, Entropy, isomorphism and equivalence in ergodic theory, Handbookof Dynamical Systems 1A (2002), 205–238.

70

BIOGRAPHICAL SKETCH

Nick has lived in Florida all his life where he has spent his time studying, teaching,

and tutoring many students. He has also enjoyed spending time with his family, surfing,

and playing his guitar. He and his wife lived together with 6 cats and 2 ducks in their

Gainesville home while Nick wrote this dissertation.

71