SOME RESULTS ON RECURRENCE AND ENTROPY

DISSERTATIO $\mathrm{N}$

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of the Ohio State University

By

Ronald Lee Pavlov Jr., B.S., $\mathrm{M}\mathrm{.}\mathrm{S}$.

$\mathrm{*}\mathrm{*}\mathrm{*}\mathrm{*}$ $\mathrm{*}$

The Ohio State University

2007

Dissertation Committee:

$\mathrm{\mapsto }\mathrm{1}\mathrm{3}\mathrm{3}\mathrm{C}\mathrm{1}\mathrm{b}\mathrm{C}\mathrm{l}\mathrm{b}\mathrm{l}\mathrm{\cup }\mathrm{1}\mathrm{1}\mathrm{\cup }\mathrm{\cup }\mathrm{1}\mathrm{1}\mathrm{1}\mathrm{1}\mathrm{1}\mathrm{1}\mathrm{l}\mathrm{b}\mathrm{b}\mathrm{C}\mathrm{C}$. Approved by

Professor Vitaly Bergelson, Advisor

Professor Alexander Leibman

Advisor

Professor Manfred Einsiedler Graduate Program in $\mathrm{M}\mathit{\xi }$

Graduate Program in Mathematics

ABSTRACT

This thesis is comprised primarily of two separate portions. In the first portion, we

exhibit, for any sparse enough increasing sequence $\mathrm{\left\{}{\mathit{p}}_{\mathit{n}}\mathrm{\right\}}$ of integers, a totally minimal,

totally uniquely ergodic, and topologically mixing system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ and $\mathit{f}\mathrm{\in }\mathit{C}\mathrm{\left(}\mathit{X}\mathrm{\right)}$ for

which the averages $\frac{\mathrm{1}}{\mathit{N}}{{\displaystyle \sum }}_{\mathit{n}\mathrm{=}\mathrm{0}}^{\mathit{N}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{{\mathit{p}}_{\mathit{n}}}\mathit{x}\mathrm{\right)}$ fail to converge on a residual set in $\mathit{X}$, answering

negatively an open question of Bergelson. We also construct here a totally minimal,

totally uniquely ergodic, and topologically mixing system $\mathrm{(}{\mathit{X}}^{\mathrm{\prime }}\mathrm{,}\mathrm{}{\mathit{T}}^{\mathrm{\prime }}\mathrm{)}$ and ${\mathit{x}}^{\mathrm{\prime }}\mathrm{\in }{\mathit{X}}^{\mathrm{\prime }}$ for

which ${\mathit{x}}^{\mathrm{\prime }}\mathrm{\notin }\mathrm{\left\{}{\mathit{T}}^{{\mathit{p}}_{\mathit{n}}}{\mathit{x}}^{\mathrm{\prime }}\mathrm{\right\}}$.

In the second portion, we study perturbations of multidimensional shifts of finite

type. Given any ${\mathbb{Z}}^{\mathit{d}}$ shift of finite type $\mathit{X}$ for $\mathit{d}\mathrm{>}\mathrm{1}$ and any word $\mathit{w}$ in the language of

$\mathit{X}$, denote by ${\mathit{X}}_{\mathit{w}}$ the set of elements of $\mathit{X}$ in which $\mathit{w}$ does not appear. If $\mathit{X}$ satisfies

a uniform mixing condition called strong irreducibility, we obtain exponential upper

and lower bounds on ${\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}\mathit{X}\mathrm{\right)}\mathrm{-}{\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}{\mathit{X}}_{\mathit{w}}\mathrm{\right)}$ dependent only on the size of $\mathit{w}$. This result

generalizes a result of Lind about $\mathbb{Z}$ shifts of finite type.

ii

To Dilip

ill

ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor Vitaly Bergelson. This the-

sis could never have been completed without his help. His infectious enthusiasm

for mathematics, along with a willingness to share his knowledge, were a constant

inspiration.

I would also like to thank my other committee members, Professors Alexander Leib-

man and Manfred Einsiedler, for agreeing to serve on my committee and for many

fruitful mathematical discussions.

Finally, I thank my friends and loved ones, without whom I would never have been

able to get as far as I have. In no particular order:

Thank you to Mom and Dad, who have always been willing to do anything to

support me and have always believed in me.

Thank you to Greg, without whose friendship and sense of humor graduate school

would have been much less bearable.

Thank you to Jasmine, who was always there for me. I can never thank you

enough for your constant love and support.

IV

VITA

2000-Present . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Teaching Associate,

The Ohio State University

1998-1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Student Instructional Associate,

The Ohio State University

2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $\mathrm{M}\mathrm{.}\mathrm{S}$. in Mathematics,

The Ohio State University

2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $\mathrm{B}\mathrm{.}\mathrm{S}$. in Mathematics,

The Ohio State University

$\mathrm{v}$

PUBLICATIONS

Research Papers

$\mathrm{•}$ Some Counterexamples In Topological Dynamics (Ergodic Theory and Dynam-

ical Systems, to appear)

$\mathrm{•}$ Perturbations of Multidimensional Shifts of Finite Type (submitted)

FIELDS OF STUDY

Major Field: Mathematics

Specialization: Ergodic Theory and Dynamical Systems

vi

TABLE OF CONTENTS

Abstract ii

Dedication . $\mathrm{i}\mathrm{i}\mathrm{i}$

Acknowledgments . 1V

Vita $\mathrm{v}$

List of Figures ix

CHAPTER PAGE

1Introduction 1

2Recurrence and convergence of ergodic averages along sparse sequences 8

2.1 Introduction 8

2.2 Some general symbolic constructions 16

2.3 Some symbolic counterexamples 31

2.4 Some general constructions on connected manifolds . 42

2.5 Some counterexamples on connected manifolds 55

2.6 A counterexample about simultaneous recurrence 61

2.7 Questions 65

3Perturbations of multidimensional shifts of finite type 67

3.1 Introduction 67

3.2 Some measure-theoretic preliminaries 84

3.3 A replacement theorem 93

3.4 The proof of the main result 114

3.5 A closer look at the main result 146

3.6 An application to an undecidability question 155

3.7 Questions 159

Vll

Bibliography

161

Vlll

LIST OF FIGURES

FIGURE PAGE

3.1 ${\mathit{f}}_{\mathrm{2}}$ 's action on a sample element of $\mathit{Y}$ 73

3.2 A portion of a sample element of $\mathit{Z}$ 76

3.3 ${\mathit{a}}_{\mathrm{4}}$ 77

3.4 ${\mathit{b}}_{\mathrm{4}}$ 77

3.5 ${\mathit{R}}_{\mathit{k}\mathrm{(}{\mathit{j}}_{\mathrm{1}}\mathrm{+}\mathit{R}\mathrm{)}\mathrm{,}\mathit{k}\mathrm{(}{\mathit{j}}_{\mathrm{2}}\mathrm{+}\mathit{R}\mathrm{)}\mathrm{,}\mathrm{\text{...}}\mathrm{,}\mathit{k}\mathrm{(}{\mathit{j}}_{\mathit{d}}\mathrm{+}\mathit{R}\mathrm{)}}$ 85

3.6 A standard replacement of $\mathit{u}$. 95

3.7 A sample element $\mathit{S}$ of ${\mathit{R}}_{\mathit{j}}$ 97

3.8 An element $\mathit{S}$ of ${\mathit{R}}_{\mathit{j}}$ associated to two standard replacements 98

3.9 The suboctants of ${\mathit{B}}_{\mathit{j}}$ 99

3.10 Elements $\mathit{S}$, ${\mathit{S}}^{\mathrm{\prime }}$ of ${\mathit{R}}_{\mathit{j}}$ whose difference is a multiple of ${\mathit{e}}_{\mathit{i}}$ tOO

3.11 An element $\mathit{S}$ of ${\mathit{R}}_{\mathit{j}}$ which contains $\mathit{O}$ 106

3.12 107

3.13 ${\mathit{B}}_{\mathit{j}}^{\mathrm{\prime }}$ 108

3.14 Intersecting occurrences of $\mathit{w}$ 116

3.15 ${\mathit{S}}_{\mathit{i}}\mathrm{\setminus }{\mathit{S}}_{\mathit{i}}^{\mathrm{\left(}\mathit{R}\mathrm{\right)}}$ 120

3.16 Disallowed and allowed pairs of overlapping ${\mathit{w}}_{\mathit{j}\mathrm{,}\mathit{d}}$ 127

3.17 A point $\mathit{x}\mathrm{\in }{\mathit{X}}_{\mathit{y}}$ 138

ix

3.18 The correspondence between copies of ${\mathrm{\Gamma }}_{\mathit{W}}$ and points in ${\mathrm{\Gamma }}_{\mathit{j}}^{\mathrm{\left(}\mathrm{2}\mathrm{\right)}}$

139

3.19 How a copy of ${\mathrm{\Gamma }}_{\mathit{W}}$ is filled if ${\mathit{b}}_{\mathit{j}\mathrm{,}\mathit{d}}\mathrm{\left(}\mathit{p}\mathrm{\right)}\mathrm{=}\mathrm{1}$ 140

3.20 ${\mathit{f}}^{\mathrm{\prime }}$

141

$\mathrm{x}$

(JIAPTER 1

INTRODUCTION

This introduction will serve as a brief mathematical and historical overview of both

of the main problems that we will examine in this thesis. Due to their somewhat

disparate natures, the two portions of this thesis will each contain a more in-depth

introduction as well. For this reason, we will for the most part relegate formal defi-

nitions to their pertinent introduction.

Ergodic theory is the study of average or long-term behavior of systems which evolve

with time. For example, one can consider a particle bouncing in a box at fixed speed.

Its position and velocity at any time can be represented by a vector in ${\mathbb{R}}^{\mathrm{6}}$. One can

then model this behavior by taking $\mathit{X}$ to be the space of all possible positions and

velocities for the particle, and $\mathit{T}$ a self-map of $\mathit{X}$ which, given a position and velocity,

gives the position and velocity one second later. This pairing of a space $\mathit{X}$ with a

self-map $\mathit{T}$ is called a dynamical system.

The more general setup is that of a group $\mathit{G}$ acting on a space $\mathit{X}$ with some sort

of structure by maps $\mathrm{\{}{\mathit{T}}_{\mathit{g}}{\mathrm{\}}}_{\mathit{g}\mathrm{\in }\mathit{G}}$ preserving this structure. In this thesis, $\mathit{G}$ is always

${\mathbb{Z}}^{\mathit{d}}$ for some $\mathit{d}\mathrm{\in }$ N. Measure-theoretic ergodic theory occurs when $\mathit{X}$ is a measure

space with a probability measure $\mathit{\mu }$ invariant under each ${\mathit{T}}_{\mathit{g}}$, and so in this case we call

1

$\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{,}\mathrm{}\mathrm{\{}{\mathit{T}}_{\mathit{g}}{\mathrm{\}}}_{\mathit{g}\mathrm{\in }\mathit{G}}\mathrm{)}$ a measure-preserving dynamical system. Topological dynam-

ics occurs when $\mathit{X}$ is a compact topological space and each ${\mathit{T}}_{\mathit{g}}$ is a homeomorphism,

and so in this case we call $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathrm{\{}{\mathit{T}}_{\mathit{g}}{\mathrm{\}}}_{\mathit{g}\mathrm{\in }\mathit{G}}\mathrm{)}$ a topological dynamical system. In ei-

ther type of system, if $\mathit{G}\mathrm{=}\mathbb{Z}$, then ${\mathit{T}}_{\mathit{n}}\mathrm{=}\mathrm{(}{\mathit{T}}_{\mathrm{1}}{\mathrm{)}}^{\mathit{n}}$ for any integer $\mathit{n}$, and so we shorten

$\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{,}\mathrm{}\mathrm{\{}{\mathit{T}}_{\mathit{n}}{\mathrm{\}}}_{\mathit{n}\mathrm{\in }\mathbb{Z}}\mathrm{)}$ to $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ and $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathrm{\{}{\mathit{T}}_{\mathit{n}}{\mathrm{\}}}_{\mathit{n}\mathrm{\in }\mathbb{Z}}\mathrm{)}$ to $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$. (Here $\mathit{T}\mathrm{=}{\mathit{T}}_{\mathrm{1}}\mathrm{.}$)

These cases are not as disjoint as they may first appear: the famed Bogoliouboff-

Krylov theorem states that any topological dynamical system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathrm{\{}{\mathit{T}}_{\mathit{g}}{\mathrm{\}}}_{\mathit{g}\mathrm{\in }\mathit{G}}\mathrm{)}$ has at

least one invariant probability measure $\mathit{\mu }$ as long as $\mathit{G}$ is an amenable group. (An

examination of amenability is beyond the scope of this thesis, but we mention that

amenable groups are a class which contain all abelian groups. We only consider

$\mathit{G}\mathrm{=}{\mathbb{Z}}^{\mathit{d}}$ in this thesis, so all topological dynamical systems examined here will possess

invariant measures.) This sometimes allows one to examine topological properties of

a system via properties of invariant measures. For instance, in Chapter 3, we will

prove some purely combinatorial properties of a ${\mathbb{Z}}^{\mathit{d}}$-action by homeomorphisms of a

Cantor set by means of studying the invariant measures of this action.

Chapter 2 deals primarily with ergodic averaging. One of the fundamental results of

ergodic theory is Birkhoff's ergodic theorem:

Theorem 1.0.1. ([Bi]) For any measure-preserving dynamical system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathrm{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$

and any $\mathit{f}\mathrm{\in }{\mathit{L}}^{\mathrm{1}}\mathrm{\left(}\mathit{X}\mathrm{\right)}$, ${\lim }_{\mathit{n}\mathrm{\to }\mathrm{\infty }}\frac{\mathrm{1}}{\mathit{n}}{{\displaystyle \sum }}_{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathit{n}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{\mathit{i}}\mathit{x}\mathrm{\right)}$ exists for $\mathit{\mu }$-almost every $\mathit{x}\mathrm{\in }\mathit{X}$.

Several results have been proven about the convergence of such averages when one

averages not along all powers of $\mathit{T}$, but only along some distinguished subset of the

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integers. ([Bou], [Bou2], [Wi]) In particular, when one averages along $\mathrm{\left\{}\mathit{p}\mathrm{\right(}\mathit{n}\mathrm{\left)}\mathrm{\right\}}$ for a

polynomial $\mathit{p}\mathrm{\left(}\mathit{n}\mathrm{\right)}$ with integer coefficients, there is the following result of Bourgain:

Theorem 2.1.12. ([Bou], p. 7, Theorem t) For any measure-preserving dynamical

system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$, for any polynomial $\mathit{q}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\in }\mathbb{Z}\mathrm{\left[}\mathit{t}\mathrm{\right]}$, and for any $\mathit{f}\mathrm{\in }{\mathit{L}}^{\mathit{p}}\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{)}$ with

$\mathit{p}\mathrm{>}\mathrm{1}$, ${\lim }_{\mathit{N}\mathrm{\to }\mathrm{\infty }}\frac{\mathrm{1}}{\mathit{N}}{{\displaystyle \sum }}_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{\mathit{q}\mathrm{\left(}\mathit{n}\mathrm{\right)}}\mathit{x}\mathrm{\right)}$ exists for $\mathit{\mu }$-almost every $\mathit{x}\mathrm{\in }\mathit{X}$.

Theorem 2.1.12 can be interpreted as follows: for any polynomial $\mathit{q}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\in }\mathbb{Z}\mathrm{\left[}\mathit{t}\mathrm{\right]}$, any

measure-preserving system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$, and any measure-theoretically "nice" func-

tion $\mathit{f}$, the set of points $\mathit{x}$ where ${\lim }_{\mathit{N}\mathrm{\to }\mathrm{\infty }}\frac{\mathrm{1}}{\mathit{N}}{{\displaystyle \sum }}_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{\mathit{q}\mathrm{\left(}\mathit{n}\mathrm{\right)}}\mathit{x}\mathrm{\right)}$ fails to converge is of mea-

sure zero, or negligible measure-theoretically. It is then natural to wonder whether or

not there is a topological parallel to this result using topological notions of "niceness"

(continuity) and negligibility (first category), and in fact such a question was posed

by Bergelson:

Question 2.1.13. ([Be], p. 51, Question 5) Assume that a topological dynamical

system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ is uniquely $\mathit{e}\mathit{r}\mathit{g}\mathit{o}\mathit{d}\mathit{i}{\mathit{c}}^{\mathrm{1}}$, and let $\mathit{p}\mathrm{\in }\mathbb{Z}\mathrm{\left[}\mathit{t}\mathrm{\right]}$ and $\mathit{f}\mathrm{\in }\mathit{C}\mathrm{\left(}\mathit{X}\mathrm{\right)}$. Is it true that for

all but a first category set of points ${\lim }_{\mathit{n}\mathrm{\to }\mathrm{\infty }}\frac{\mathrm{1}}{\mathit{n}}{{\displaystyle \sum }}_{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathit{n}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{\mathit{p}\mathrm{\left(}\mathit{i}\mathrm{\right)}}\mathit{x}\mathrm{\right)}$ exists?

One of our results is a (quite negative) answer to Question 2.1.13:

Theorem 2.1.15. For any increasing sequence $\mathrm{\left\{}{\mathit{p}}_{\mathit{n}}\mathrm{\right\}}$ of integers with upper Banach

density $\mathit{z}\mathit{e}\mathit{r}{\mathit{o}}^{\mathrm{2}}$, there exists a totally minimal, totally uniquely ergodic, and topologically

1A topological dynamical system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ is uniquely ergodic if there exists exactly one T-

invariant measure $\mathit{\mu }$.

$\mathrm{2}\mathrm{A}$ set $\mathit{A}\mathrm{\subseteq }\mathbb{N}$ has upper Banach density zero if $\lim {\sup }_{\mathit{n}\mathrm{\to }\mathrm{\infty }}{\sup }_{\mathit{m}\mathrm{\in }\mathbb{N}}\frac{\mathrm{\left|}\mathrm{\right\{}\mathit{m}\mathrm{,}\mathit{m}\mathrm{+}\mathrm{1}\mathrm{,}\mathrm{\text{...}}\mathrm{,}\mathit{m}\mathrm{+}\mathit{n}\mathrm{-}\mathrm{1}\mathrm{\}}\mathrm{\cap }\mathit{A}\mathrm{|}}{\mathit{n}}\mathrm{=}\mathrm{0}$.

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mixing topological dynamical system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ and a continuous function $\mathit{f}$ on $\mathit{X}$ with

the property that $\frac{\mathrm{1}}{\mathit{N}}{{\displaystyle \sum }}_{\mathit{n}\mathrm{=}\mathrm{0}}^{\mathit{N}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{{\mathit{p}}_{\mathit{n}}}\mathit{x}\mathrm{\right)}$ fails to converge for a residual set of $\mathit{x}$.

We also examine recurrence along distinguished subsets of the integers, motivated

primarily by the following result:

Theorem 2.1.17. ( $\mathrm{\left[}\mathrm{B}\mathrm{e}\mathrm{L}\mathrm{\right]}$, p. 14, Corollary 1.8) For any $\mathit{d}\mathrm{\in }\mathbb{N}$, any $\mathit{m}\mathit{i}\mathit{n}\mathit{i}\mathit{m}\mathit{a}{\mathit{l}}^{\mathrm{3}}$ topolog-

ical dynamical system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathrm{\{}{\mathit{T}}_{\mathit{v}}{\mathrm{\}}}_{\mathit{v}\mathrm{\in }{\mathbb{Z}}^{\mathit{d}}}\mathrm{)}$ and any polynomials ${\mathit{q}}_{\mathrm{1}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$, ${\mathit{q}}_{\mathrm{2}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$, $\mathrm{\text{...}}$, ${\mathit{q}}_{\mathit{d}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\in }\mathbb{Z}\mathrm{\left[}\mathit{t}\mathrm{\right]}$

with ${\mathit{q}}_{\mathit{i}}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}$ for $\mathrm{1}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{i}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{d}$, for a residual set of $\mathit{x}\mathrm{\in }\mathit{X}$ there exists a sequence

$\mathrm{\left\{}{\mathit{n}}_{\mathit{i}}\mathrm{\right\}}$ of positive integers such that ${\mathit{T}}_{\mathit{q}\mathrm{\left(}{\mathit{n}}_{\mathit{i}}\mathrm{\right)}{\mathit{e}}_{\mathit{i}}}\mathit{x}\mathrm{\to }\mathit{x}$ for $\mathrm{1}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{i}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{d}$, where $\mathrm{\{}{\mathit{e}}_{\mathit{i}}{\mathrm{\}}}_{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{d}}$ is the

standard orthonormal basis of ${\mathbb{R}}^{\mathit{d}}$.

In particular, this implies that for any minimal topological dynamical system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$

and any polynomial $\mathit{q}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\in }\mathbb{Z}\mathrm{\left[}\mathit{t}\mathrm{\right]}$ with $\mathit{q}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}$, the set of $\mathit{x}\mathrm{\in }\mathit{X}$ with $\mathit{x}\mathrm{\in }\mathrm{\{}{\mathit{T}}^{\mathit{q}\mathrm{\left(}\mathit{n}\mathrm{\right)}}\mathit{x}{\mathrm{\}}}_{\mathit{n}\mathrm{\in }\mathbb{N}}$

is residual. The following result shows that for $\mathit{q}$ with degree at least two, this residual

set is not necessarily all of $\mathit{X}$.

Theorem 2.1.18. For any increasing sequence $\mathrm{\left\{}{\mathit{p}}_{\mathit{n}}\mathrm{\right\}}$ of integers with upper Banach

density zero, there exists a totally minimal, totally uniquely ergodic, and topologically

mixing topological dynamical system (X,$\mathrm{}$T) and an uncountable set A $\mathrm{\subset }\mathit{X}$ such that

for every x $\mathrm{\in }\mathit{A}$, the sequence $\mathrm{\left\{}{\mathit{T}}^{{\mathit{p}}_{\mathit{n}}}\mathit{x}\mathrm{\right\}}$ does not have x as a limit point, i.e. there is

no sequence of positive integers $\mathrm{\left\{}{\mathit{n}}_{\mathit{i}}\mathrm{\right\}}$ such that ${\mathit{T}}^{{\mathit{p}}_{{\mathit{n}}_{\mathit{i}}}}$x converges to x.

Another consequence of Theorem 2.1.17 is that for any commuting minimal homeo-

morphisms $\mathit{T}$ and $\mathit{S}$ of a compact space $\mathit{X}$, there exists a residual set of $\mathit{x}$ for which

sA topological dynamical system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ is minimal if there exist no nonempty proper closed

$\mathit{T}$-invariant subsets of $\mathit{X}$.

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there exists a sequence of positive integers $\mathrm{\left\{}{\mathit{n}}_{\mathit{i}}\mathrm{\right\}}$ such that ${\mathit{T}}^{{\mathit{n}}_{\mathit{i}}}\mathit{x}\mathrm{\to }\mathit{x}$ and ${\mathit{S}}^{{\mathit{n}}_{\mathit{i}}}\mathit{x}\mathrm{\to }\mathit{x}$.

The following theorem shows that for some systems, it is the case that this residual

set of $\mathit{x}$ is not all of $\mathit{X}$.

Theorem 2.1.21. There exists a totally minimal topological dynamical system (X,$\mathrm{}$T)

and a point x $\mathrm{\in }\mathit{X}$ such that for any positive integers r $\mathrm{\ne }\mathit{s}$, any sequence $\mathrm{\left\{}{\mathit{n}}_{\mathit{i}}\mathrm{\right\}}$ of

integers satisfying ${\mathit{T}}^{\mathit{r}{\mathit{n}}_{\mathit{i}}}\mathit{x}\mathrm{\to }\mathit{x}$ and ${\mathit{T}}^{\mathit{s}{\mathit{n}}_{\mathit{i}}}\mathit{x}\mathrm{\to }\mathit{x}$ is eventually zero.

The systems which we construct to prove Theorems 2.1.15 and 2.1.18 are symbolic

dynamical systems. A symbolic dynamical system is defined by first choosing a

finite set $\mathit{A}$, called the alphabet. $\mathrm{\Omega }\mathrm{=}{\mathit{A}}^{\mathit{G}}$ endowed with the product topology is a

compact space, and for any $\mathit{g}\mathrm{\in }\mathit{G}$, we may define ${\mathit{\sigma }}_{\mathit{g}}$ the shift homeomorphism by

$\mathrm{\left(}{\mathit{\sigma }}_{\mathit{g}}\mathit{\omega }\mathrm{\right)}\mathrm{\left(}\mathit{h}\mathrm{\right)}\mathrm{=}\mathit{\omega }\mathrm{\left(}\mathit{h}\mathit{g}\mathrm{\right)}$ for $\mathit{\omega }$ $\mathrm{\in }$ Q. Any closed set $\mathit{X}\underset{\mathrm{‾}}{\mathrm{\subset }}\mathrm{\Omega }$ has a topology induced by $\mathrm{\Omega }$, and

if $\mathit{X}$ is invariant under each ${\mathit{T}}_{\mathit{g}}$, then $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathrm{\{}{\mathit{\sigma }}_{\mathit{g}}{\mathrm{\}}}_{\mathit{g}\mathrm{\in }\mathit{G}}\mathrm{)}$ is a topological dynamical system

which we call a symbolic dynamical system.

In Chapter 3, we examine symbolic dynamical systems exclusively. There, we consider

a specific type of symbolic dynamical system called a shift of finite type. A ${\mathbb{Z}}^{\mathit{d}}$-shift

of finite type is a symbolic dynamical system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathrm{\{}{\mathit{\sigma }}_{\mathit{v}}{\mathrm{\}}}_{\mathit{v}\mathrm{\in }{\mathbb{Z}}^{\mathit{d}}}\mathrm{)}$ where $\mathit{X}$ is defined by

specifying a finite set ${F}$ of finite words (a word is a function from a finite subset of

${\mathbb{Z}}^{\mathit{d}}$ to $\mathit{A}$) and taking $\mathit{X}$ to be the set of all elements of $\mathrm{\Omega }$ in which none of the words

in ${F}$ appear. The shift of finite type $\mathit{X}$ specified by a finite set ${F}$ of words in this

way is denoted by ${\mathrm{\Omega }}_{{F}}$. For example, the set of all biinfinite sequences of zeroes and

ones in which no two ones occur consecutively is a shift of finite type, as is the set of

5

all ${\mathbb{Z}}^{\mathrm{2}}$ arrays of zeroes and ones in which no three-by-four blocks of all zeroes or all

ones appear.

We are particularly interested in the effects of forbidding a particular word from a

shift of finite type $\mathit{X}$. Given any word $\mathit{w}$, define ${\mathit{X}}_{\mathit{w}}$ to be the set of elements of $\mathit{X}$ in

which $\mathit{w}$ does not appear. Then clearly ${\mathit{X}}_{\mathit{w}}$ is a subset of $\mathit{X}$, and so the topological

entropy ${\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}{\mathit{X}}_{\mathit{w}}\mathrm{\right)}$ of ${\mathit{X}}_{\mathit{w}}$ is not greater than the topological entropy ${\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}\mathit{X}\mathrm{\right)}$ of $\mathit{X}$. It

is natural to wonder how much the topological entropy drops by when $\mathit{w}$ is removed,

as it is a sort of measure of how important $\mathit{w}$ is to the information-retaining capacity

of $\mathit{X}$. In [L], Lind proved the following: (the condition $\mathit{w}\mathrm{\in }{\mathit{L}}_{{\mathrm{\Gamma }}_{\mathit{n}}}\mathrm{\left(}\mathit{X}\mathrm{\right)}$ means that

$\mathit{w}\mathrm{\in }{\mathit{A}}^{\mathrm{[}\mathrm{1}\mathrm{,}\mathrm{\text{...}}\mathrm{,}\mathit{n}{\mathrm{]}}^{\mathit{d}}}$ and that $\mathit{w}$ appears in some element of $\mathit{X}\mathrm{.}$)

Theorem 3.1.16. ([L], p. 360, Theorem 3) For any topologically transitive Z-shift

of finite type $\mathit{X}\mathrm{=}{\mathrm{\Omega }}_{{F}}$ with positive topological entropy ${\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}\mathit{X}\mathrm{\right)}$, there exist constants

${\mathit{C}}_{\mathit{X}}$, ${\mathit{D}}_{\mathit{X}}$, and ${\mathit{N}}_{\mathit{X}}$ such that for any $\mathit{n}\mathrm{>}$ ${\mathit{N}}_{\mathit{X}}$ and any word $\mathit{w}\mathrm{\in }{\mathit{L}}_{{\mathrm{\Gamma }}_{\mathit{n}}}\mathrm{\left(}\mathit{X}\mathrm{\right)}$, if we denote

by ${\mathit{X}}_{\mathit{w}}$ the shift of finite type ${\mathrm{\Omega }}_{{F}\mathrm{u}\mathrm{\left\{}\mathit{w}\mathrm{\right\}}}$, then

$\frac{{\mathit{C}}_{\mathit{X}}}{{\mathit{e}}^{{\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}\mathit{X}\mathrm{\right)}\mathit{n}}}\mathrm{<}{\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}\mathit{X}\mathrm{\right)}\mathrm{-}{\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}{\mathit{X}}_{\mathit{w}}\mathrm{\right)}\mathrm{<}\frac{{\mathit{D}}_{\mathit{X}}}{{\mathit{e}}^{{\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}\mathit{X}\mathrm{\right)}\mathit{n}}}$.

Our main result is a generalization of Theorem 3.1.16 for ${\mathbb{Z}}^{\mathit{d}}$-shifts of finite type

which satisfy a mixing condition called strong irreducibility. (We formally define

strong irreducibility in Definition 3.1.19.)

Theorem 3.1.22. For any d $\mathrm{>}$1 and any strongly irreducible ${\mathbb{Z}}^{\mathit{d}}$-shift X $\mathrm{=}{\mathrm{\Omega }}_{{F}}$

of finite type with uniform filling length R and positive topological entropy ${\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}\mathit{X}\mathrm{\right)}$,

6

there exist constants ${\mathit{N}}_{\mathit{X}}\mathrm{\in }\mathbb{N}$ and ${\mathit{D}}_{\mathit{X}}\mathrm{\in }\mathbb{R}$ such that for any $\mathit{n}\mathrm{>}$ ${\mathit{N}}_{\mathit{X}}$ and any word

$\mathit{w}\mathrm{\in }{\mathit{L}}_{{\mathrm{\Gamma }}_{\mathit{n}}}\mathrm{\left(}\mathit{X}\mathrm{\right)}$, if we denote by ${\mathit{X}}_{\mathit{w}}$ the shift of finite type ${\mathrm{\Omega }}_{{F}\mathrm{u}\mathrm{\left\{}\mathit{w}\mathrm{\right\}}}$, then

$\frac{\mathrm{1}}{{\mathit{e}}^{{\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}\mathit{X}\mathrm{\right)}\mathrm{(}\mathit{n}\mathrm{+}\mathrm{4}\mathrm{4}\mathit{R}\mathrm{+}\mathrm{7}\mathrm{0}{\mathrm{)}}^{\mathit{d}}}}\mathrm{<}{\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}\mathit{X}\mathrm{\right)}\mathrm{-}{\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}{\mathit{X}}_{\mathit{w}}\mathrm{\right)}\mathrm{<}\frac{{\mathit{D}}_{\mathit{X}}}{{\mathit{e}}^{{\mathit{h}}^{\mathit{t}\mathit{o}\mathit{p}}\mathrm{\left(}\mathit{X}\mathrm{\right)}\mathrm{(}\mathit{n}\mathrm{-}\mathrm{2}\mathit{R}{\mathrm{)}}^{\mathit{d}}}}$.

One way in which ${\mathbb{Z}}^{\mathit{d}}$-shifts of finite type are more difficult to deal with for $\mathit{d}\mathrm{>}\mathrm{1}$ is

that given a finite collection ${F}$ of patterns, it is undecidable whether or not the shift

of finite type $\mathit{X}$ induced by ${F}$ is even nonempty. Our methods yield a situation in

which this question can be answered:

Theorem 3.6.1. For any alphabet A, there exist F, G $\mathrm{\in }\mathbb{N}$ such that for any m $\mathrm{>}\mathrm{0}$

and any finite set of words ${{F}}_{\mathit{m}}\mathrm{=}\mathrm{\{}{\mathit{w}}_{\mathit{k}}\mathrm{\in }{\mathit{L}}_{{\mathrm{\Gamma }}_{{\mathit{n}}_{\mathit{k}}}}\mathrm{(}\mathit{X}\mathrm{)}\mathrm{}$: $\mathrm{1}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{k}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{m}\mathrm{\}}$ satisfying ${\mathit{n}}_{\mathrm{1}}\mathrm{>}\mathit{G}$

and ${\mathit{n}}_{\mathit{k}}\underset{\mathrm{‾}}{\mathrm{>}}\mathit{F}\mathrm{(}{\mathit{n}}_{\mathit{k}\mathrm{-}\mathrm{1}}{\mathrm{)}}^{\mathrm{4}{\mathit{d}}^{\mathrm{2}}}$ for $\mathrm{1}\mathrm{<}\mathit{k}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{m}$, ${\mathrm{\Omega }}_{{{F}}_{\mathit{m}}}\mathrm{\ne }\mathrm{\varnothing }$.

7

(JIAPTER 2

RECURRENCE AND CONVERGENCE OF ERGODIC

AVERAGES ALONG SPARSE SEQUENCES

2.1 Introduction

In this chapter, we are concerned with the convergence of averages of the form

$\frac{\mathrm{1}}{\mathit{N}}{{\displaystyle \sum }}_{\mathit{n}\mathrm{=}\mathrm{0}}^{\mathit{N}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{{\mathit{p}}_{\mathit{n}}}\mathit{x}\mathrm{\right)}$ for an increasing sequence of integers $\mathrm{\left\{}{\mathit{p}}_{\mathit{n}}\mathrm{\right\}}$. We begin with some

definitions.

Definition 2.1.1. A measure-preserving dynamical system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{,}\mathrm{}\mathrm{\{}{\mathit{T}}_{\mathit{g}}{\mathrm{\}}}_{\mathit{g}\mathrm{\in }\mathit{G}}\mathrm{)}$

consists of a measure space $\mathit{X}$, a probability measure $\mathit{\mu }$ with $\mathit{\sigma }$-algebra 8 of measurable

sets, and a group action $\mathrm{\{}{\mathit{T}}_{\mathit{g}}{\mathrm{\}}}_{\mathit{g}\mathrm{\in }\mathit{G}}$ of transformations ${\mathit{T}}_{\mathit{g}}$ : $\mathit{X}\mathrm{\to }\mathit{X}$ with $\mathit{\mu }\mathrm{\left(}{\mathit{T}}_{\mathit{g}}^{\mathrm{-}\mathrm{1}}\mathit{A}\mathrm{\right)}\mathrm{=}$

$\mathit{\mu }\mathrm{\left(}\mathit{A}\mathrm{\right)}$ for all $\mathit{A}\mathrm{\in }\mathit{B}$.

Definition 2.1.2. A measure-preserving dynamical system (X,$\mathrm{}$B,$\mathrm{}\mathit{\mu }\mathrm{,}\mathrm{}\mathrm{\left\{}{\mathit{T}}_{\mathit{g}}{\mathrm{\}}}_{\mathit{g}\mathrm{\in }\mathit{G}}\mathrm{\right)}$ is er-

godic if any set A satisfying ${\mathit{T}}_{\mathit{g}}\mathit{A}\underset{\mathrm{‾}}{\mathrm{\subset }}$ A for all g $\mathrm{\in }\mathit{G}$ has $\mathit{\mu }\mathrm{\left(}\mathit{A}\mathrm{\right)}\mathrm{\in }${0,1}.

Definition 2.1.3. A topological dynamical system (X,$\mathrm{}\mathrm{\left\{}{\mathit{T}}_{\mathit{g}}{\mathrm{\}}}_{\mathit{g}\mathrm{\in }\mathit{G}}\mathrm{\right)}$ consists of $\mathit{a}$

compact topological space X and a group action $\mathrm{\{}{\mathit{T}}_{\mathit{g}}{\mathrm{\}}}_{\mathit{g}\mathrm{\in }\mathit{G}}$ of homeomorphisms ${\mathit{T}}_{\mathit{g}}$ :

X $\mathrm{\to }\mathit{X}$.

8

Definition 2.1.4. Given a topological dynamical system (X,$\mathrm{}\mathrm{\left\{}{\mathit{T}}_{\mathit{g}}{\mathrm{\}}}_{\mathit{g}\mathrm{\in }\mathit{G}}\mathrm{\right)}$, a Borel prob-

ability measure $\mathit{\mu }$ on X is called ergodic if (X,$\mathrm{}\mathit{B}\mathrm{\left(}\mathit{X}\mathrm{\right)}$, $\mathit{\mu }$, $\mathrm{\left\{}{\mathit{T}}_{\mathit{g}}{\mathrm{\}}}_{\mathit{g}\mathrm{\in }\mathit{G}}\mathrm{\right)}$ is an ergodic

measure-preserving dynamical system, where $\mathit{B}\mathrm{\left(}\mathit{X}\mathrm{\right)}$ is the Borel $\mathit{\sigma }$-algebra of X.

In this chapter, all dynamical systems have $\mathit{G}\mathrm{=}\mathbb{Z}$, so as already described we

will use the notations $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ and $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ for measure-preserving and topological

dynamical systems respectively.

Definition 2.1.5. A topological dynamical system (X,$\mathrm{}$T) is minimal if for any

closed set K with ${\mathit{T}}^{\mathrm{-}\mathrm{1}}\mathit{K}\underset{\mathrm{‾}}{\mathrm{\subset }}\mathit{K}$, K $\mathrm{=}\mathrm{\varnothing }$ or K $\mathrm{=}\mathit{X}$. (X,$\mathrm{}$T) is totally minimal if

(X,$\mathrm{}{\mathit{T}}^{\mathit{n}}\mathrm{)}$ is minimal for every n $\mathrm{\in }$ O.

Definition 2.1.6. A topological dynamical system (X,$\mathrm{}$T) is uniquely ergodic if

there is only one Borel measure $\mathit{\mu }$ on X such that $\mathit{\mu }\mathrm{\left(}\mathit{A}\mathrm{\right)}\mathrm{=}\mathit{\mu }\mathrm{\left(}{\mathit{T}}^{\mathrm{-}\mathrm{1}}\mathit{A}\mathrm{\right)}$ for every Borel

set $\mathit{A}\underset{\mathrm{‾}}{\mathrm{\subset }}$ X. (X,$\mathrm{}$T) is totally uniquely ergodic if (X,$\mathrm{}{\mathit{T}}^{\mathit{n}}\mathrm{)}$ is uniquely ergodic for

every n $\mathrm{\in }\mathbb{N}$.

Definition 2.1.7. A topological dynamical system (X,$\mathrm{}$T) is topologically mixing

if for any open sets U, $\mathit{V}\underset{\mathrm{‾}}{\mathrm{\subset }}\mathit{X}$, there exists N $\mathrm{\in }\mathbb{N}$ such that for any n $\mathrm{>}$N,

$\mathit{U}\mathrm{\cap }{\mathit{T}}^{\mathit{n}}\mathit{V}\mathrm{\ne }\mathrm{\varnothing }$.

Definition 2.1.8. For any set $\mathit{A}\underset{\mathrm{‾}}{\mathrm{\subset }}\mathbb{N}$, the upper Banach density of A is defined

by

${\mathit{d}}^{\mathrm{*}}\mathrm{\left(}\mathit{A}\mathrm{\right)}\mathrm{=}\lim \sup \sup \frac{\mathrm{\left|}\mathrm{\right\{}\mathit{m}\mathrm{,}\mathit{m}\mathrm{+}\mathrm{1}\mathrm{,}\mathrm{\text{...}}\mathrm{,}\mathit{m}\mathrm{+}\mathit{n}\mathrm{-}\mathrm{1}\mathrm{\}}\mathrm{\cap }\mathit{A}\mathrm{|}}{\mathit{n}}$.

$\mathit{n}\mathrm{\to }\mathrm{\infty }\mathit{m}\mathrm{\in }\mathbb{N}$

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Definition 2.1.9. For any set $\mathit{A}\underset{\mathrm{‾}}{\mathrm{\subset }}\mathbb{N}$, the upper density of A is defined by

$\overline{\mathit{d}}\mathrm{\left(}\mathit{A}\mathrm{\right)}\mathrm{=}{\lim }_{\mathit{n}\mathrm{\to }}{\sup }_{\mathrm{\infty }}\frac{\mathrm{\left|}\mathrm{\right\{}\mathrm{1}\mathrm{,}\mathrm{\text{...}}\mathrm{,}\mathit{n}\mathrm{\}}\mathrm{\cap }\mathit{A}\mathrm{|}}{\mathit{n}}$.

Definition 2.1.10. For a set $\mathit{A}\underset{\mathrm{‾}}{\mathrm{\subset }}\mathbb{N}$, the density of A is defined by

$\mathit{d}\mathrm{\left(}\mathit{A}\mathrm{\right)}\mathrm{=}{\lim }_{\mathit{n}\mathrm{\to }\mathrm{\infty }}\frac{\mathrm{\left|}\mathrm{\right\{}\mathrm{1}\mathrm{,}\mathrm{\text{...}}\mathrm{,}\mathit{n}\mathrm{\}}\mathrm{\cap }\mathit{A}\mathrm{|}}{\mathit{n}}$

if this limit exists.

Definition 2.1.11. Given a topological dynamical system (X,$\mathrm{}$T) and a T-invariant

Borel probability measure $\mathit{\mu }$, a point x $\mathrm{\in }\mathit{X}$ is (T,$\mathrm{}\mathit{\mu }\mathrm{)}$-generic if for every f $\mathrm{\in }\mathit{C}\mathrm{\left(}\mathit{X}\mathrm{\right)}$,

${\lim }_{\mathit{n}\mathrm{\to }\mathrm{\infty }}\frac{\mathrm{1}}{\mathit{n}}{\displaystyle \sum }_{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathit{n}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{\mathit{i}}\mathit{x}\mathrm{\right)}\mathrm{=}{\displaystyle \int }\mathit{f}\mathit{d}\mathit{\mu }$.

In the measure-preserving setup, there are several positive results about convergence

of averages of the form $\frac{\mathrm{1}}{\mathit{N}}{{\displaystyle \sum }}_{\mathit{n}\mathrm{=}\mathrm{0}}^{\mathit{N}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{{\mathit{p}}_{\mathit{n}}}\mathit{x}\mathrm{\right)}$, including this theorem of Bourgain:

Theorem 2.1.12. ([Bou], p. 7, Theorem t) For any measure-preserving dynamical

system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$, for any polynomial $\mathit{q}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\in }\mathbb{Z}\mathrm{\left[}\mathit{t}\mathrm{\right]}$, and for any $\mathit{f}\mathrm{\in }{\mathit{L}}^{\mathit{p}}\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{B}\mathrm{,}\mathrm{}\mathit{\mu }\mathrm{)}$ with

$\mathit{p}\mathrm{>}\mathrm{1}$, ${\lim }_{\mathit{N}\mathrm{\to }\mathrm{\infty }}\frac{\mathrm{1}}{\mathit{N}}{{\displaystyle \sum }}_{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathit{N}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{\mathit{q}\mathrm{\left(}\mathit{n}\mathrm{\right)}}\mathit{x}\mathrm{\right)}$ exists for $\mathit{\mu }$-almost every $\mathit{x}\mathrm{\in }\mathit{X}$.

The following question regarding a possible topological version of Theorem 2.1.12 was

posed by Bergelson:

Question 2.1.13. ([Be], p. 51, Question 5) Assume that a topological dynamical

system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ is uniquely ergodic, and let $\mathit{p}\mathrm{\in }\mathbb{Z}\mathrm{\left[}\mathit{t}\mathrm{\right]}$ and $\mathit{f}\mathrm{\in }\mathit{C}\mathrm{\left(}\mathit{X}\mathrm{\right)}$. Is it true that for

all but a first category set of points ${\lim }_{\mathit{n}\mathrm{\to }\mathrm{\infty }}\frac{\mathrm{1}}{\mathit{n}}{{\displaystyle \sum }}_{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathit{n}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{\mathit{p}\mathrm{\left(}\mathit{i}\mathrm{\right)}}\mathit{x}\mathrm{\right)}$ exists?

to

Bergelson added the hypothesis of unique ergodicity because it is a classical result

that a system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ is uniquely ergodic with unique $\mathit{T}$-invariant measure $\mathit{\mu }$ if and

only if for every $\mathit{x}\mathrm{\in }\mathit{X}$ and $\mathit{f}\mathrm{\in }\mathit{C}\mathrm{\left(}\mathit{X}\mathrm{\right)}$, ${\lim }_{\mathit{n}\mathrm{\to }\mathrm{\infty }}\frac{\mathrm{1}}{\mathit{n}}{{\displaystyle \sum }}_{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathit{n}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{\mathit{i}}\mathit{x}\mathrm{\right)}\mathrm{=}{{\displaystyle \int }}_{\mathit{X}}\mathit{f}\mathit{d}\mathit{\mu }$, and so in

the topological setup this is a natural assumption to make about $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ to achieve

the desired result.

However, Bergelson was particularly interested in the convergence of these averages

to the "correct limit," i.e. ${{\displaystyle \int }}_{\mathit{X}}\mathit{f}\mathit{d}\mathit{\mu }$ where $\mathit{\mu }$ is the unique $\mathit{T}$-invariant measure on

$\mathit{X}$. To have any hope for such a result, it also becomes necessary to assume ergod-

icity for all powers of $\mathit{T}$ in order to avoid some natural counterexamples related to

distribution (mod $\mathit{k}$) of $\mathit{p}\mathrm{\left(}\mathit{n}\mathrm{\right)}$ for positive integers $\mathit{k}$. For example, if $\mathit{p}\mathrm{\left(}\mathit{n}\mathrm{\right)}\mathrm{=}{\mathit{n}}^{\mathrm{2}}$, $\mathit{T}$ is

the permutation on $\mathit{X}\mathrm{=}\mathrm{\{}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{2}\mathrm{\}}$ defined by $\mathit{T}\mathit{x}\mathrm{=}\mathit{x}\mathrm{+}\mathrm{1}$ (mod 3), $\mathit{\mu }$ is normalized

counting measure on $\mathit{X}$, and $\mathit{f}\mathrm{=}\mathit{\chi }\mathrm{\left\{}\mathrm{0}\mathrm{\right\}}$, then $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ is obviously uniquely ergodic with

unique invariant measure $\mathit{\mu }\mathrm{=}\frac{{\mathit{\delta }}_{\mathrm{0}}\mathrm{+}{\mathit{\delta }}_{\mathrm{1}}\mathrm{+}{\mathit{\delta }}_{\mathrm{2}}}{\mathrm{3}}$, but

${\lim }_{\mathit{n}\mathrm{\to }\mathrm{\infty }}\frac{\mathrm{1}}{\mathit{n}}{\displaystyle \sum }_{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathit{n}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{\mathit{p}\mathrm{\left(}\mathit{i}\mathrm{\right)}}\mathit{x}\mathrm{\right)}\mathrm{=}$ $\mathrm{i}\mathrm{f}\mathit{x}\mathrm{=}\mathrm{1}\mathrm{i}\mathrm{f}\mathit{x}\mathrm{=}\mathrm{0}\mathrm{i}\mathrm{f}\mathrm{f}\mathit{x}\mathrm{=}\mathrm{2}\mathrm{\text{'}}$, .and

To avoid such examples, we would need $\mathit{T}$ to be totally ergodic as well as uniquely

ergodic, and so it makes sense to assume total unique ergodicity to encompass both

properties. Bergelson's revised question then looks like this:

Question 2.1.14. Assume that a topological dynamical system (X,$\mathrm{}$T) is totally uniquely

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ergodic with unique $\mathit{T}$-invariant measure $\mathit{\mu }$, and let $\mathit{p}\mathrm{\in }\mathbb{Z}\mathrm{\left[}\mathit{t}\mathrm{\right]}$ and $\mathit{f}\mathrm{\in }\mathit{C}\mathrm{\left(}\mathit{X}\mathrm{\right)}$. Is it true

that for all but a first category set of points ${\lim }_{\mathit{n}\mathrm{\to }\mathrm{\infty }}\frac{\mathrm{1}}{\mathit{n}}{{\displaystyle \sum }}_{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathit{n}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{\mathit{p}\mathrm{\left(}\mathit{i}\mathrm{\right)}}\mathit{x}\mathrm{\right)}\mathrm{=}{{\displaystyle \int }}_{\mathit{X}}\mathit{f}\mathit{d}{\mathit{\mu }}^{\mathit{l}}\mathrm{.}$?

We answer Questions 2.1.13 and 2.1.14 negatively in the case where the degree of

$\mathit{p}$ is at least two, and in fact prove some slightly more general results. The level of

generality depends on what hypotheses we place on the space $\mathit{X}$. In particular, we

can exhibit more counterexamples in the case where $\mathit{X}$ is a totally disconnected space

than we can in the case where $\mathit{X}$ is a connected space such as ${\mathbb{T}}^{\mathit{k}}$.

Theorem 2.1.15. For any increasing sequence $\mathrm{\left\{}{\mathit{p}}_{\mathit{n}}\mathrm{\right\}}$ of integers with upper Banach

density zero, there exists a totally minimal, totally uniquely ergodic, and topologically

mixing topological dynamical system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ and a continuous function $\mathit{f}$ on $\mathit{X}$ with

the property that $\frac{\mathrm{1}}{\mathit{N}}{{\displaystyle \sum }}_{\mathit{n}\mathrm{=}\mathrm{0}}^{\mathit{N}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{{\mathit{p}}_{\mathit{n}}}\mathit{x}\mathrm{\right)}$ fails to converge for a residual set of $\mathit{x}$.

Theorem 2.1.16. For any increasing sequence $\mathrm{\left\{}{\mathit{p}}_{\mathit{n}}\mathrm{\right\}}$ of integers with the property

that for some integer $\mathit{d}$, ${\mathit{p}}_{\mathit{n}\mathrm{+}\mathrm{1}}\mathrm{<}\mathrm{(}{\mathit{p}}_{\mathit{n}\mathrm{+}\mathrm{1}}\mathrm{-}{\mathit{p}}_{\mathit{n}}{\mathrm{)}}^{\mathit{d}}$ for all suffiffifficiently large $\mathit{n}$, there ex-

ists a totally minimal, totally uniquely ergodic, and topologically mixing topological

dynamical system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ and a continuous function $\mathit{f}$ on $\mathit{X}$ with the property that

$\frac{\mathrm{1}}{\mathit{N}}{{\displaystyle \sum }}_{\mathit{n}\mathrm{=}\mathrm{0}}^{\mathit{N}\mathrm{-}\mathrm{1}}\mathit{f}\mathrm{\left(}{\mathit{T}}^{{\mathit{p}}_{\mathit{n}}}\mathit{x}\mathrm{\right)}$ fails to converge for a residual set of $\mathit{x}$. In addition, the space $\mathit{X}$ is

a connected $\mathrm{2}\mathit{d}\mathrm{+}$ 9-manifold.

We note that Theorems 2.1.15 and 2.1.16 answer Question 2.1.13 negatively for $\mathit{p}$

with degree at least two, since the sequence ${\mathit{p}}_{\mathit{n}}\mathrm{=}\mathit{p}\mathrm{\left(}\mathit{n}\mathrm{\right)}$ for any nonlinear $\mathit{p}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\in }\mathbb{Z}\mathrm{\left[}\mathit{t}\mathrm{\right]}$

satisfies the hypotheses of both theorems.

Theorems 2.1.15 and 2.1.16 are about nonconvergence of ergodic averages along cer-

tain sequences of powers of $\mathit{x}$. We also prove two similar results about nonrecurrence

12

of points. As motivation, we note that a minimal system has the property that every

point is recurrent. In other words, if $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ is minimal, then for all $\mathit{x}\mathrm{\in }\mathit{X}$, it is the

case that $\mathit{x}\mathrm{\in }\mathrm{\{}{\mathit{T}}^{\mathit{n}}\mathit{x}{\mathrm{\}}}_{\mathit{n}\mathrm{\in }\mathbb{N}}$. If $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathit{T}\mathrm{)}$ is totally minimal, then all points are recurrent

even along infinite arithmetic progressions: for any nonnegative integers $\mathit{a}$, $\mathit{b}$, and for

all $\mathit{x}\mathrm{\in }\mathit{X}$, $\mathit{x}\mathrm{\in }\mathrm{\{}{\mathit{T}}^{\mathit{a}\mathit{n}\mathrm{+}\mathit{b}}\mathit{x}{\mathrm{\}}}_{\mathit{n}\mathrm{\in }\mathbb{N}}$. It is then natural to wonder if the same is true for other

sequences of powers of $\mathit{T}$, and in this vein there is the following result of Bergelson

and Leibman, which is a corollary to their Polynomial van der Waerden theorem:

Theorem 2.1.17. ( $\mathrm{\left[}\mathrm{B}\mathrm{e}\mathrm{L}\mathrm{\right]}$, p. 14, Corollary 1.8) For any $\mathit{d}\mathrm{\in }\mathbb{N}$, any minimal topolog-

ical dynamical system $\mathrm{(}\mathit{X}\mathrm{,}\mathrm{}\mathrm{\{}{\mathit{T}}_{\mathit{v}}{\mathrm{\}}}_{\mathit{v}\mathrm{\in }{\mathbb{Z}}^{\mathit{d}}}\mathrm{)}$ and any polynomials ${\mathit{q}}_{\mathrm{1}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$, ${\mathit{q}}_{\mathrm{2}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$,