ABSTRACT the Specification Property and Chaos In

ABSTRACT

The Speciﬁcation Property and Chaos in Multidimensional Shift Spaces and General Compact Metric Spaces

Reeve Hunter, Ph.D.

Advisor: Brian E. Raines, D.Phil.

Rufus Bowen introduced the speciﬁcation property for maps on a compact metric space. In this dissertation, we consider some implications of the speciﬁcation

d property for Zd-actions on subshifts of ΣZ as well as on a general compact metric space. In particular, we show that if σ : X X is a continuous Zd-action with → d a weak form of the speciﬁcation property on a d-dimensional subshift of ΣZ , then

σ exhibits both ω-chaos, introduced by Li, and uniform distributional chaos, introduced by Schweizer and Smítal. The ω-chaos result is further generalized for some broader, directional notions of limit sets and general compact metric spaces with uniform expansion at a ﬁxed point. The Speciﬁcation Property and Chaos in Multidimensional Shift Spaces and General Compact Metric Spaces

Reeve Hunter, B.A.

A Dissertation

Approved by the Department of Mathematics

Lance L. Littlejohn, Ph.D., Chairperson

Submitted to the Graduate Faculty of Baylor University in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

Approved by the Dissertation Committee

Brian E. Raines, D.Phil., Chairperson

Nathan Alleman, Ph.D.

Will Brian, D.Phil.

Markus Hunziker, Ph.D.

David Ryden, Ph.D.

Accepted by the Graduate School August 2016

J. Larry Lyon, Ph.D., Dean

Page bearing signatures is kept on ﬁle in the Graduate School. Copyright c 2016 by Reeve Hunter

LIST OF FIGURES vi

ACKNOWLEDGMENTS vii

DEDICATION viii

1 Introduction 1

2 Preliminaries 4

2.1 Dynamical Systems ...... 4

2.2 Notions of Chaos ...... 5

2.3 The Speciﬁcation Property ...... 13

2.4 Symbolic Dynamics and Shift Spaces ...... 20

2.5 Zd-Actions and Multidimensional Shift Spaces ...... 22

d 3 Chaos in Subshifts of ΣZ Via the Weak Speciﬁcation Property 28

3.1 General ω-Limit Sets for Zd-Actions ...... 28

d 3.2 Minimal Sets in Subshifts of ΣZ ...... 29

3.3 ω-Chaos ...... 37

3.4 Directional Limit Sets ...... 41

3.5 Directional Limit Chaos ...... 43

3.6 Distributional Chaos ...... 44

3.7 Some Implications ...... 51

iv 4 Speciﬁcation and ω-Chaos in a General Compact Metric Space 53

4.1 Some Deﬁnitions and Constructions ...... 54

4.2 Constructing a Set Exhibiting ω-Chaos ...... 58

5 Future Work 63

5.1 More with ω-Chaos ...... 63

5.2 Relations Between Various Types of Chaos for Zd-Actions ...... 64 5.3 Tiling Spaces ...... 64

5.4 Maps on Dendrites ...... 66

BIBLIOGRAPHY 70

v LIST OF FIGURES

2.1 Tracing orbits with the speciﬁcation property ...... 16

2 2.2 An example of a point in ΣZ ...... 25

d 2.3 The shift action for ΣZ ...... 25

3.1 Block gluing ...... 31

3.2 Gluing multiple patches together ...... 33

d 3.3 “Exploded” point from ΣZ ...... 34

2 3.4 ω-chaos: Building shells in ΣZ ...... 39

3 3.5 ω-chaos: Building shells in ΣZ ...... 40

3.6 Directional limit sets ...... 42

3.7 Directional limit sets for multiple directions ...... 43

2 3 3.8 Uniform DC1: Building shells in ΣZ and ΣZ ...... 47

d 3.9 Uniform DC1: Stripped shells in ΣZ ...... 48

4.1 ω-chaos: Deﬁning the sets U0 and U1 ...... 55

4.2 ω-chaos: Using the speciﬁcation property to construct points . . . . . 57

4.3 ω-chaos: Finding limit points with controlled orbits ...... 59

5.1 An example of a Penrose tiling ...... 65

5.2 Graphs, trees, and dendrites ...... 67

5.3 A dendritic Julia set ...... 68

vi ACKNOWLEDGMENTS

I would like to thank Brian Raines for giving me a wonderful introduction to Topology, an excellent example of understated mastery in the classroom, and the gift of agreeing to be my advisor. I appreciate your continual encouragement and steadfastness throughout my time here. Thanks for your valuable insights, your endless patience, and your genuine concern for me—both academically and personally.

To my colleagues at Baylor: your varied perspectives helped me to break out of the narrow view I often found myself developing in both teaching and research.

In particular, I would like to say thank you to Nathan and Tim for your help and companionship over the past ﬁve years. When things seemed especially bleak in the dark halls of Sid Richardson, we had Anamanaguchi’s “Meow” to thank for lifting our spirits. Come to think of it, any time was a good time for “Meow.”

Thank you to my family for your support during my time in grad school. To

Mom, Dad, Zellyn, Bevin, Trent, Leah, Delvyn, Charity, and Cheryl: though the times we have to spend together are too few, the many visits, phone calls, and conversations have been a precious, constant presence in my life.

Finally, I must thank my wife, Jessie, for agreeing together with me to venture on this journey. Thank you for all your support, both practical and otherwise, your willingness to work towards a common goal when things seemed insurmountable, and most of all for your friendship.

vii To Chuzzlewit

viii CHAPTER ONE

Introduction

Topological dynamics, along with the related ﬁelds of ergodic theory and differentiable dynamics, has its roots in celestial mechanics. In 1609, Johannes Kepler, using data from the Danish astronomer Tycho Brahe, oﬀered a fairly accurate ge- ometric description of the movement of planets in our solar system, [59]. In 1687,

Isaac Newton presented general, mathematically rigorous principles describing the interaction of any two celestial bodies, [78]. Generalizations of Newton’s results to more than two bodies proved diﬃcult to pin down. Modern topological dynamics, ergodic theory, and diﬀerentiable dynamics can all be traced back to Henri Poincaré’s work on the three body problem, the study of how three celestial bodies interact under the laws of physics. For a comprehensive history of Poincaré’s exploration of this problem, see [15].

Rather than focusing on the general structure of solutions to differential equations based on their initial value, Poincaré chose instead to consider the big picture behavior and interactions of all possible individual solutions. In today’s terminology, he was considering the orbit structure of a system rather than focusing on a single orbit. The Poincaré Recurrence Theorem is the first major theorem in dynamical systems that sprung from this view. For some interesting historical notes and references on this theorem, see [18]. Poincaré’s work had a profound influence on George Birkhoff who, in the early 1900’s, continued to apply this new approach while emphasizing the importance of discrete dynamical systems in understanding continuous differential equations, [20]. A nice explanation of the differences between

1 the traditional diﬀerential equations approach and the dynamical systems approach is outlined by Akin in his survey of Topological Dynamics, [3].

Gustav Arnold Hedlund continued in the vein of Birkhoﬀ’s work bringing this new view of dynamical systems to a wider audience. Seeking to unify the discrete and continuous time views of dynamical systems, Hedlund with his student Walter

Gottschalk addressed both views along with the general approach of topological group actions acting on space. In the mid 1950’s, these ideas and others were presented in the first work dedicated to the topological view of dynamics, [51]. As well as introducing the term “topological dynamics” to distinguish the study from the analytical perspective, Hedlund was also influential in the subfield of symbolic dynamical systems.

With an interest in the chaotic properties of systems, one goal of this dissertation is to explore two of the notions mentioned in the previous paragraph: symbolic dynamics and group actions. Chaos theory can be traced back to Poincaré and the idea of sensitive dependence on initial conditions (more on this in Chapter Two).

We focus on two notions formulated in the early 1990’s. In particular, we address

d distributional chaos and ω-chaos in the context of subshifts of ΣZ with a weak form of the speciﬁcation property and ω-chaos in the context of general compact metric spaces with the speciﬁcation property. In Chapter Three, we address Zd-subshifts.

d Theorem 3.8. Assume X ΣZ is a nondegenerate subshift with the weak speciﬁ- ⊂ cation property. Then σ has ω-chaos.

d Theorem 3.15. If X ΣZ is a nondegenerate subshift with the weak speciﬁcation ⊂ property then (X, σ) is uniformly distributionally chaotic of type 1.

2 Theorem 3.8 is further broadened to include some directional variants of ω- chaos. In Chapter Four we prove a result for general compact metric spaces.

Theorem 4.6. Assume that X is a compact metric space with f : X X a surjective → map with the speciﬁcation property. Assume s X is a ﬁxed point such that there ∈ is η > 0, λ > 1 such that if 0 < d(s, y) < η, then d(f(s), f(y)) λd(s, y). Then f is ≥ ω-chaotic.

In Chapter Two, we give some historical context for the problems presented in this dissertation as well as all necessary preliminary deﬁnitions and theorems. In

Chapter Three, we introduce generalized ω-limit sets and prove that the speciﬁcation

d property implies ω-chaos for any ΣZ -subshifts. We show further that the same assumption implies uniform distributional chaos of type 1. It is known that an expansive dynamical system with the speciﬁcation property has ω-chaos. In Chapter

Four, we show that speciﬁcation along with the weakened assumption of uniform expansion at a ﬁxed point is enough to imply ω-chaos. Finally, we present some areas of further interest and ideas for future work in Chapter Five.

3 CHAPTER TWO

Preliminaries

2.1 Dynamical Systems

Throughout this dissertation, unless otherwise stated, X will be a compact metric space f : X X a continuous map. A dynamical system, (X, f), is a → topological space together with a map f : X X. We use N to denote the set of → natural numbers and ω = N0 to denote the natural numbers with zero. We use Z to denote the set of integers and R to denote the set of real numbers. For any n N0 and ∈ x X, we denote the n-fold composition of f with itself by f n(x) where f 0(x) = x. ∈ We denote by f −1(x) the preimage of x under f, i.e. f −n(x) = y X : f n(y) = x . { ∈ } In the case that f is invertible, we write f −n(x) = y rather than f −n(x) = y . { } For any x X, we call the set of all forward iterates of x under f, f n(x): ∈ { + n N0 , the forward orbit of x under f, denoted Orbf (x). In the case that f is ∈ } −n invertible, we call the set of points f (x): n N0 the backward orbit of x under { ∈ } n − f and the set of points f (x): n Z the orbit of x under f, denoted Orbf (x) { ∈ } and Orbf (x), respectively. When the context is clear, we may omit the subscript in any of these abbreviations, e.g. Orbf (x) = Orb(x).

If f p(x) = x for some p N, then we say x is periodic. If p is the smallest ∈ number such that f p(x) = x, we say x has period p. If f(x) = x, we call x a fixed point. Denote by Pern(f) the set of all periodic points of period n for f. Then we write Per(f) = S Per (f). We say x is eventually periodic if f n(x) is periodic for n∈N n + some n N0. Note that Orb(x) is finite if, and only if, x is periodic, and Orb (x) ∈ is finite if, and only if, x is eventually periodic.

4 A point x is recurrent for f if, for any open set U x, there is n N such 3 ∈ that f n(x) U. More generally, we call a point x non-wandering for f if for any ∈ open set U x there is a point y U and an n N such that f n(y) U. The set 3 ∈ ∈ ∈ of non-wandering points for f is denoted by Ω(f) or simply Ω when the context is clear. Periodicity, as well as the more generalized notions of a point “returning” to its starting position, are important to the study of dynamical systems. For instance, the topological entropy (see Deﬁnition 2.1) of a continuous map on a compact metric space is completely determined by the behavior of the non-wandering points, [4,26].

The ω-limit set of x, denoted by ωf (x), or ω(x) is the set of accumulation points of Orb+(x), often written as T f k(x): k > n . A point y is in ω(x) if, and only n∈N { }

nk if, there is an increasing sequence of natural numbers, (nk), such that f (x) y. →

When f is invertible, the α-limit set of x is deﬁned as αf (x) = α(x) = ωf −1 (x), the

ω-limit set of x under the map f −1. For any x X, ω(x) is a closed invariant subset ∈ of X.

2.2 Notions of Chaos

Various notions of predictability and complexity are important to the study of dynamical systems. Already mentioned are notions of periodicity and recurrence— notions that describe repetitive behavior of a dynamical system under iteration. We now introduce some notions of complexity, beginning with the ideas of mixing and entropy borrowed from ergodic theory (i.e. measurable dynamics) and moving on to topological notions of chaos.

Transitivity, Weakly Mixing, Mixing, and Exactness

The various concepts of mixing presented here are meant to capture the essence of physical mixing of gases and liquids important to the study of thermodynamics.

5 Roughly speaking, any given small region of a system should eventually “mix” with any other given small region under some process. These “mixing” notions ensure that a system can not be decomposed into smaller subsystems that do not interact—every open set interacts with every other open set under the process of iteration. We begin with the weakest notion and move toward the strongest.

For a dynamical system (X, f), we say that f is (topologically) transitive if for any open sets U, V X, there is n N such that f n(U) V = . Furthermore, ⊆ ∈ ∩ 6 ∅ (X, f) is (topologically) weakly mixing if the product (X X, f f) is transitive. × × Equivalently, (X, f) is weakly mixing if for every open U, V, W X, there is n N ⊆ ∈ such that f n(U) W = and f n(V ) W = . We say (X, f) is (topologically) ∩ 6 ∅ ∩ 6 ∅ n mixing if for any open U, V X, there is N N0 such that f (U) V = for all ⊂ ∈ ∩ 6 ∅ n N. Finally, (X, f) is (topologically) exact if for any open U X, there is n N ≥ ⊆ ∈ such that f n(U) = X, i.e. any open set eventually maps onto the whole space. The following implications are straightforward:

Exact Mixing Weakly Mixing Transitivity ⇒ ⇒ ⇒ In addition, a point x is said to be a transitive point for f if it has a dense orbit, i.e.

Orbf (x) = X. When X is a compact metric space and f is surjective, the existence of a transitive point is equivalent to topological transitivity.

Entropy

In the 1940’s, Claude E. Shannon developed the concept of entropy in the context of information theory, [97]—his work at Bell Labs focused on eﬃciency of telecommunication systems. Inspired by information theory in the late 1950’s,

Andrey Kolmogorov together with his student Yakov Sinai formulated a notion of metric entropy (or measure-theoretic entropy) for a dynamical system, [60, 100].

6 The ﬁrst deﬁnition of topological entropy, mimicking Kolmogorov-Sinai entropy but relying solely on the topological structure of a space, was introduced by Roy Adler,

Alan Konheim, and M. H. McAndrew in the mid 1960’s, [2]. In the early 1970’s,

Rufus Bowen and Efim Dinaburg later reformulated independently this definition for metric spaces, [29, 43]. This latter definition was proved equivalent to the original definition of topological entropy in [30]. Although generalizations of this definition apply to non-compact spaces, [31], in what follows we assume X is a compact metric space. We give a definition following Bowen and Dinaburg.

Given a dynamical system (X, f), we say a set E X is (n, )-separated ⊆ if for every x, y E with x = y we have that there is 0 i < n such that ∈ 6 ≤ d(f i(x), f i(y)) . Let s(n, ) be the maximum cardinality of an (n, )-separated ≥ set in X.

Deﬁnition 2.1. The (topological) entropy of a function f acting on a compact metric space X is given by 1 h(f) = lim lim sup log s(n, ) →0 n→∞ n We may deﬁne entropy using spanning sets rather than separating sets. We say E X is (n, )-spanning if for every x X, there is y E such that ⊆ ∈ ∈ d(f i(x), f i(y)) < for all 0 i < n. Let r(n, ) be the minimum cardinality of ≤ an (n, )-spanning set in X. Then 1 h(f) = lim lim sup log r(n, ) →0 n→∞ n Intuitively, one may think of s(n, ) as the number of orbit segments of length n needed to approximate any orbit segment of length n up to an accuracy of .

Then h(f) can be viewed as the asymptotic growth rate of essentially distinct orbit types. It is therefore a measure of complexity of (X, f) with range R+ . ∪ {∞} Topological entropy is preserved under topological conjugacy and may therefore 7 be used to distinguish dynamical systems exhibiting diﬀerent entropies. Positive topological entropy is sometimes referred to as topological chaos, [87].

Li-Yorke Chaos

Although the word “chaos” had some limited usage in mathematics already, the 1975 paper of Tien Yien Li and James Yorke brought the term (as it is currently understood) into the world of dynamics, [69]. The main theorem of the paper states that, for continuous maps of the interval, the existence of a periodic point of period three implies periodic points of any given period (a special case of Sharkovski˘ı’s

Theorem, [98]) as well as an uncountable set of points that exhibits “chaotic” behavior under f which we now describe.

n n Two points x, y X are said to be proximal if lim infn→∞ d(f (x), f (y)) = 0 ∈ n n and distal if lim infn→∞ d(f (x), f (y)) = 0. 6

Deﬁnition 2.2. A set S X is scrambled if for all x, y S with x = y we have ⊆ ∈ 6 n n (1) lim infn→∞ d(f (x), f (y)) = 0, and

n n (2) lim supn→∞ d(f (x), f (y)) > 0.

A function is said to chaotic in the sense of Li and Yorke (brieﬂy LY-chaotic) if there exists an uncountable scrambled set.

Li and Yorke’s original deﬁnition had the additional assumption that for any p X periodic, lim sup d(f n(x), f n(p)) > 0 for all x S, i.e. S should contain ∈ n→∞ ∈ no asymptotically periodic points. Since the existence of two such points in S con- tradicts either condition (1) or (2) above, the set S may contain at most one such point, z, so that S z witnesses LY-chaos. The additional assumption is therefore \{ } unnecessary.

8 Devaney Chaos

One of the earliest notions of chaotic behavior is sensitive dependence on initial conditions. The basic idea is that minute changes in a systems beginning state can have profound, unforeseen effects as time moves forward. Poincaré noticed this behavior as he studied the three body problem. Furthermore, Poincaré in his lectures before the Société de Psychologie in Paris accurately predicted this phenomenon’s influence in meteorology, [84]. Half a century later, Edward Lorenz encountered problems in weather modeling when small rounding error produced massive changes in results, [71]. Lorenz’s findings and coining of the term “butterfly effect,” along with writings of James Gleick, Ivar Ekeland, and Michael Crichton, [40,45,48,107], aided to bring ideas of chaos into the world of popular science and pop culture.

A dynamical system (X, f) has sensitive dependence on initial conditions

(brieﬂy SDIC) if there is δ > 0 such that for any x X and any neighborhood ∈ n n U x, there is y U and n N0 with d(f (x), f (y)) > δ. 3 ∈ ∈

Deﬁnition 2.3 ([42]). A dynamical system (X, f) is called chaotic in the sense of

Devaney (brieﬂy Devaney chaotic), if the following three conditions hold:

(1) f is transitive,

(2) periodic points are dense in X,

(3) and f has sensitive dependence on initial conditions.

For spaces that are inﬁnite, Banks, Brooks, Cairns, Davis, and Stacey have shown that condition (3) is unnecessary, [12]. Nevertheless, SDIC is often still in- cluded in Devaney’s deﬁnition as it appeared in 1986—see for instance [14].

9 Distributional Chaos

In 1994, Berthold Schweizer and Jaroslav Smítal put forth a new notion of chaos for interval maps borrowing ideas from the theory of probabilistic measure spaces, [95]. They later generalized this result to a general compact metric space formulating three distinct versions presented here, [11].

Let (X, f) be a dynamical system. For any x, y X and any n N, deﬁne ∈ ∈ (n) 1 i i Φ () = i : d(f (x), f (y)) < and i < n xy n { } (n) So Φxy () is the percentage of iterates of x and y that are within of one another when considering only the ﬁrst n iterates.

As in the earlier work of Schweizer and Abe Sklar, [94], we introduce distribution functions on X. Let

(n) Φxy() = lim inf Φxy (), and n→∞

∗ (n) Φxy() = lim sup Φxy (). n→∞ These non-decreasing functions are called the lower and upper (distance) distribution

∗ functions, respectively. Note that Φxy() = Φxy() = 0 for < 0 and that Φxy() = Φ∗ () = 1 for diam(X). xy ≥

Deﬁnition 2.4. Let (X, f) be a compact metric space together with continuous function on X. A pair of points x, y X is called distributionally chaotic of type 1, 2, ∈ and 3 (brieﬂy DC1, DC2, and DC3) if

∗ (DC1) Φ 1 and Φxy() = 0 for some > 0, xy ≡ ∗ ∗ (DC2) Φ 1 and Φxy < Φ for some > 0, or xy ≡ xy ∗ (DC3) Φxy < Φxy for some > 0, respectively. For each i 1, 2, 3 , the system (X, f) is called DCi if there exists ∈ { } an uncountable set D such that each pair of distinct x, y D is DCi. If in (DC1) ∈

10 above does not depend on the choice of x, y D, then we say (X, f) is uniformly ∈ distributionally chaotic of type 1 (brieﬂy uniformly DC1 ).

From the deﬁnition, we have

Uniformly DC1 DC1 DC2 DC3. ⇒ ⇒ ⇒ Note that in [11], as well as elsewhere in the literature, a chaotic pair rather than an uncountable set is enough to classify (X, f) as distributionally chaotic.

It is sometimes easier to consider the requirements of (DC1) in terms of sub- sequences. Notice that if Φ∗ 1, then for some strictly increasing subsequence of xy ≡ naturals, (ni)i∈N, we have that, for any > 0, the percentage of the ﬁrst ni iterates of x and y that are within of one another tends to 1 as i . Intuitively, if we → ∞ look at a carefully selected sequence of initial iterates of x and y, it looks as if x and y stay close together most of the time under iteration. On the other hand, if

Φxy() = 0 for some > 0, then there is some other strictly increasing subsequence of naturals, (nj)j∈N, for which the percentage of the ﬁrst nj iterates of x and y that are at least apart tends to 1 as j . Then, there is a sequence of initial iterates → ∞ of x and y on which it seems x and y stay apart most of the time.

Omega Chaos

In 1993 Shihai Li introduced ω-chaos, a form of chaos that combines the ideas of Li and Yorke with ω-limit sets, [68]. Li wished to present a form of chaos similar to LY-chaos that was equivalent to positive topological entropy for continuous maps of the interval. While this equivalence does hold for interval maps, it is not true for a general compact metric space, [52,54].

Deﬁnition 2.5. A set Ω X is called an ω-scrambled set for f if for any x, y Ω ⊆ ∈ with x = y we have 6 11 (1) ω(x) ω(y) is uncountable, \ (2) ω(x) ω(y) = , ∩ 6 ∅ (3) ω(x) contains a point that is not periodic.

The map f has ω-chaos if there is an uncountable ω-scrambled set.

Some Relationships Between the Various Notions of Chaos

Some other notions of chaos appearing in the literature but not discussed here are the chaos of Wiggins, [111], Martelli, [72], Block and Coppel, [21], Auslander and Yorke, [7], various other versions of Li-Yorke chaos, [35,61,83,105,106], stronger versions of Devaney chaos, [65,79], and a few weaker versions of ω-chaos, [52].

The relationships between various types of chaos have been a major area of study for the past few decades. For the case when X is homeomorphic to the compact interval, I = [0, 1], these relationships have been studied extensively—see for instance [104], [47], and the fairly comprehensive presentation in [89] which also addresses generalizations to graph maps. We brieﬂy mention that for maps of the interval, the following are equivalent: positive topological entropy, Devaney chaos,

ω-chaos, and the existence of a DC1, DC2, DC3, or ω-scrambled pair. All of the previous properties imply Li-Yorke chaos, but the converse does not hold.

In a general compact metric space, the implications between diﬀerent types of chaos mentioned are weaker. There are no implications, for instance, between positive topological entropy, Devaney chaos, ω-chaos, and the existence of a DC1 pair. On the other hand, all of these properties imply the existence of a LY scrambled pair. It is interesting to note that ω-chaos does not, in general, imply an uncountable

LY-scrambled set, [82]—this is the case even for subshifts of ΣZ (see Section 2.4). An

12 excellent, short reference for more on the relationship between chaotic properties in a general compact metric space is [79], and some recent results are outlined in [67].

2.3 The Speciﬁcation Property

Before we introduce the definition of the specification property, Definition 2.9, we outline some of the context for the significance of the property. Here we assume some basic knowledge of measure spaces. For an introduction, see [46]. Ergodic

Theory is the study of dynamical systems together with an invariant measure. Let

X be a compact metric space, µ a measure on X, and T : X X a measure- → preserving homeomorphism, i.e. µ(T −1(B)) = µ(B) for all B X measurable. If ⊆ T is measure-preserving, we say that µ is invariant under T . We call the triple

(X, µ, T ) a measure-theoretic dynamical system. Many of the results mentioned in this section in fact hold when T is a measure preserving continuous map rather than a homeomorphism.

Denote by M(X) the measures on X. Viewed as a space of linear functionals with the weak topology, M(X) is a compact metric space. For T : X X a → homeomorphism, we may denote by MT (X) M(X) the set of T -invariant measures ⊂ on X. The structure of the subset MT (X) is at the heart of ergodic theory.

A set B X is T -invariant if T −1(B) = B almost everywhere. A measure- ⊆ theoretic dynamical system (X, µ, T ) is ergodic if for every measurable, T -invariant

B X, one of the following holds: ⊆ (1) µ(B) = 0, or

(2) µ(B) = 1.

We say that µ is ergodic with respect to T and that T is ergodic with respect to

µ. In a way, ergodicity in a measure-theoretic context is analogous to transitivity

13 in a strictly topological context. Ergodicity means that X cannot be decomposed into the disjoint union of two invariant sets each of which has positive measure. See

Chapter Five of [41].

Theorem 2.6 (Krylov-Bogolyubov, [62]). The set MT (X) is non-empty.

Proposition 2.7 (See [41, Proposition 5.7]). The set of ergodic measures is a (nonempty) Gδ subset of MT (X).

As mentioned in Section 2.2, topological entropy was based on its measure- theoretic predecessor which we now deﬁne. A partition of X is a collection (Ai)i∈I P of at most countably many disjoint sets such that i∈I µ(Ai) = 1. For partitions

α = (Ai)i∈I and β = (Bj)j∈J , define α β = Ai Bj : i I, j J . Similarly, ∨ { ∩ ∈ ∈ } n j j for partitions α1, . . . , αn, we may define αk. Let T (α) = (T (Ai))i∈I and, for ∨k=1 b −k a, b Z with a b, let (α)a = a≤k≤bT (α). define the entropy of the partition α ∈ ≤ ∨ by X Hµ(α) = µ(A) log µ(A). − A∈α For a partition α with finite entropy, define the entropy of α with respect to T by

1 N−1 hµ(α, T ) = lim Hµ (α)0 . N→∞ N

Deﬁnition 2.8. We deﬁne the measure-theoretic entropy of (X, µ, T ) (or entropy of

T with respect to µ) by

hµ(T ) = sup hµ(α, T ): α is a partition with H(α) < . { ∞}

When Adler, Konheim, and McAndrew ﬁrst introduced topological entropy, they conjectured the following relationship between topological entropy and measure- theoretic entropy:

h(T ) = sup hµ(T ). ( ) µ∈MT (X) †

14 This equality, called the variational principle, was later proved by Dinaburg, [43], and Goodman, [49]. Clearly then, topological entropy bounds measure-theoretic entropy—this was earlier proved by Goodwyn, [50]. Since all µ MT (X) have ∈ support in ΩT (X), the non-wandering points of T , [41, 6.19], we have as a corollary to the variational principle that

h(T ) = h(T Ω), | i.e. topological entropy is equivalent to the entropy restricted to the non-wandering set.

The variational principle, ( ), brings up a natural question: given a compact † metric space X, for what transformations T do there exist measures m MT (X) ∈ such that

hm(T ) = sup hµ(T ) = h(T )? µ∈MT (X)

In other words, when does there exist a measure in MT (X) that actually achieves the maximum entropy? A measure m MT (X) has maximal entropy (or is an ∈ equilibrium state) if it maximizes the entropy. The set of such measures, which may be empty, is denoted Mmax(T ). A dynamical system (X,T ) is called intrinsically ergodic if Mmax(T ) = 1. In the next few paragraphs, we introduce some suﬃcient | | conditions for a dynamical system to be intrinsically ergodic.

The speciﬁcation property for a map on a compact metric space was introduced by Bowen in 1971, [30].

Deﬁnition 2.9. We say that f : X X has the speciﬁcation property provided for → every δ > 0 there is some Nδ N such that for s 2 and for any s points x1, . . . , xs ∈ ≥ ∈

X and any sequence of natural numbers a1 b1 < a2 b2 < < as bs with ≤ ≤ ··· ≤ ai bi−1 Nδ for 2 i s and any period p bs + Nδ there is a periodic point − ≥ ≤ ≤ ≥

15 j j x X of period p such that d(f (x), f (xi)) < δ for ai j bi and 1 i s. The ∈ ≤ ≤ ≤ ≤ map has the weak speciﬁcation property if the above holds with s = 2.

Essentially, the speciﬁcation property allows one to trace an arbitrary number of orbit segments as close as one likes with a periodic point as long as the number of iterates between tracing orbit segments is suﬃciently large. See Figure 2.1.

x δ x1

Figure 2.1: An example of a point x that follows, within δ, speciﬁed orbit segments of x1, x2, and x3, respectively.

We say a homeomorphism T is expansive if there is δ > 0 such that for every x, y X with x = y, there is n Z such that d(T n(x),T n(y)) > δ. Bowen showed ∈ 6 ∈ that, for systems (X,T ) that are expansive and have the speciﬁcation property, the topologically entropy may be computed by the asymptotic growth rate of periodic points of each period. In addition, he constructed an ergodic measure with full support exhibiting maximal entropy. In fact, this measure is the unique measure of maximal entropy, so we have the following theorem:

16 Theorem 2.10 (Bowen, [30,32]). If (X,T ) is expansive and has the speciﬁcation property, then (X,T ) is intrinsically ergodic.

One particular area of interest for ergodic theory is differentiable dynamical systems. More specifically, the study of differentiable maps that are structurally stable on compact manifolds has been a fruitful area of study. Stability in this context means that slight perturbations of the map ϕ that keep some desired number of derivatives close to those of ϕ will not change the topological structure of orbits.

See Chapter Twenty Three of [41].

Let M be a compact manifold without boundary and denote by Diff(M) the set of Ck diffeomorphisms, ϕ : M M. Stephen Smale defined a subset of Diff(M) of → structurally stable maps that were later called Morse-Smale diffeomorphisms, [103].

This set, in fact, characterized all structurally stable diﬀeomorphisms on one dimensional manifolds. Motivated by ﬂows on compact manifolds with negative curvature,

D. V. Anosov introduced another class of structurally stable diﬀeomorphisms later called Anosov diﬀeomorphisms, [6].

Hoping to unify both Morse-Smale and Anosov diﬀeomorphism under a common characterization, Smale introduced a condition which he called Axiom A, [101,

102]. Axiom A diffeomorphisms contain both Morse-Smale and Anosov diffemor- phisms and are made structurally stable with the addition of one other structural assumption of Morse-Smale diffeomorphisms, [85].

For M a compact manifold and x M, denote by TxM the tangent space at x ∈ and let TM = x∈M TxM denote the tangent bundle. Then for Φ Diff(M) induces ∪ ∈ a map DΦ: TM TM called the tangent map, a vector bundle homeomorphism → mapping TxM to TΦ(x)M. A Reimannian structure on TM is defined if for any x M there exists a positive definite symmetric bilinear form on TxM depending ∈ 17 smoothly on x. Denote by the corresponding norm on TxM. We call K M k · k ⊆ + − hyperbolic if there is a splitting of the tangent bundle to K, TK M = E E , and ⊕ constants c > 0 and λ > 1 so that, for all n N, ∈ DΦn(v) cλ−n v for v E+, and k k ≤ k k ∈ DΦn(v) cλn v for v E−. k k ≥ k k ∈

Definition 2.11. A diffeomorphism Φ: M M satisfies Axiom A if →

(1) the periodic points are dense in ΩΦ(M) = Ω, and

(2) Ω is hyperbolic.

For surfaces, hyperbolicity of Ω implies that the periodic points are dense in

Ω, [77]. This is not the case for higher dimensional manifolds, however, [63].

Let Φ: M M be an Axiom A diﬀeomorphism on a compact manifold. → Smale showed that the non-wandering set of Φ decomposes into a ﬁnite number of disjoint closed Φ-invariant sets, called basic sets, each of which is topologically transitive under Φ, [102]. Bowen showed that basic sets decompose further into a

ﬁnite union of disjoint closed sets, called elementary parts, that map cyclically onto one another, each of which is topologically mixing under some iterate of Φ, [30].

k Then, for the appropriate k N0, Φ restricted to each elementary part is expansive ∈ (in the hyperbolic sense) and has the speciﬁcation property. Along with the existence of a Markov partition, [27,28], these properties allow one to say much about Axiom

A diﬀeomorphisms using symbolic dynamics.

Although the speciﬁcation property is a strong property for a dynamical system, we see, then, that it arises naturally in an important class of diﬀerentiable maps on manifolds implying a unique equilibrium state for such maps. In addition,

Hofbauer has shown that every continuous piecewise monotone map of the interval

18 has the weak speciﬁcation property, [58]. For continuous maps of the interval, Blokh has shown that the speciﬁcation property is equivalent to topological mixing, [22].

Blokh later generalized this result to topological graphs, [23]. More recently, Buzzi characterized the speciﬁcation property for piecewise continuous, piecewise monotone maps of the interval with a ﬁnite number of pieces, [37].

b For f : X X, δ > 0, and a, b Z , , we call xi i=a a δ-pseudo- → ∈ ∪ {−∞ ∞} { } orbit if d(f(xi), xi+1) < δ for a i < b. We say (X, f) has the shadowing property ≤ if, for all > 0, there is δ > 0 so that every δ-pseudo-orbit is -shadowed by some

b i x X, i.e., for xi a δ-pseudo-orbit, there is x X with d(f (x), xi) < for ∈ { }i=a ∈ a i b. ≤ ≤ Bowen showed that any system with the shadowing property that is also weakly mixing has the (non-periodic) specification property. If, in addition, the system is expansive, then the periodic specification property holds, [30]. To read more about the specification property and host of other similar properties, Kwietniak, Łącka, and Oprocha have a nice survey paper available, [64].

For shift spaces (see Section 2.4), weak speciﬁcation is equivalent to speci-

fication (if we ignore the periodicity requirement), and there is a straightforward, sensible characterization of these properties. In 2009, Lampart and Oprocha showed that, in this setting, the weak specification property implies ω-chaos. The assumption of specification is a natural one as mixing shifts of finite type have specification, [41, Proposition 21.2].

A large portion of this dissertation is dedicated to generalizing a result of

Lampart and Oprocha for subshifts of ΣZ, defined in Section 2.4. In particular, it is shown that a weak form of the specification property implies ω-chaos in the context of multidimensional shift spaces, defined in Section 2.5. In addition to this

19 generalization, it is shown that the same assumptions imply uniform distributional chaos.

2.4 Symbolic Dynamics and Shift Spaces

Many of the following definitions are repeated in Section 2.5 with slightly different terminology befitting the context. A short introduction to classic symbolic dynamics, however, is presented here in order to develop an appropriate intuition for what is to come. We give a very brief history of symbolic dynamics here and introduce some helpful terminology.

Symbolic dynamics arose as a means to study physical phenomena modeled by diﬀerential equations. Named by Marston Morse and Gustav Hedlund in 1938,

[74], symbolic dynamics was used at least 40 years earlier by Jacques Hadamard to examine geodesic ﬂows on surfaces of negative curvature, [53]. Since then, symbolic dynamics has become a valuable area of study in its own right having applications in data transmission and storage. For an introduction, see [34] or [36]. For a text dedicated to the theory and its applications, see [70].

Let Σ be a ﬁnite set, called the alphabet, endowed with the discrete topology.

Let ΣZ denote the space of maps x : Z Σ with the product topology. The metric → Z −n Z on Σ , d is deﬁned by d(x, y) = 2 where n = min i : xi = yi . Points in Σ {| | 6 } are bi-inﬁnite sequences of symbols from Σ indexed by Z. For simplicity, consider

Σ = 0, 1 . Then points in ΣZ may be written as follows: { } x = 1001001001001 , ··· ··· y = 1011001001110 , and ··· ··· z = 0101110010011 , ··· ···

20 where the bold symbols represent the symbol in the zeroth coordinate. With the described metric, sequences that agree on a large central block are considered close.

−4 1 0 With x, y, and z as given, d(x, y) = 2 = 16 while d(x, z) = 2 = 1.

Z Z Denote by σ the shift map, σ :Σ Σ , deﬁned coordinatewise by σ(x)i = → xi+1. This map shifts each coordinate to the left by one. For example, if

x = 1001001001001 , ··· ··· σ(x) = 0010010010010 . ··· ··· The space ΣZ with the shift map is called the full shift on n symbols where n = Σ . | | By 2Z, we mean the full shift on two symbols, i.e. ΣZ where Σ = 0, 1 . { } In Chapter Four, we will also consider 2ω = ΣN0 , inﬁnite strings of 1’s and 0’s

ω indexed by ω = N0. We call (2 , σ) the full one-sided shift on two symbols. In this case, the shift map deletes one coordinate and shifts each symbol over by one. For example we may have

x = 1001001100100 , ··· σ(x) = 001001100100 . ··· A subset X ΣZ is invariant under the shift map if σ(X) = X. If X ΣZ ⊆ ⊆ is closed and invariant, it is called a subshift of ΣZ.A block or word over Σ is a ﬁnite sequence of symbols from Σ. Given Σ, deﬁne the words of length n by

n Bn = x0x1 xn−1 : xi Σ , 0 i < n . Let B∞ = n∈ Bn, any possible ﬁnite { ··· ∈ ∀ ≤ } ∪ N0 string of symbols from Σ. A sequence x ΣZ contains a word w of length w = n ∈ | | if there is some i Z so that xi+j = wj for all 0 j < n. ∈ ≤ Z Let F B∞ be ﬁnite and let XF = x Σ : x contains no words from F . ⊂ { ∈ }

We call F the set of forbidden words for XF. Then XF is a closed, invariant subset

Z Z of Σ . We call XF Σ a subshift of finite type (briefly SFT). One may also choose ⊆ to define shifts of finite type using “admissible” words rather than forbidden words.

21 Shifts of ﬁnite type have some nice properties. In particular, the allowable strings of symbols appearing as points in the space may be encoded in a square matrix, A, with nonnegative integer entries. The entropy of a shift of ﬁnite type,

(X, σ), can then be computed explicitly based on the spectrum of A. If A is prim- itive, then the associated subshift, (X, σ), is mixing and therefore has speciﬁcation property, [41].

2.5 Zd-Actions and Multidimensional Shift Spaces

We introduce here some terminology for a Zd-action acting on a compact metric space. A version of the speciﬁcation property in this context is introduced. We give some brief historical notes and then introduce multidimensional shifts and subshifts along with a generalization of the shift map from Section 2.4.

Let (X, ρ) be a compact metric space, d 1, and T a continuous action of ≥ Zd on X. So T : X Zd X, and we denote T (x, t) by T t(x) for t Zd. If × → ∈ T 0 is another Zd-action on the compact metric space (Y, µ), we say T and T 0 are topologically semiconjugate if there is a continuous surjection ϕ : X Y such that → ϕ T t = T 0t ϕ for all t Zd and all x X. We say T 0 is a factor of T . If ϕ · · ∈ ∈ is a homeomorphism, we call φ a topological conjugacy and say that T and T 0 are topologically conjugate. A point x X is periodic if its orbit, T t : t Zd , is ﬁnite. ∈ { ∈ } The set Per(T ) is the set of all periodic points in X under the action T .

Definition 2.12. Let X be a compact metric space. A Zd-action T has the specification property, briefly SP, if for all δ > 0, there is Nδ N such that for every collection of ∈ d rectangles Q1,Q2,...,Qk in Z satisfying

dist(Qi,Qj) = min t s ∞ Nδ t∈Qi k − k ≥ s∈Qj

22 d for 1 i < j k and every subgroup Λ Z with dist(Qi + q, Qj) Nδ for ≤ ≤ ⊂ ≥

1 i, j k and q Λ 0 , and for every collection of points x1, x2, . . . , xk in ≤ ≤ ∈ \{ } t t X, there is y X satisfying ρ(T y, T xj) < δ for all t Qj, 1 j k, and with ∈ ∈ ≤ ≤ T ty = y for every t Λ. If T satisfies the above with k = 2, we say T has the weak ∈ specification property, briefly WSP.

As in the case for a continuous map on X, the speciﬁcation property allows one to “trace orbit segments.” In this case, an “orbit” is indexed by Zd rather than

Z or N0. Then “orbit segments” may be visualized as blocks indexed by rectangles

d of coordinates in Z rather than strings of coordinates in Z or N0. For any finite number of blocks sufficiently far apart from one another, one may find a periodic point that appears to mimic closely some prescribed behavior when we restrict our attention to the specified blocks.

When referring to SP or WSP in this context, we will drop the periodicity requirement as this condition is not needed for the proofs for Zd-subshifts. For convenience, a simpler deﬁnition of the weak speciﬁcation property without the periodicity requirement is stated here.

Deﬁnition 2.13.A Zd-action T has the (non periodic) weak speciﬁcation property if

d for all δ > 0, there is Nδ N such that for every pair of rectangles Q1,Q2 Z ∈ ⊂ satisfying dist(Q1,Q2) > Nδ and for every pair of points x1, x2 X, there is y X ∈ ∈ t t satisfying ρ(T y, T xj) < δ for all t Qj, j = 1, 2. ∈

The specification property for Zd-actions was first considered by Ruelle in his 1973 paper, [88]. Generalizing results of Bowen, Ruelle considered Zd-actions satisfying both expansiveness and specification. In his paper, Ruelle shows that in such a system, the maximal measure-theoretic entropy (as a function on invariant

23 probability measures) is equal to the topological entropy of the system—topological entropy in Ruelle’s paper is a special case of the more general concept of pressure.

In addition, the entropy is carried by the periodic points of the system, meaning that only periodic points need to be considered in order to calculate the entropy.

Ruelle’s original definition varies slightly from the one used here and is, in fact, stronger than even Definition 2.12. The simplification is for convenience, and we need not rely on the more general definitions to establish the desired results. The definition we use, Definition 2.13, is equivalent to the property of block gluing. See for instance [33]. The intuition for the name “block gluing” may be evident already, and should be made more transparent later when we introduce Lemma 3.4.

As mentioned, the compact metric spaces we will be concerned with are generalizations of ΣZ. The group action will be a generalization of σ, the usual shift map on ΣZ.

Denote by Σ a ﬁnite set, called the alphabet, with the discrete topology and

d let ΣZ denote the space of maps x : Zd Σ with the product topology. For → d t = (t1, t2, ..., td) Z , we let t = max ti : 0 i d , which is called the ∈ | | { ≤ ≤ } d inﬁnity norm. We may then use the metric ρ on ΣZ deﬁned by ρ(x, y) = 2−n where

d K n = min t : xt = yt . For K Z , we refer to a Σ as a coloring of K or {| | 6 } ⊆ ∈ a K-pattern (analogous to a word, w, for ΣZ). If K is a cube in Zd, we call a K-

K pattern a block. If J K with a Σ , then we may also consider a J , the coloring ⊂ ∈ | of J induced by a restricted to the set J. Two patterns a ΣK and b ΣK+t are ∈ ∈ d Z congruent if as = bs+t for every t K. For x Σ , we say x contains the pattern ∈ ∈ K d a Σ if xs+t and at are congruent for some t Z and all s K. ∈ ∈ ∈

24 d Points in ΣZ look like an inﬁnite array of symbols indexed by the integer lattice Zd. For simplicity, consider Σ = 0, 1 . See Figure 2.2 for an example of a { } 2 point in ΣZ .

1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0

2 Figure 2.2: An example of a point in ΣZ . The bold symbol signiﬁes the coordinate indexed by the origin. Points are close if they agree on a large central block.

d d We will consider the shift action, σ :ΣZ Zd ΣZ , which is deﬁned × → d coordinatewise by σ(x, t)s = xs+t for t Z . See Figure 2.3. We will denote ∈ t σ(x, t)s by σ (x)s. 1 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 1 1 0 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 01 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 0 0 0 10 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 1 1 1 0 10 0 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 01 0 1 0 0 1 0 0 0 0 1

d 1 0 0 0 1 1 1 1 1 0 0 Figure 2.3: An example of x ΣZ on the left1 and σt(x) on the right. In this case, t = (1, 2) ∈ with xt circled on the left. After shifting,1 the0 symbol0 1 in0 coordinate0 0 t1is now0 1 centered1 at (0, 0), circled on the right. Each symbol is colored to help emphasize how the coordinates of x are shifted under σ. 25 d A subset X ΣZ is invariant under the shift action if σt(X) = X for all ⊆ d t Zd. If X ΣZ is closed and invariant, it is called a d-dimensional subshift ∈ ⊆ d or Zd-subshift of the full shift ΣZ . Given a ﬁnite alphabet, Σ, and a ﬁnite set

K Zd, we may define a collection F ΣK of colorings of K. If F is finite, then ⊂ ⊆ d Z XF = x Σ : x contains no patterns from F is a d-dimensional shift of finite { ∈ } type defined by F. We call F the set of forbidden patterns for XF. As in the case for classic subshifts, a shift of finite type may also be defined in terms of “admissible patterns” rather than forbidden ones.

d A subshift X ΣZ is strongly irreducible if there is N N such that for all ⊆ ∈ d J, K Z with dist(J, K) N and any y1, y2 X, there is x X with x J = y1 J ⊂ ≥ ∈ ∈ | | and x K = y2 K . Note J and K need not be rectangles. In fact, they need not be | | bounded in Zd. Note that if X is a strongly irreducible subshift, this implies the (non-periodic) specification property. It is not known whether a strongly irreducible subshift necessarily has a periodic point. However, the existence of just one periodic point in a strongly irreducible subshift implies that periodic points are dense, [38]. For a strongly irreducible subshift with a dense set of periodic points to imply the strong specification property, one would still need to show that you can find periodic points of any period larger than some N N. ∈ Many basic properties of subshifts in dimension one—that is, subshifts of ΣZ— do not translate well into higher dimensions. For instance, as mentioned earlier, topological entropy for a shift of finite type when d = 1 may be explicitly computed based on the spectral properties of an associated matrix, A. For d > 1, the case is much more difficult, [55, 56]. In fact, given a set of forbidden words, it may be undecidable as to whether the associated shift of finite type is nonempty, [19,86]. For

26 this reason, one often needs to impose additional restrictions or address particular classes of subshifts in order to make real headway.

27 CHAPTER THREE

d Chaos in Subshifts of ΣZ Via the Weak Speciﬁcation Property

The ﬁrst main result of this chapter generalizes a result of Lampart and

Oprocha showing that a weak form of the speciﬁcation property implies ω-chaos and uniform distributional chaos for Zd-actions in the context of multidimensional

d shift spaces, i.e., subshifts of ΣZ . The second main result shows that, with the same assumptions, the action is also uniformly distributionally chaotic of type 1.

The first section introduces a generalization of the ω-limit set that is well- defined for continuous Zd-actions. In Section 3.2, an uncountable family of uncountable minimal sets is constructed using the weak specification property. Making use of this result, Section 3.3 contains the proof that WSP implies ω-chaos in subshifts of

d ΣZ . Some notions of directional limit sets are addressed in Sections 3.4 and 3.5. In

d Section 3.6, it is shown that WSP in subshifts of ΣZ implies uniform distributional chaos of type 1.

3.1 General ω-Limit Sets for Zd-Actions Much of this chapter deals heavily with the idea of limit sets. Unfortunately, there is no yet agreed upon notion of how ω-limit sets should be deﬁned when considering Zd-actions on a space. Oprocha ﬁrst introduced some generalized notion of ω-limit sets in [80]. The limit sets of Oprocha will be addressed in Section 3.4.

For now, we will use the following general deﬁnition introduced by Meddaugh and

Raines, [73].

d Definition 3.1. For x X ΣZ , define the ω-limit set of x to be the set ∈ ⊆ d t ω(x) = y X : > 0,M N, t Z with t > M and ρ(y, σ (x)) < . { ∈ ∀ ∈ ∃ ∈ | | } 28 So y ω(x) if, and only if, any central block of symbols in y appears infinitely ∈ often as a patch in x. Note that the above definition is not a strict generalization of the usual definition of an ω-limit set. In the case where d = 1 where our context is

ΣZ, the above deﬁnition is equivalent to ! ! \ [ \ f n(x): n j f −n(x): n j , { ≥ } { ≥ } j∈N j∈N the union of the usual ω-limit and α-limit sets. As mentioned, some other notions of limit sets— ones that may seem more akin to the classical deﬁnition of an ω-limit set—in a later section.

d 3.2 Minimal Sets in Subshifts of ΣZ

In order to prove the upcoming Theorem 3.3, we will rely on the existence of an uncountable collection of uncountable, disjoint minimal sets in 2Z = 0, 1 Z. { }

d Deﬁnition 3.2. We call A X invariant if f(A) A. A set M ΣZ is minimal ⊂ ⊂ ⊂ if it is nonempty, closed, and invariant under σ and contains no proper subset with these properties.

It is a well known fact that M is minimal if, and only if, every point x of M is uniformly recurrent, [34]; moreover if x is uniformly recurrent then ω(x) is minimal.

Then ω(x) = M for all x M with M minimal. It is possible to construct an un- ∈ countable collection of points Λ 2Z that are uniformly recurrent with the property ⊆ that if x, y Λ and x = y then ω(x) ω(y) = and each ω(x) is uncountable, [75]. ∈ 6 ∩ ∅ Thus there are uncountably many disjoint minimal subsets of 2Z. We will use these

d points in what follows to ﬁnd an uncountable collection of points in ΣZ , each of whose ω-limit sets contains an uncountable minimal set.

Theorem 3.3. For Σ nondegenerate, there exists an uncountable, disjoint collection

d of uncountable minimal sets in ΣZ , the full Zd-shift on Σ. 29 Proof. Without loss of generality, assume 0, 1 Σ. For y 0, 1 Z, deﬁne { } ⊆ ∈ { } d Z Z d ϕ : 0, 1 Σ coordinatewise by ϕ(y)t = yt1 . Recall that t Z is a d- { } → ∈ tuple, (t1, t2, ..., td), so t1 here denotes the ﬁrst coordinate of t. For example, if y = 001101100 ΣZ, we have ··· · · · ∈ . .

001101100

d ϕ(y) = 001101100 ΣZ . ··· · · · ∈ 001101100 . .

Note that ϕ is a bicontinuous injection onto the set

n d o Z Z = x Σ : xt = xs whenever t1 = s1 . ∈

Let Mα α∈A be an uncountable, disjoint collection of uncountable minimal sets in { } 0, 1 Z. { }

Fix α A and consider ϕ(Mα) = Nα. We show that Nα is minimal. Note ∈ t d that Nα is compact and therefore closed. Let x σ (Nα) for some t Z . Then ∈ ∈ t d −1 there is z Nα so that σ (z) = x, i.e., xs+t = zs for all s Z . Let y = ϕ (z). ∈ ∈

t1 0 0 Then σ (y) = y Mα with ϕ(y )s = xs for all s. Then Nα is closed and invariant. ∈ −1 Suppose N ( Nα is invariant as well as closed. Then ϕ (N) ( Mα is closed and invariant, a contradiction to Mα minimal. So Nα is minimal.

Note that for α, β A, α = β, ϕ(Mα) ϕ(Mβ) = . Then we have that ∈ 6 ∩ ∅

ϕ(Mα) α∈A = Nα α∈A is an uncountable, disjoint collection of uncountable mini- { } { } d mal sets in ΣZ .

Recall the deﬁnition of the (non-periodic) weak speciﬁcation property for Zd- actions acting on a compact metric space.

30 Definition 2.13.A Zd-action T on a space X has the (non periodic) weak specification property (briefly WSP) if for all δ > 0, there is Nδ N such that for every pair of ∈ d rectangles Q1,Q2 Z satisfying dist(Q1,Q2) > Nδ and for every pair of points ⊂ t t x1, x2 X, there is y X satisfying ρ(T y, T xj) < δ for all t Qj, j = 1, 2. ∈ ∈ ∈

The following lemma is essentially an equivalent deﬁnition of the (non-periodic) weak speciﬁcation property. The property described in the statement of the lemma is called block gluing (See Figure 3.1).

1 1 1 0 1 0 0 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 0 1 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 0 1 0 1 N0 1 0 1 0 1 1 0 1 1 0 1 0 0 0 1 ≥1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 0 1 1

Figure 3.1: Lemma 3.4 allows us to think of the weak speciﬁcation in terms of “block d gluing.” Two rectangular patterns are guaranteed to appear in some point of ΣZ provided d their coordinate are at least N apart in Z .

d Lemma 3.4. Let X ΣZ be a nondegenerate subshift. Then σ has the (non-periodic) ⊂ weak speciﬁcation property if, and only if, there is N N such that for any two ∈ d rectangles Q1,Q2 Z with d(Q1,Q2) N and any two points x1, x2 X, there is ⊂ ≥ ∈ y X with y Q = x1 Q and y Q = x2 Q . ∈ | 1 | 1 | 2 | 2

31 Proof. We prove the reverse implication ﬁrst. Suppose there is N N such that for ∈ d any two rectangles Q1,Q2 Z with dist(Q1,Q2) N and any two points x1, x2 ⊂ ≥ ∈

X, there is y X with y Q1 = x1 Q1 and y Q2 = x2 Q2 . Let 0 < δ < 1. Choose l N ∈ | | | | ∈ −l d such that 2 < δ and let Nδ = N + 2l. Let Q1,Q2 Z with dist(Q1,Q2) > Nδ ⊂ 0 0 and let x1, x2 X. Let Q = Q1 + t : t l and Q = Q2 + t : t l . Then ∈ 1 { | | ≤ } 2 { | | ≤ } 0 0 dist(Q ,Q ) > Nδ 2l = N. By our assumption, there is y X with y Q0 = x1 Q0 1 2 − ∈ | 1 | 1 t t −l and y Q0 = x2 Q0 . Then ρ(σ (y), σ (xj)) < 2 < δ for all t Qj, j = 1, 2. So X has | 2 | 2 ∈ WSP.

Now suppose σ has WSP. Letting δ = 1, the argument follows.

d The next lemma deals with sequences of rectangles in Z , Qi i∈ , with each { } N member of the sequence a suﬃcient distance from the complex hull of the union of the previous rectangles in the sequence. We show that given any such sequence of rectangles and any colorings of those rectangles, WSP implies there is a point that agrees with the colorings on each rectangle. This lemma will be used later on to construct points. See Figure 3.2

d Lemma 3.5. Let X ΣZ be a nondegenerate subshift with the weak speciﬁcation ⊂ d property and N witnessing WSP. Suppose we have rectangles Qi i∈ in Z with { } N conv[( 0

Then there is z X such that z Q = xi Q . ∈ | i | i

Proof. Let Qi i∈ and xi i∈ be as in the assumptions of the lemma. Let y1 = x1. { } N { } N d By WSP, there is y2 such that y2 Q1 = x1 Q1 and y2 Q2 = x2 Q2 . For A Z , denote | | | | ∈ by conv(A) the convex hull of A. Deﬁne P2 = y2 conv[Q ∪Q ]. Suppose yi−1 and | 1 2

Pi−1 are deﬁned for some i > 2. By WSP, there is yi such that yi P = yi−1 and | i−1 yi Q = xi Q . Deﬁne Pi = yi conv[Q ∪P ]. | i | i | i i−1 32 A B

Figure 3.2: Each rectangle above represent a patch of symbols from Σ. Lemma 3.5 and Lemma 3.4 together allow one to “glue” multiple rectangular patches together provided the next patch in the sequence lies suﬃciently far outside of the convex hull of the previous patches.

In this manner, we may deﬁne inductively the sequence of points yi i∈ . Let { } N z be a limit point of yi . Then z Qi = xi Qi for all i N. { } | | ∈

Note that, without loss of generality, any xi in the assumptions of Lemma

3.5 could have been constructed in the above manner. This gives us a nice way to

“patch” all of Zd with speciﬁed symbols, provided the gaps between each patch are

d large enough. In the next lemma, we do just that. For X ΣZ with WSP, we take ⊆ a point y and “shatter” it. We spread out the symbols by some uniform factor and

d use the previous lemma to ﬁnd a point in ΣZ that ﬁlls in the gaps. See Figure 3.3.

d Lemma 3.6. Let X ΣZ be a nondegenerate subshift with the weak speciﬁcation ⊂ d property. Let N witness WSP. Then for all y ΣZ and k N with k N, there is ∈ ∈ ≥ d x X such that xkt = yt for all t Z . ∈ ∈

Proof. Choose k N and let M N. Without loss of generality, suppose all ≥ ∈ symbols from Σ appearing in y also appear in some point of X. We will construct 33 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 0 0 1 01 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 00 00 0 0 0 0 0 0 0 1 1 1 1 1 0 0 10 11 1 10 1 1 0 1 0 1 10 1 1 1 1 0 1 1 0 01 1 01 0 01 0 1 0 0 0 0 1 11 1 01 0 00 0 1 1 1 0 0 0 010 1 00 0 10 1 1 1 1 0 0 0 0 01 0 01 1 00 0 0 0 1 1 1 0 01 0 110 010 1 1 1 1 0 0 0 0 10 01 101 1 10 1 1 0 0 0 1 1 11 0 11 1 010 00 1 1 0 0 1 1 10 00 000 00 100 0 0 0 1 1 0 1 1 11 0 10 1 11 0 1 1 1 0 0 1 1 10 11 01 001 00 10 1 0 0 0 0 1 0 100011011 1 10 0 0 0 0 0 10 1 11 00 00 0 0 1 1 1 0 1 0 1 1 1 0 11 00 101000000 0 1 1 1 1 00 01 001 00 11 1 1 1 0 0 0 0 0 10 1 00 00 11 11 011 0 0 1 1 0 010110110 1 00 0 1 0 1 0 0 01 0 00 11 00 0 00 0 1 1 1 1 1 0 00 0 01 01100100 0 0 0 0 0 00 0 010 1 10 1 11 1 0 0 0 1 0 0 0 01 0 10 10 10 1 1 1 1 1 0 0 0 001 0 101 0 0 0 0 1 0 0 1 01 0 01 0 01 0 0 0 1 1 0 0 00 1 1 111 1 00 0 0 0 1 00 0 01 0 01 0 1 1 0 0 0 1 1 1 0 11 1 0 01 0 0 0 1 0 1 1 11 1 1 1 0 0 0 0 0 0 0 11 11 1 0 0 0 1 1 1 1 1 10 1 1 1 0 0 1 0 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 01 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0

d Figure 3.3: In Lemma 3.6, we take a point, y, from a subshift, X, of ΣZ and shift every coordinate outward from the origin by a common factor. We show that if X has WSP, there is a point in x that agrees with the “exploded” version of y on the right while ﬁlling in the gaps. inductively a sequence of points in X any limit of which satisﬁes the requirements

M M of the theorem. By WSP, there is x X such that (x )kt = yt whenever t = ae1 1 ∈ 1 d for all M a M where e1 is the usual basis vector in Z . Likewise, there is − ≤ ≤ M M x X such that (x )kt = yt whenever t = a1e1 + a2e2 for all M a1, a2 M. 2 ∈ 2 − ≤ ≤ M Now suppose xn−1 is deﬁned for n < d. By an application of Lemma 3.5, there is

M M Pn x such that (x )kt = yt when t = aiei for any M ai M. Inductively, n n i=1 − ≤ ≤ M we may deﬁne xd . We may deﬁne such a point for each M N. Taking a limit ∈ M point of the sequence (xd )M∈N completes the proof.

We now use the previous lemma to construct an uncountable, disjoint collection of uncountable minimal sets in X whenever σ has WSP. The basic idea is to take an

d uncountable collection of uncountable minimal sets in ΣZ , the full Zd-shift, apply

d Lemma 3.6 to form new sets in X ΣZ , and then use the resulting collection ⊆ 34 in X to generate minimal sets disjoint from uncountably many other minimal sets generated in the same fashion.

d Theorem 3.7. Suppose X ΣZ is a nondegenerate subshift with the weak spec- ⊂ iﬁcation property. There exists an uncountable, disjoint collection of uncountable minimal sets in X.

Proof. Consider the group action σ˜ generated by σNe , σNe , . . . , σNe where ei is { 1 2 d } the usual ith basis vector, i.e., σ˜t(x) = σNt(x) for all t Zd. For the purposes of this ∈ proof, we will abuse notation and denote σ˜ by σN .

Without loss of generality, suppose 0, 1 Σ such that both 0 and 1 each { } ⊆ appear in some coordinate of a point in X. Consider

d Z = x X : xNt 0, 1 , t Z . ∈ ∈ { } ∀ ∈ This is the set of points having only zeros or ones in any coordinate in the NZd

d lattice. Note that Z is closed and invariant under σN . Let Y = 0, 1 Z , the full { } shift on two symbols, and deﬁne γ : Z Y coordinatewise by γ(z)t = zNt for all → t Zd. By Lemma 3.6, γ is surjective. Note that σNt(z) = σt(γ(z)). Then γ is a ∈ semiconjugacy witnessing (Z, σN ) and (Y, σ) semiconjugate.

N Z σ Z

γ γ

Y σ Y

By Theorem 3.3, there exists Mα α∈A, an uncountable, disjoint collection of { } d Z −1 uncountable minimal sets in Y = 0, 1 . For each α A, we consider γ (Mα) { } ∈ ⊆ −1 d X. Notice that γ (Mα) = x X : there is y Mα such that xNt = yt, t Z { ∈ ∈ ∀ ∈ } is closed and invariant under σN . By Zorn’s lemma, there exists a subset M 0 α ⊆ −1 N 0 0 γ (Mα) minimal under σ . Now Mα is uncountable, as otherwise γ(Mα) ( Mα is

35 invariant under σ in Y , a contradiction to Mα minimal. Furthermore, For a fixed s Zd, observe that for all t Zd, ∈ ∈ Nt s 0 s Nt 0 s 0 σ (σ (Mα)) = σ σ (Mα) = σ (Mα) , so that any shift of M 0 is invariant under σN . For each α A, define such a set M 0 . α ∈ α As a consequence of the definition, we then have M 0 M 0 = for α, β A with α ∩ β 6 ∅ ∈ α = β. 6 Our next step is to use M 0 to construct sets that are minimal under the { α}α∈A N 0 action of σ rather than just σ . We begin with the following observation. Since Mα is invariant under σN , we have that

[ t 0 [ t 0 σ (Mα) = σ (Mα) . t∈Zd |t|

d t 0 0 Fix α A and t Z with 0 < t < N. Note that if σ (Mα) Mβ = for ∈ ∈ | | ∩ 6 ∅ some β = α, then since both σt(M 0 ) and M 0 are closed and invariant under σN with 6 α β M 0 minimal, σt(M 0 ) = M 0 . Then, for any given t, there is at most one β = α such β α β 6 that σt(M 0 ) M 0 = . In fact, we have that α ∩ β 6 ∅ β A : σt(M 0 ) M 0 = , for all 0 < t < N (2N 1)2. |{ ∈ α ∩ β 6 ∅ | | }| ≤ − Since A is uncountable, we may then assume without loss of generality that σt(M 0 ) α ∩ 0 d Mβ = for all α = β and all t Z with t < N. To see this, it is enough to note ∅ 6 ∈ | | that a partition of an uncountable set into ﬁnite sets is necessarily uncountable.

For each α A, let ∈ [ t 0 [ t 0 Wα = σ (Mα) = σ (Mα). t∈Zd |t|

36 Fix α A. We show now that Wα is minimal under σ. Suppose there is ∈ 0 W Wα nonempty, closed, and invariant under σ. Then M W = is nonempty, ⊆ α ∩ 6 ∅ N 0 N closed, and invariant under σ . Since Mα is minimal under σ , it must be the case that M 0 W = M 0 . Then, since W is invariant under σ, we have α ∩ α [ t 0 [ t 0 σ (W M ) = σ (M ) = Wα W. ∩ α α ⊆ t∈Zd t∈Zd So W = Wα and Wα is minimal under σ. Then Wα α∈A is an uncountable, disjoint { } collection of uncountable minimal sets in X.

d In Section 3.3, we use these results to show that if a subshift X ΣZ , has ⊆ the weak speciﬁcation property, then σ exhibits ω-chaos. In particular, we rely on

Theorem 3.7 as well as a careful construction of points in X to build an uncountable

ω-scrambled set.

3.3 ω-Chaos

d Recall that for X ΣZ a subshift, we say σ has ω-chaos if there is an ⊆ uncountable set Ω X such that, for all x, y Ω, ⊂ ∈ (1) ω(x) ω(y) is uncountable, \ (2) ω(x) ω(y) = , and ∩ 6 ∅ (3) ω(x) Per(σ) = . \ 6 ∅

d Theorem 3.8. Assume X ΣZ is a nondegenerate subshift with the weak speciﬁca- ⊂ tion property. Then σ has ω-chaos, i.e., there is an uncountable ω-scrambled set for

σ.

Proof. Let N witness WSP for σ. By Theorem 3.7, let Mα α∈A Mz be an { } ∪ { } uncountable, disjoint collection of uncountable minimal sets with a point z Mz. ∈

37 Let p1 = q1 = 0 and, for i N, deﬁne inductively pi+1 = qi + N and qi+1 = ∈ pi+1 + i. We then have

p1 = 0 q1 = 0

p2 = q1 + N = N q2 = p2 + 1 = N + 1

p3 = q2 + N = 2N + 1 q3 = p3 + 2 = 2N + 3

p4 = q3 + N = 3N + 3 q4 = p4 + 3 = 3N + 6 (i 2)(i 1) (i 1)(i) p5 = (i 1)N + − − q5 = (i 1)N + − − 2 − 2 . . . .

d Let S1 = 0 Z . { } ⊂ d Si = t Z : pi t qi and either tl qi−1 or tl pi for 1 l d . { ∈ ≤ | | ≤ | | ≤ | | ≥ ≤ ≤ }

See Figures 3.4 and 3.5. One may think of the Si’s as fragmented layers or shells

d of coordinates in Z . For instance, Si+1 contains Si within its convex hull, yet

Si Si+1 = for all i. Note that if t Si, then t is contained in a cube Q Si with ∩ ∅ ∈ ⊂ th side length i. For all i N, we will refer to the set Si as the i shell. ∈

For each α A, we now describe a process of constructing a point xα X ∈ ∈ such that (Mα Mz) ω(xα) and Mβ ω(xα) = for any β A with β = α. Let ∪ ⊂ ∩ ∅ ∈ 6 yα Mα. By WSP and Lemma 3.5 (this lemma allows us to glue convex shapes ∈ i together), we may choose inductively the sequence (xα)i∈N of points in X requiring that, for j N, ∈ 1 (xα)0 = z0,   2j−1 (xα )t if t q2j−1 2j | | ≤ (xα )t =  (yα)t if t S2j,  ∈

38 S3 S3 S3

S2 S2 S2

S3 S2 S1 S2 S3

S2 S2 S2

S3 S3 S3

Figure 3.4: An example of S1, S2, and S3 for the case d = 2 and N = 4. Each rectangle d represents a block of coordinates in ΣZ .

  2j (xα )t if t q2j 2j+1 | | ≤ (xα )t =  zt if t S2j+1.  ∈ Let x be a limit point of (xi ) . Then α α i∈N   zt if t S2j−1  ∈ (xα)t =  (yα)t if t S2j,  ∈ for all j N. In other words, when i is odd, xα Si = z Si , and when i is even, ∈ | | xα S = yα S . | i | i

To see that Mα ω(xα), note that there is an increasing sequence of natural ⊂ d numbers (ji)i∈ and a sequence (t(i))i∈ with t(i) Z such that (t(i) + s) S2ji N N ∈ ∈ t(i) for all s i. Consider the sequence of points (yi)i∈ = (σ (xα))i∈ . Taking a | | ≤ N N limit point y of (yi)i∈N, note that, by the construction of xα, y is also a limit point

39 Figure 3.5: An example of S1, S2, and a portion of S3 for the case d = 3 and N = 4. Each d rectangle represents a block of coordinates ΣZ .

of yα. Since Mα is minimal, ω(yα) = ω(y) = Mα. Since y ω(xα), we must also ∈ have ω(xα) Mα. A similar argument shows Mz ω(xα). ⊃ ⊂ Suppose for the purpose of contradiction that for some β A, β = α, we have ∈ 6

Mβ ω(xα) = . Since Mβ is minimal, we then have Mβ ω(xα). Let yβ Mβ. ∩ 6 ∅ ⊆ ∈

Then yβ ω(xα) and any central block of yβ appears inﬁnitely often as a pattern in ∈ d xα. In particular, for any k N, we may ﬁnd a cube Q Z of side length 2k + N ∈ ⊂ d such that, for all r Q, r qk and such that xα Q = yβ Q+s for some s Z . ∈ | | ≥ | | ∈ Notice that due to the size of Q and our choice of r, we must have that Q intersects

0 some Si on a cube Q of side length at least k. Then, by the construction of xα, if

0 0 Q is contained in an even shell, yα Q0 = yβ Q0+s, and if Q is contained in an odd | | shell, z Q0 = yβ Q0+s. | |

40 This implies either

−b(k−1)/2c −b(k−1)/2c dist(Mα,Mβ) 2 or dist(Mz,Mβ) 2 ≤ ≤ where dist(A, B) = infx∈A,y∈B ρ(x, y) . As this is true for any k > 0, we have a { } contradiction since Mα, Mβ, and Mz are disjoint. We have shown, then, that for

β A with β = α, Mβ ω(xα) = . ∈ 6 ∩ ∅

For each α A, deﬁne a point xα in the prescribed method. Then the set ∈

Ω = xα α∈A is an ω-scrambled set under the shift action. To see this note, that for { }

α = β we have ω(xα) ω(xβ) contains the uncountable set Mα, an uncountable set 6 \ that contains no periodic points. We also have Mz ω(xα) ω(xβ). ∈ ∩

3.4 Directional Limit Sets

As mentioned earlier, there is no straightforward generalization of the usual

d ω-limit set of a point in ΣZ as there are many directions to consider when applying a shift to points in the space. The ﬁrst limit sets considering the many directions available in higher dimensional subshifts were introduced by Oprocha, [80], based on earlier ideas of semigroup limit sets, [39, 51, 93]. In his paper, Oprocha considered limit sets restricting the action to each quadrant of Zd—we address this brieﬂy a little later. A generalization of this notion was later given by Meddaugh and Raines, [73], and is presented here.

(d) d d We will call D = η Z : gcd ηi = 1 the set of directions in Z . { ∈ { } }

Deﬁnition 3.9. For η D(d) and x X, deﬁne the η-type limit set of x to be ∈ ∈ d t Lη(x) = y X : > 0,M N, t Z with t η > M and ρ(y, σ (x)) < { ∈ ∀ ∈ ∃ ∈ · } where denotes the usual dot product. ·

In other words, if one considers an increasing sequence of natural number (Mi)

d and a sequence (t(i))i∈ with t(i) Z such that t(i) η > Mi for all i, then any N ∈ · 41 t(i) limit point of σ (x) will be in Lη(x). See Figure 3.6 for an idea of what shifts of x we are considering in constructing the limit set Lη(x).

· η

d Figure 3.6: For a ﬁxed η Z and M N, the coordinates in the shaded regions above d ∈ ∈ represent t Z for which t η > M. ∈ ·

Definition 3.10. For a finite set E D(d) and x X, define the E-type limit sets of ⊂ ∈ x to be

d t LE+ (x) = y X : > 0,M N, t Z with min t η > M and ρ(y, σ (x)) < { ∈ ∀ ∈ ∃ ∈ η∈E | · | } and

d t LE− (x) = y X : > 0,M N, t Z with max t η > M and ρ(y, σ (x)) < . { ∈ ∀ ∈ ∃ ∈ η∈E | · | }

An example of the coordinates in Zd considered for E-type limit sets is given in Figure 3.7. Note that if LE+ is empty, then so is the corresponding limit set. The limit sets of Oprocha mentioned earlier are L + type limit sets where E = aiei : E { d ai 1, 1 for 1 i d where ei 0≤i≤d are the usual basis vectors in Z . For ∈ {− } ≤ ≤ } { } Oprocha, then, the directions considered for a limit set all belonged to one orthant of Zd—the n-dimensional analogue to a quadrant of Z2.

42 t t

· · η η 1 1 = =

M M

t t η2 η2 · = M · = M

2 2 η η η1 η1

LE+ (x) LE− (x)

Figure 3.7: For E = η1, η2 and M N, the coordinates in the shaded regions above d { } ∈ represent t Z for which minη∈E t η > M and maxη∈E t η > M, respectively. ∈ | · | | · |

In some ways, E-type limit sets may be a more accurate generalization of the

Z usual ω-limit set for the d = 1 case. For instance, if E = e1 , then for x X Σ , { } ∈ ⊆

+ − ω(x) = Le1 (x) = LE (x) = LE (x), were ω(x) above represents the usual ω-limit set of x. Recall that our generalization of an ω-limit set to higher dimensions is actually the union of the α-limit and ω- limit sets when considering the d = 1 case. In the directional limit notation just introduced, if we choose E = aiei : ai 1, 1 for 1 i d , then for x X { ∈ {− } ≤ ≤ } ∈ ⊂ d ΣZ , we have

ω(x) = LE− (x) where ω(x) here represents the generalized notion of Raines and Meddaugh introduced in Section 3.1.

3.5 Directional Limit Chaos

We now consider a generalization of ω-chaos for any ﬁnite set of directions in

D(d). The result of Theorem 3.8 generalizes for this broader notion of directional limit chaos.

43 d (d) Deﬁnition 3.11. For X = ΣZ and a ﬁnite set E D , we say σ has L ± limit type ⊂ E chaos if there is an uncountable set Ω X such that, for all x, y Ω, ⊂ ∈

(1) L ± (x) L ± (y) is uncountable, E \ E

(2) L ± (x) L ± (y) = , and E ∩ E 6 ∅

(3) L ± (x) Per(σ) = E \ 6 ∅

Our proof of the next theorem relies on the following lemma.

d Lemma 3.12 (Meddaugh-Raines, [73]). For E D(d) ﬁnite and x X ΣZ , the ⊂ ∈ ⊆ sets LE± (x) are closed and invariant under σ.

d Theorem 3.13. Assume X ΣZ is a nondegenerate subshift with the weak speciﬁca- ⊂ (d) tion property and E D ﬁnite with L ± (x) = for all x X. Then σ has L ± ⊂ E 6 ∅ ∈ E limit type chaos.

Proof. By Lemma 3.12, if x M with M minimal under the action of σ, L ± (x) = ∈ E

ω(x) = M whenever LE± is nonempty. With this in mind, the proof is almost identical to that of Theorem 3.8.

3.6 Distributional Chaos

Distributional chaos for a map, f : X X, has a natural generalization to → continuous actions of Zd on X. Let (X, ρ) be a compact metric space, T : t T t a → continuous action of Zd on X. For any x, y X and any n N, deﬁne ∈ ∈

(n) 1 d t t Φxy () = t Z : ρ(T x, T y) < and t < n 2n 1 d ∈ | | (n) | − | So Φxy () is the percentage of shifts of x and y that are within of one another when considering all possible shifts that have magnitude less than n.

44 Let

(n) Φxy() = lim inf Φxy (), and n→∞

∗ (n) Φxy() = lim sup Φxy (). n→∞ These non-decreasing functions are called the lower and upper (distance) distribution

∗ functions, respectively. Note that Φxy() = Φxy() = 0 for < 0 and that Φxy() = Φ∗ () = 1 for diam(X). xy ≥

Deﬁnition 3.14. Let (X,T ) be a compact metric space together with a Zd-action T on X. A pair of points x, y X is called distributionally chaotic of type 1, 2, and 3 ∈ (brieﬂy DC1, DC2, and DC3) if

∗ (DC1) Φ 1 and Φxy() = 0 for some > 0, xy ≡ ∗ ∗ (DC2) Φ 1 and Φxy < Φ for some > 0, and xy ≡ xy ∗ (DC3) Φxy < Φxy for some > 0, respectively. For each i 1, 2, 3 , the system (X,T ) is called DCi if there exists ∈ { } an uncountable set D such that each pair of distinct x, y D is DCi. If in (DC1) ∈ above does not depend on the choice of x, y D, then we say (X, f) is uniformly ∈ distributionally chaotic of type 1 (brieﬂy uniformly DC1).

The proof of the next theorem begins in a similar manner to the proof of

Theorem 3.8. To prove DC1, however, the percentage of coordinates that we have control over at each step in our construction needs to be signiﬁcantly larger than that of the previous step. For this reason, the pi’s and qi’s are deﬁned in a slightly more complicated manner in the proof that follows.

d Theorem 3.15. If X ΣZ is a nondegenerate subshift with the weak speciﬁcation ⊂ property, then (X, σ) is uniformly distributionally chaotic of type 1.

45 Proof. Let N witness WSP for σ. Let p1 = 0 and, for i N, deﬁne inductively ∈ qi = i pi and pi+1 = qi + N. We then have ·

p1 = 0 q1 = 0

p2 = N q2 = 2N

p3 = 2N + N = 3N q3 = 3(3N) = 9N

p4 = 9N + N = 10N q4 = 4(10N) = 40N

p5 = 40N + N = 41N q5 = 5(41N) = 205N . . . .

d Let S1 = 0 Z . For i 2, define { } ⊂ ≥ d Si = t Z : pi t qi and either tl qi−1 or tl pi for 1 l d . { ∈ ≤ | | ≤ | | ≤ | | ≥ ≤ ≤ } These are precisely the shells from Theorem 3.8 with one modification: these shells have widths growing at a much faster rate. See Figure 3.8. For each i 2 we will ≥ d consider the cube Qi = conv(Si) = s Z : s qi and the set Si Qi as { ∈ | | ≤ } ⊂ previously defined. In fact, for k N we will consider Si(k) Qi, a proper subset ∈ ⊂ of Si, defined by

d Si(k) = t Z :(t + s) Si for all s k . { ∈ ∈ | | ≤ }

One may think of Si(k) as Si with a layer of coordinates (of width k) stripped oﬀ.

See Figure 3.9. Notice that Si(k) may be empty for only ﬁnitely many i N. Note, ∈ too, that

d d d Qi = conv(Si) = (2qi + 1) , and Si(k) > 2 (qi + 1 pi 2k) , | | | | | | − − whenever Si(k) is nonempty. The inequality on the right states that Si(k) is at least as large as its corners, of which there are 2d many.

46 d = 2 d = 3

Figure 3.8: An example of S1, S2, and S3 for the cases d = 2, 3 and N = 3. For the d = 3 case, a portion of S2 and S3 are removed to achieve a cutaway view.

We now compare the size of Qi and Si(k) as i for ﬁxed k N. We will → ∞ ∈ show that d d Si(k) 2 (qi + 1 pi 2k) L = lim | | lim − −d = 1. i→∞ Qi ≥ i→∞ (2qi + 1) | | Since, for i suﬃciently large, 0 < Si(k) < Qi , we certainly have that 0 L 1. | | | | ≤ ≤

Recall that qi = i pi so that · d d 2 (qi + 1 pi 2k) L lim − −d ≥ i→∞ (2qi + 1) d d 2 (i pi + 1 pi 2k) = lim · − −d i→∞ (2i pi + 1) · d 2(i pi + 1 pi 2k) = lim · − − i→∞ 2i pi + 1 · d 2i pi 2 pi + 2 4k = lim · − · − i→∞ 2i pi + 1 · d 2i pi 2 pi 2 4k = lim · + lim − · + lim − i→∞ 2i pi + 1 i→∞ 2i pi + 1 i→∞ 2i pi + 1 · · · = [1 + 0 + 0]d = 1.

47 S3(4) S3(4) S3(4)

S3(4) S3(4)

S3(4) S3(4) S3(4)

Figure 3.9: An example of S3(4) for the case d = 2 and N = 3 with Si for i 3 displayed ≤ faintly in the background. In this case, S1(4) and S2(4) are empty.

Then, in fact, L = 1. This tells us that, for a ﬁxed k N, the size of Si(k) is large ∈ relative to the size of Qi Si(k) as i gets large. We will use this fact a little later. ⊃ Before we get there, we ﬁrst construct a distributionally scrambled set.

N By WSP and Lemma 3.5, for each α 0, 1 and any x0, x1 X, we may ∈ { } ∈ i choose inductively the sequence (bα)i∈ of points in X requiring that, for j N, N ∈ 1 (bα)0 = xα1 ,   j (bα)t if t qj, i.e. t Qj j+1 | | ≤ ∈ (bα )t =  (xα )t if t Sj+1.  j+1 ∈ i Let yα be a limit point of (bα)i∈ . Then (yα)t = (xαj )t whenever t Sj for all j N. N ∈ ∈ N For any α 0, 1 we may deﬁne such a yα. ∈ { }

48 N We will consider a subset A 0, 1 which we now deﬁne. Let Mγ γ∈G be ⊂ { } { } an uncountable collection of disjoint minimal sets in 0, 1 N. For each γ G, choose { } ∈

α(γ) Mγ. We will consider the set A = α(γ) γ∈G. ∈ { }

Note that, since Mγ γ∈G are closed and disjoint, for α, β A, α = β, for all { } ∈ 6 j N we must have that σj(α) = σj(β). In other words, no two strings of zeros and ∈ 6 ones in our set agree in every coordinate after some ﬁnite initial segment. We would like to eliminate the possibility of two strings disagreeing in every coordinate after a

ﬁnite initial segment of coordinates, as well. Denote by α the point in 0, 1 N such { } that αi = αi for all i N, i.e. α disagrees with α in every coordinate. Note that for 6 ∈ each α A, the set β A : σj(α) = σj(β) for some j N is at most countable. ∈ { ∈ ∈ } Then, without loss of generality, we may in fact assume that for each α A, there ∈ is no β A and j N such that σj(α) = σj(β). To see this, it is enough to note ∈ ∈ that a partition of an uncountable set into countable sets is necessarily uncountable.

This ensures that any α, β A with α = β will agree in inﬁnitely many coordinates ∈ 6 and disagree in inﬁnitely many coordinates.

Let M0,M1 X be disjoint minimal sets and let x0 M0 and x1 M1. For ⊂ ∈ ∈ each α A, we may now construct a point yα in the previously described method, ∈ i.e., such that (yα)t = (xαj )t whenever t Sj for all j N. Let D = yα : α A be ∈ ∈ { ∈ } a set of points constructed in this manner. We will show that D is an uncountable, uniformly distributionally scrambled set.

−n Recall that our metric is deﬁned by ρ(x, y) = 2 where n = min t : xt = yt , {| | 6 } so x and y are within at least 2−n if they agree on a central block of radius n. Note that since M0 M1 = with M0 and M1 closed in a compact space, there are ∩ ∅ −l l N and such that 0 < < 2 < dist(M0,M1). Choose k N such that ∈ ∈ 2−k < < 2−l, and note that if x and y agree on a central block of radius k, we

49 −k must have that dist(x, y) < 2 < . Note, too, that if instead we have x M0 and ∈

d t1 t2 y M1, then, for any t1, t2 Z , we have σ (x) M0 and σ (y) M1. Therefore ∈ ∈ ∈ ∈

t1 t2 −l −k d dist(σ (x), σ (y)) > 2 > 2 for any t1, t2 Z . In other words, any shifts of x ∈ and y will disagree in some coordinate within a central block of size at most l.

Choose yα, yβ D and choose two increasing sequences of naturals, (in)n∈ ∈ N and (im)m∈ , such that αin = βin and αim = βim for all n, m N. Then, by the way N 6 ∈ we have deﬁned yα and yβ, we have that

yα S = xα S = xβ S = yβ S , and | in in | in in | in | in

yα S = xα S = xβ S = yβ S . | im im | im 6 im | im | im

In other words, yα and yβ are defined in the same way on Sin and differ on Sim for all m, n N. The extent to which yα and yβ differ is dependent on how x0 and x1 ∈ differ. More precisely, we have

t t −k dist(σ (yα), σ (yβ)) < 2 < , whenever t Si (k) and ∈ n −l t t < 2 < dist(σ (yα), σ (yβ)), whenever t Si (k). ∈ m

So, for i (in)n∈ such that Si(k) = , we have ∈ N 6 ∅ 1 Φ(qi+1)() = 0 t q : ρ(σt(y ), σt(y )) < yαyβ d i α β 2qi + 1 { ≤ | | ≤ } | | Si(k) | | d ≥ 2qi + 1 | d | d 2 (qi + 1 pi 2k) > − −d . (2qi + 1) Then it must be that

Φ∗ () = lim sup Φ(n) () yαyβ yαyβ n→∞

lim sup Φ(qin +1)() yαyβ ≥ in→∞

50 d d 2 (qi + 1 pi 2k) lim − −d = 1. ≥ i→∞ (2qi + 1)

On the other hand, for i (im)m∈ such that Si(k) = , we have ∈ N 6 ∅ 1 Φ(qi+1)() = 0 t q : ρ(σt(y ), σt(y )) < yαyβ d i α β 2qi + 1 { ≤ | | ≤ } | |d 2qi + 1 Si(k) | | − |d | ≤ 2qi + 1 |d | d 2 (qi + 1 pi 2k) < 1 − −d . − (2qi + 1) In this case, we have

(n) Φyαyβ () = lim inf Φy y () n→∞ α β

lim inf Φ(qim +1)() yαyβ ≤ im→∞ d d 2 (qi + 1 pi 2k) lim 1 − −d ≤ i→∞ − (2qi + 1)

= 1 1 = 0. −

As α and β were chosen arbitrarily and was not dependent on our choice of yα or yβ, the desired result is obtained. That is, the set D = yα : α A witnesses { ∈ } d (X, σ) ΣZ is uniformly distributionally chaotic of type 1. ⊆

3.7 Some Implications

Although ω-chaos does not in general imply Li-Yorke chaos for subshifts, a

DC1 scrambled set also exhibits LY-chaos.

d Corollary 3.16. Suppose X ΣZ is a subshift with the weak speciﬁcation property. ⊆ Then (X, σ) is chaotic in the sense of Li and Yorke.

d Recall that a subshift X ΣZ is strongly irreducible if there is N N such ⊆ ∈ d that for all J, K Z with dist(J, K) N and any y1, y2 X, there is x X with ⊂ ≥ ∈ ∈ x J = y1 J and x K = y2 K . Since X strongly irreducible implies the (non-periodic) | | | | speciﬁcation property, combining results of the last few sections along with the

51 previous corollary, we have shown several results that apply to strongly irreducible

Zd-subshifts.

d Corollary 3.17. Suppose X ΣZ is a strongly irreducible subshift. Then ⊆ (1) σ has ω-chaos,

d (2) if E Z is ﬁnite with LE± (x) = for all x X, then σ has LE± limit type ⊂ 6 ∅ ∈ chaos,

(3) (X, σ) is uniformly distributionally chaotic of type 1, and

(4) (X, σ) exhibits Li-Yorke chaos.

52 CHAPTER FOUR

Speciﬁcation and ω-Chaos in a General Compact Metric Space

In this chapter we partially answer a question raised by Lampart and Oprocha,

[66]. We show that if f is a surjection with the speciﬁcation property on a compact metric space, X, and we have uniform expansion near a ﬁxed point, then f has

ω-chaos. The main theorem is stated as follows:

We prove this by encoding subsystems of (X, f) using subshifts from 2ω =

0, 1 N0 , the full (one-sided) shift on two symbols (see Section 2.4). { } The question of Lampart and Oprocha we address is the following: “does every map with the weak specification (or its stronger version) posses an ω-scrambled set?” While this question remains open, we show that the answer is yes for systems with uniform expansion near a fixed point. Some sort of expansivity is a natural assumption, as known examples of systems exhibiting the specification property are typically expansive, [41]. In the next section we give some preliminary definitions and constructions. In Section 4.2, we carefully construct some subsystems of (X, f) via an encoding of subshifts of (2ω, σ) and proceed to prove the main theorem.

53 4.1 Some Deﬁnitions and Constructions

In order to show f is ω-chaotic, we will exhibit an uncountable omega scrambled set. The construction of this set will rely on the fact mentioned in Section 3.2 that 2ω has an uncountable set of points whose ω-limit sets form an uncountable collection of minimal sets. In fact it was stated that this is true for 2Z, but the same reasoning applies to 2ω. We will prove our main theorem via a careful encoding of points from 2ω into our dynamical system, (X, f), with the speciﬁcation property and uniform expansion near a ﬁxed point.

ω We begin by ﬁxing periodic points t0, t1 X. Then, for each α 2 , we will ∈ ∈ construct a point xα X whose iterates under f get close to t0 or t1 in a pattern ∈ determined by the coordinates of α. The point xα will also, under iterations of the map f, get close to the ﬁxed point, s, mentioned in the theorem.

For the arguments and lemmas that follow, we will assume f : X X is a → continuous surjection with the speciﬁcation property on a compact metric space. In addition, assume that there is a ﬁxed point s such that there is some η > 0 and

λ > 1 such that if 0 < d(s, y) < η then d(s, f(y)) λd(s, y). ≥

Since f has the speciﬁcation property, there are points t0, t1 X periodic with ∈ periods p and q, respectively, and t0 = t1, t0 = s = t1. 6 6 6 Let > 0 such that

i j B2(s),B2(f (t0)),B2(f (t1)) 0≤i

Since f is uniformly continuous, choose δ0 > 0 such that if d(y, z) < δ0, then d(f i(y), f i(z)) < , for all 0 i pq. Let 0 < δ < min η, , δ0 . Let N witness spec- ≤ ≤ { } i i iﬁcation property for δ. Let U0 = B2(f (t0)) 0≤i

Then Bδ(s),U0,U1 is a disjoint collection in X. See Figure 4.1. { }

54 s η

t0 2 δ t1

U0 U1

Figure 4.1: The sets U0 and U1 (depicted with dashed lines) generated by the choice of periodic points t0 and t1.

Deﬁnition 4.1. Choose α 2ω. We will consider the following sequence of points in ∈ X:

Φ(α) = (tα0 , s, tα0 , tα1 , s, tα0 , tα1 , tα2 , s . . .) where the indices on the α’s follow (0, 0, 1, 0, 1, 2, 0, 1, 2, 3 ...). Notice that any initial segment of α is repeated inﬁnitely often as indices. We will refer to Φ(α) as the pattern induced by α and denote the jth term of Φ(α) by Φ(α)j. So Φ(α)j is either t0, t1, or s for any given j N. ∈

Next, using the speciﬁcation property we will construct a point xα X such ∈ that xα’s iterates follow the pattern of Φ(α) in the correct order. The iterates of xα

will stay within δ of iterates of each tαi for 2N iterates. For the ﬁrst s that appears in Φ(α), we want the iterates of xα to remain within δ of s for 4N many iterations.

55 The kth instance of s in Φ(α) corresponds to iterates of xα staying within δ of s for

4kN many consecutive iterations.

To this end, we recursively construct a sequence of integers

0 = p1 < q1 < p2 < q2 < p3 < q3 < , ··· so that

p1 = 0 q1 = 2N iterates of xα are near tα0

p2 = q1 + N q2 = p2 + 4N iterates of xα are near s

p3 = q2 + N q3 = p3 + 2N iterates of xα are near tα0

p4 = q3 + N q4 = p4 + 2N iterates of xα are near tα1

2 p5 = q4 + N q5 = p5 + 4 N iterates of xα are near s 2nd time . . . .

Continuing in this manner, we deﬁne pi and qi for all i N. Note the following ∈ association between terms of Φ(α) and each pj for j N. ∈

Φ(α) = tα0 s tα0 tα1 s tα0 tα1 tα2 s . . .

p1 p2 p3 p4 p5 p6 p7 p8 p9 ...

We are now ready to deﬁne xα.

ω Lemma 4.2. For each α 2 , there is a point xα X such that ∈ ∈ i i−pj d f (xα) , f (Φ(α)j) δ, pj i < qj, j N0 ≤ ≤ ∈

i Proof. Since s is a ﬁxed point, we have that f (s) = s, for all i N0. Then, by the ∈ speciﬁcation property, we have the following:

1 There is xα such that

i 1 i d f x , f (tα ) < δ, p1 i < q1. α 0 ≤ 2 There is xα such that

i 2 i d f x , f (tα ) < δ, p1 i < q1 and α 0 ≤

56 i 2 d f x , s < δ, p2 i < q2. α ≤ 3 There is xα such that

i 3 i d f x , f (tα ) < δ, p1 i < q1, α 0 ≤ i 3 d f x , s < δ, p2 i < q2, and α ≤

i 3 i−p3 d f x , f (tα ) < δ, p3 i < q3. α 0 ≤ k Generally, there is xα such that

i k i−pj d f x , f (Φ(α)j) < δ, pj i < qj, α ≤ for j k. See Figure 4.2. ≤

t x1 : 0 α 2N ··· t x2 : 0 ? s α 2N N 4N ··· t t x3 : 0 s 0 α 2N N 4N N 2N ··· t t t x4 : 0 s 0 1 α 2N N 4N N 2N N 2N ···

5 t0 s t0 t1 s xα: 2N N 4N N 2N N 2N N 42N ··· . . .

Figure 4.2: Constructing points xα that shadow orbit segments of t0, t1 and s according to Φ(α), the pattern induced by α.

k Then x is a sequence of points in Bδ (tα ). Denote a limit point of this α k∈N 0 ω sequence xα and, for each α 2 , note that we may deﬁne such an xα. Then xα has ∈ the desired property.

Note that in the above proof we use the fact that f is a surjective mapping

k to make sure iterates of xα fall within δ of Φ(α)j rather than some iterate of Φ(α)j. Similar strategy is used elsewhere in this chapter.

57 4.2 Constructing a Set Exhibiting ω-Chaos

In order to prove the Main Theorem, we need to prove a few simpler lemmas regarding the points we constructed in the previous section and the points that appear in their ω-limit sets.

ω Lemma 4.3. For any α 2 , s ωf (xα). ∈ ∈

Proof. Our goal is to show that if many consecutive iterates of xα are within δ <

η of s, then xα must, in fact, limit to s. We use the fact that s has uniform

ω expansivity. Let α 2 and xα as in Lemma 4.2. Note that λd(s, y) d(s, f(y)) ∈ ≤ whenever d(s, y) < η. Suppose λnd(s, y) d(s, f n(y)) with d(s, f n(y)) < η. Then ≤ λn+1d(s, y) λd(s, f n(y)) d(s, f n+1(y)). ≤ ≤ Suppose for all 0 i n we have d(s, f i(y)) < η. Then ≤ ≤ λnd(s, y) d(s, f n(y)) < η ≤ 1 η d(s, y) d(s, f n(y)) < . ≤ λn λn η i Let 0 < ν < η and let m N such that m < ν. Then if d(s, f (y)) δ < η, ∈ λ ≤ η for all i m, d(s, y) < m < ν. Let n N such that qn pn m, 2N. Then ≤ λ ∈ − ≥

i pn d(f (xα), s) δ < η for pn i < qn and so d(f (xα), s) < ν. Since this holds for ≤ ≤ any ν with 0 < ν < η, s ωf (xα). ∈

Next, we will choose β ωσ(α). We will then deﬁne a point zβ whose iterates ∈ get close to iterates of t0 and t1 in a pattern determined by the coordinates of β—we do not mean Φ(β) here, but rather a pattern considering only t0 and t1 depending on the coordinates of β while not including the ﬁxed point s. See Figure 4.3.

To this end, let c0 = 0. Then deﬁne di = ci + 2N and ci+1 = di + N for i N0. ∈ So we have

c0 = 0 d0 = 2N

58 tβ0 tβ1 tβ2 tβ3 tβ4 zβ: 2N N 2N N 2N N 2N N 2N ···

Figure 4.3: The point zβ ω(xα) shadows orbit segments of t0, t1 according to the ∈ coordinates of β ω(α). ∈

c1 = 3N d1 = 5N

c2 = 6N d2 = 8N

c3 = 9N d3 = 11N . . . .

ω ω Lemma 4.4. Suppose α 2 with β ωσ(α) 2 . Then there is a point zβ ∈ ∈ ⊆ ∈

i i−cj ωf (xα) X such that d f (zβ), f (tβj ) δ for all cj i < dj for all j N0. ⊆ ≤ ≤ ∈

ω Proof. Let α 2 and let β ωσ(α). Then there is some increasing sequence of ∈ ∈

rn natural numbers (rn)n∈ such that σ (α)[0,n) = β[0,n) for all n N. Here α[0,n) N ∈ r denotes the initial segment of α of length n. Note that if qk pk 4 N from some − ≥ k, r N, then Φ(α)r = s and Φ(α)r+j = s for 0 < j r + 1. Let k1 N such that ∈ 6 ≤ ∈

r1 qk1−1 pk1−1 4 N. Suppose kn−1 is deﬁned. Let kn N such that kn > kn−1 and − ≥ ∈

rn+n qk −1 pk −1 4 N. n − n ≥

Note that for all n N, Φ(α)kn+rn+j = tβj for 0 j < n. Letting in = pkn+rn , ∈ ≤

in in consider (f (xα)) . Noting that d (f (xα), tβ ) δ for all n, let zβ be a limit n∈N 0 ≤

in i i−cj point of (f (xα)) . Then zβ has the property that d f (zβ), f (tβ ) δ for n∈N j ≤ all cj i < dj for all j N0. ≤ ∈

So for any β ωσ(α), we may deﬁne zβ ωf (xα) X such that zβ follows ∈ ∈ ⊆ iterates of t0 and t1 according to the coordinates of β. We now show a converse of sorts. We show that if zβ also belongs to ωf (xγ), then β ωσ(γ). ∈

59 ω Lemma 4.5. Suppose α, γ 2 , β ωσ(α) and xα and xγ are deﬁned as in Lemma ∈ ∈

4.2. Suppose further that zβ ωf (xα) as in Lemma 4.4. Then, if zβ ωf (xγ), we ∈ ∈ must have β ωσ(γ). ∈

Proof. Recall that we chose > 0 so that

i j B2(s),B2(f (t0)),B2(f (t1)) 0≤i

i i sets. We then let U0 = B2(f (t0)) 0≤i

ω Suppose α, γ 2 are ﬁxed and let β ωσ(α). Deﬁne xα, xγ, and zβ ωf (xα) ∈ ∈ ∈ as in Lemmas 4.2 and 4.4. Suppose zβ ωf (xγ). Then there is some increasing ∈

rn sequence of naturals, (rn)n∈ , such that f (xγ) zβ. Let M N. Using uniform N → ∈ continuity, choose 0 < ξ < δ such that if d(x, y) < ξ, d(f i(x), f i(y)) < δ for all

rn 0 i 3MN. Fix n such that d (f (xγ) , zβ) < ξ. Then, since zβ has the property ≤ ≤

i i−cj that d f (zβ), f (tβj ) δ for all cj i < dj for j N0, we must also have that ≤ ≤ ∈

i rn i−cj d f (f (xγ)), f (tβ ) 2δ for all cj i < dj for 0 j < M. To see this, note j ≤ ≤ ≤

i rn that cj−1, dj−1 < 3jN for all j N. Less strictly, we have f (f (xγ)) Uβj for all ∈ ∈ cj i < dj for 0 j < M. ≤ ≤

rn Suppose pk rn < qk for some k N. According to Φ(α)k, this means f (xγ) ≤ ∈

rn is in one of U0, U1, or Bδ(s). Since f (xγ) B2δ(tβ ) Uβ , it must be the case ∈ 0 ⊂ 0

i pk i cj rn that d (f (f (xγ)) , f (tβ )) δ for c0 i < d0. Likewise, since f (f (xγ)) Uβ 0 ≤ ≤ ∈ j

i pk i−cj for 0 j < M, we must have that d f (f (xγ)) , f (tβ ) δ for cj i < dj, ≤ j ≤ ≤

i i−pk+j 0 j < M. Equivalently, d f (xγ), f (tβ ) δ for pk+j i < qk+j, 0 j < ≤ j ≤ ≤ ≤ k M. Then Φ(γ)k+j = tβ for 0 j < M, i.e., σ (γ)[0,M) = β[0,M). j ≤

60 pk Suppose instead that qk−1 rn < pk for some k N. Then, since f (xγ) ≤ ∈ ∈

i i−pk+j Uβ , a similar argument shows that d f (xγ), f (tβ ) δ for pk+j i < qk+j, 0 j ≤ ≤ k 0 j < M 1, and hence σ (γ)[0,M−1) = β[0,M−1). ≤ −

Since this is true for all M N, we must have that β ωσ(γ). Hence, if ∈ ∈ zβ ωf (xγ), then β ωσ(γ). ∈ ∈

Using the previous lemmas, we are now ready to prove the main theorem.

Unlike the main theorems of Chapter Three, we do not rely on the existence of an uncountable, disjoint collection of uncountable minimal sets in X. We do, however, rely on the existence of such a collection in 2ω.

ω Proof. Let A 2 such that ωσ(α) α∈A is an uncountable family of uncountable ⊂ { } minimal sets with ωσ(α) ωσ(γ) = , for all α, γ A, α = γ. Then for each α A we ∩ ∅ ∈ 6 ∈ may deﬁne a point xα as in Lemma 4.2. By Lemmas 4.4 and 4.5, for each β ωσ(α), ∈ there is a point zβ ωf (xα) such that zβ / ωf (xγ) for all γ A α . To see this, ∈ ∈ ∈ − { } recall that Lemma 4.5 states that zβ ωf (xγ) implies β ωσ(γ), a contradiction as ∈ ∈

ωσ(α) wσ(γ) = . Note that the collection zβ : β ωσ(α) is uncountable. Note, ∩ ∅ { ∈ } too, that since ωσ(α) is minimal, β ωσ(α) is not periodic. Hence, by construction, ∈ zβ is not periodic for all β ωσ(α). Let Ω = xα α∈A. Then for any xα, xγ Ω, we ∈ { } ∈ have ωf (xα) ωf (xγ) is uncountable. Moreover, ωf (xα) ωf (xγ) s by Lemma 4.3. \ ∩ 3

Finally, we have that ωf (xα) Per(f) = . Hence f is ω-chaotic. \ 6 ∅

61 It is reasonably clear that the assumption that s be a ﬁxed point can be weakened to the assumption of an expanding periodic orbit. The question remains whether a weaker form of the speciﬁcation property (with or without any sort of expansivity assumption) can achieve the same result. It would be nice if the expansivity condition could be done away with completely.

62 CHAPTER FIVE

Future Work

5.1 More with ω-Chaos

Suppose we have two systems (X, f) and (Y, g) that are semiconjugate via a map π : X Y as pictured. → f X X

π π

g Y Y

If (Y, g) exhibits ω-chaos, it is not true in general that (X, f) does, as well.

Along with introducing ω-chaos in [68], Li proved the following theorem.

Theorem 5.1 (Li). If f is countable-to-one semiconjugate to g via π : X Y , then → if (Y, g) contains an uncountable ω-scrambled set Ω with \ ω(x) = , 6 ∅ x∈Ω then so does (X, f). Moreover, one can construct such an ω-scrambled set for (X, f) from π−1(Ω).

In the paper of Lampart and Oprocha, [66], they prove the following two theorems.

Theorem 5.2 (Lampart, Oprocha). Assume that Y Zd is a subshift with the weak ⊆ speciﬁcation property and y Y not transitive. Additionally assume that (X, f) is ∈ an extension of (Y, σ) via a factor map π : X Y and the set → [ B = ωf (x) x∈π−1({y}) contains at most countably many minimal sets. Then f is ω-chaotic.

63 Theorem 5.3 (Lampart, Oprocha). Assume that Y Zd is a subshift with the weak ⊆ speciﬁcation property and y Y not transitive. Additionally assume that (X, f) ∈ is an extension of (Y, σ) via a factor map π : X Y and π−1( y ) is at most → { } countable. Then f is ω-chaotic.

It seems likely that similar results also hold in the case of Zd-actions, or, more

d speciﬁcally, for subshifts of ΣZ . This would show that the results of Chapter Three can be extended to a much broader class of spaces than simply Zd-subshifts.

5.2 Relations Between Various Types of Chaos for Zd-Actions In 2005, Huo Yun Wang and Jin Cheng Xiong presented the following theorem with proofs of several implications, [110].

Theorem 5.4 (Wang, Xiong). Assume that X ΣZ is a (one-sided) subshift of ﬁnite ⊆ type. Then the following are equivalent.

(1) h(σ) > 0, i.e. (X, σ) has positive entropy,

(2) (X, σ) is chaotic in the sense of Li and Yorke,

(3) (Z, σ Z ) is chaotic in the sense of Devaney for some closed invariant subset | Z X, and ⊆ (4) (X, σ) is distributionally chaotic of type 1.

It would be interesting to see which of these results generalize to the case

d subshifts of ﬁnite type in ΣZ .

5.3 Tiling Spaces

Related to symbolic dynamics is the theory of tiling spaces. In fact, the theory of tiling spaces is a generalization of sequence spaces and therefore contains all of Zd symbolic dynamics, [112]. Rather than a Zd-action, however, tiling spaces

64 are paired with a continuous rather than a discrete action. The study of tiling spaces has some surprising applications. In particular, certain types of aperiodic tilings exhibit patterns similar to quasicrystals, crystal-like structures that have rigid structure but lack the usual periodic symmetries of crystals, [96,99]. An example of a quasicrystalline structure realized as a tiling is the Penrose tiling, shown in Figure

5.1.

2 Figure 5.1: A patch of a Penrose tiling. An actual tiling continues forever to tile all of R . A tiling space might consist of all limit points of translations of the above tiling.

Informally, one builds a tiling space from a collection of tiles, usually poly- hedra, with some restrictions on which tiles can be positioned next to others. One may then tile Rd (think of R2 for simplicity) by placing tiles side by side until the entire space is ﬁlled up with no gaps between tiles. If x describes such a tiling, Rd

d acts on x by translation, Tt(x): x x t for t R . A tiling space is a collection 7→ − ∈ of tilings closed under translation. Two tilings are close if they agree (up to small

65 translation) on a large ball centered at the origin. For a more formal introduction, see [91].

Questions of decidability—when may a set of tiles tile Rd?—arise in the theory of tiling and in Zd-symbolics, [19, 109]. Certain kinds of tilings, those built from a ﬁnite collection of polyhedral tiles meeting face to face along with some weak topological requirements, can be viewed as direct analogues with Zd subshifts. Take for instance the following theorem of Sadun and Williams, [92].

Theorem 5.5 (Sadun-Williams). A tiling space XT that satisﬁes the above mentioned hypotheses is homeomorphic to a tiling space X whose tiles are marked d-cubes or equivalently to the d-fold suspension of a Zd subshift. The space X is deﬁned by local matching rules if and only if XT is.

Through this relationship, information can be passed from a shift space to a tiling space and vice versa. In addition, there are natural ways to view tiling spaces as inverse limits on d-dimensional branch manifolds, [5, 16, 17, 81, 90]. It would be interesting to see how some known results for chaos in discrete shift spaces (or inverse limit spaces) can inform the behavior of continuous translation on tiling spaces and vice versa. In particular, it may be that results from Chapter Three could be applied in the context of tiling spaces.

5.4 Maps on Dendrites

A finite graph is a compact, connected metric space that can be written as the union of finitely many arcs each pair of which is either disjoint, or intersects in one or both of their endpoints. A tree is uniquely arcwise connected finite graph—a finite graph with no simple closed curves. A dendrite is a generalization of a tree, and can be defined as a compact, connected, locally connected metric space containing no

66 simple closed curves, [76]. We call a point x such that X x has more than two \{ } connected components a branchpoint of X. Intuitively, one may think of a (locally connected) tree having inﬁnitely many branchpoints. See Figure 5.2.

Figure 5.2: From left to right, examples of a graph, tree, and dendrite. The three dots in the third picture represent an inﬁnite progression of diminishing arcs limiting down to a point.

While a finite tree (or graph) may be composed of a finite number of arcs and vertices, a dendrite may have infinitely many of each. In fact, branch points in a dendrite may be dense! Such dendrites arise, for instance, as Julia sets of certain complex quadratic functions, f : C C. See Figure 5.3. → Stewart Baldwin has shown that the dynamics on dendrites arising as Julia sets of complex quadratic functions may be represented by a (non-Hausdorff) symbolic shift space, [9,10]. Averbeck and Raines recently used this characterization to show that the dynamics on dendritic Julia sets exhibit uniform distributional chaos of type 1, [8]. It would be interesting to investigate the possibility of characterizing other classes of continuous maps on dendrites in a similar fashion.

As mentioned in Section 2.3, for continuous maps of the interval, Alexander

Blokh showed that the speciﬁcation property is equivalent to mixing, [22,25]. Later, he generalized these results to topological graphs, [23, 24]. In fact, Blokh showed

67 Figure 5.3: An example of a Julia set that is a dendrite. Image by Alexis Monnerot- Dumaine [GFDL or CC-BY-SA-3.0], via Wikimedia Commons. that for graphs, if the map is just transitive then either the system is conjugate to an irrational rotation of the circle, or the system has the relative speciﬁcation property. If (X, f) has the relative speciﬁcation property, then there is a regular periodic decomposition (minimally overlapping closed sets) D = (D0,...,Dn−1) whose union

m is X so that (Di, f D ) has the speciﬁcation property for each 0 i < n. Many | i ≤ of the results mentioned here rely on the existence of a terminal regular periodic decomposition—in essence, the system cannot be reasonably further broken down into further pieces that map cyclically onto one another in a nice way. See [13] for a precise deﬁnition.

Some partial extensions of these results to dendrites have been made, [44], although the implications are slightly weaker. In fact, Hoehn and Mouron have constructed an example of a weak mixing map on a dendrite which is not mixing,

68 [57]. In addition, the map has a unique periodic point (a fixed point), [1]. Then transitivity on dendrite comes no where near to implying the specification property or even the relative specification property. In addition, it has recently been shown that there is a transitive dendrite map admitting an infinite sequence of refining regular periodic decompositions, [108]. Then the methods Blokh relied on for the case of graph maps do not apply to all dendrites.

It would be interesting, still, to explore classes of dendrites for which transitivity may still imply relative speciﬁcation or for which mixing and speciﬁcation are equivalent. The structure of end points seem to pose a problem when attempt- ing to tame the behavior of maps on dendrites. Perhaps considering appropriate restrictions on the set of end points may be a good start toward positive results.

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