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Structure and randomness in dynamical systems : different forms of equicontinuity and sensitivity García-Ramos, Felipe 2015

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Structure and randomness in dynamical systems: different formsof equicontinuity and sensitivitybyFelipe Garc´ıa-RamosB.Math., Universidad Nacional Autonoma de Mexico, 2008M.Sc. in Mathematics, University of British Columbia, 2010a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Mathematics)The University of British Columbia(Vancouver)April 2015c© Felipe Garc´ıa-Ramos, 2015AbstractWe study topological and measure theoretic forms of mean equicontinuity and mean sensitivityfor dynamical systems. With this we characterize well known notions like systems with discretespectrum, almost periodic functions, and subshifts with regular extensions. We also study the limitbehaviour of µ-equicontinuous cellular automata.In this thesis we prove a conjecture from [55] (see Corollary 2.3.18); this was independentlysolved by Li-Tu-Ye in [45]. In Chapter 3 we answer questions from [8].iiPrefaceThis thesis is based on three manuscripts written by the author Felipe Garcia-Ramos which weresubmitted for publication (one accepted to Ergodic Theory and Dynamical Systems): Chapter 2 isbased on [22] and [20], Chapter 3 is based on [21]. The material in Chapters 2.1.1, 2.1.2, and 2.3.3is part of a paper in preparation with Brian Marcus.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Mean sensitivity, mean equicontinuity and spectral properties . . . . . . . . . . 72.1 Measure theoretical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 Almost periodic and µ−mean sensitive functions . . . . . . . . . . . . . . . . 82.1.2 µ− f−Mean expansivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Sequence entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Topological results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.1 Mean equicontinuity and mean sensitivity . . . . . . . . . . . . . . . . . . . . 142.2.2 Diam-mean equicontinuity and diam-mean sensitivity . . . . . . . . . . . . . 182.2.3 Almost automorphic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.4 Topological sequence entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.5 Counter-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Topological and measure theoretical . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.1 µ−Mean equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.2 µ−Mean sensitive pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29iv2.3.3 µ− f−Mean equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.4 µ−Mean equicontinuity points . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Limit behaviour of µ−equicontinuous cellular automata . . . . . . . . . . . . . . 383.1 Equicontinuity and local periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.2 Topological equicontinuity and local periodicity . . . . . . . . . . . . . . . . . 403.1.3 Measure theoretical equicontinuity and local periodicity . . . . . . . . . . . . 423.2 Weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 φ−Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58vAcknowledgmentsI would like to thank deeply my mathematical Senseis who not only taught me mathematics butalso taught me how to enjoy mathematics (in order of appearance):• My parents: who allowed me to think independently and showed me not to be afraid ofmistakes.• Gareth Shearman: I have not seen him since 2001 but I will never forget him and hisclasses. He introduced me to physics and is still my teaching role model.• Mo´nica Clapp: the person that introduced me to analysis; the main reason why I got deeplyinto Mathematics. A person with a great heart that is rarely not smiling.• Brian Marcus: it was once said that Brian tries to bring the best in people. I hope that oneday I can support people the way he does. I hope that I can continue to enjoy mathematics,mathematicians and people in general the way he does. A person that once decided to hugeveryone at a conference out of happiness.I also learned ergodic theory and dynamical systems theory from great people who are also greatmathematicians, these include: Karl Petersen, Tomasz Downarowicz, Boris Solomyak, XiangdongYe, Ronnie Pavlov (who introduced me to Brian and symbolic dynamics), Tom Meyerovitch, Nis-hant Chandgotia and Raimundo Bricen˜o. I thank them for their time and patience.I would like to thank my wife for reminding me to enjoy life at moments with work pressure.Finally I would like to thank CONACyT for supporting me with a PhD fellowship.viDedicationI dedicate this thesis to Sandra my sister.viiChapter 1IntroductionThe theory of dynamical systems emerged as the study of how time affected physical systems inthe long term. In the mathematical theory, time can be replaced by an action on a space. The ab-straction of dynamical systems has provided many benefits. It has provided a way to study spatialstructure as well as time behaviour with the same tools, and it has provided interesting connec-tions with many other fields such as, Functional Analysis, Information Theory, Thermodynamics,Statistical Physics, Number Theory, Differential Geometry and Group Theory.The study of dynamical systems can be divided into Smooth Dynamics, Topological Dynamicsand Ergodic Theory. The work of this thesis is related mostly to Topological Dynamics and ErgodicTheory (measurable dynamics). A topological dynamical system (TDS) is a compact metric setwith a continuous action. A measure preserving system (MPS) is a probability space with a measurepreserving action. Similar properties for dynamical systems can be described in a purely topologicaland in a purely measure theoretical way (e.g. mixing, entropy, discrete spectrum). During my PhDstudies my main research interest was the following: the comparison of results involving similarmeasure theoretical and topological properties and the study of hybrid properties, that is, propertiesthat use topology and measure theory. In particular I am interested in studying and comparingdichotomies between chaos/randomness/unpredictability and order/structure/rigidity.Vancouver is a very rainy city. During my first year of Ph.D. I entertained myself by doingmodels of how raindrops behave on windows. I wanted the models to capture the two differentphases that I could see. When there is very little rain, most of the raindrops don’t move; thesystem is very predictable. When it is raining hard, the system looks chaotic and unpredictable.I presented my first seminar talk on this topic and in [19] I prove that a certain kind of discretesystems can be decomposed into products of chaotic elements (shifts) and predictable elements(identities). After this I began studying different ways in which predictability and unpredictabilitycould be expressed in topological and measure theoretical terms.11.1 Chapter 2A topological dynamical system (TDS) is a pair (X,T ), where X is a compact metric space(a.k.a. phase space), and T : X → X is a continuous transformation (we will later study moregeneral dynamical systems). Many definitions stated in this section will be stated later to makethe thesis easier to read.Let X = S1 = [0, 1) be the unit circle with normalized angles. The map defined by Tx :=x+ θmod 1 represents a rotation by θ (see Figure 1.1). We have that (X,T ) is a TDS.Figure 1.1: RotationWithin a small neighbourhood it is very easy to predict how every point in it will behave.This system is predictable. A way to define predictability on topological dynamical systems iswith equicontinuity. A TDS is equicontinuous if for every ε > 0 there exists δ > 0 such that ifd(x, y) ≤ δ then d(T ix, T iy) ≤ ε for all i ∈ Z+. In other words, points that start close will alwaysstay close. The prototype for an equicontinuous TDS is a rotation on a compact abelian group(e.g. the circle rotation).Again let X = S1 = [0, 1) be the unit circle with normalized angles. The transformation definedby Tx := 2xmod 1 is a TDS known as the doubling map (see Figure 1.2). We do not have a goodprediction of how points in a small neighbourhood behave. This map is chaotic. A weak formof chaos is called sensitivity (or sensitive dependence on initial conditions). A TDS is sensitiveif there exists ε > 0 such that for every open set A ⊂ X there exists x, y ∈ A and i ∈ Z+ suchthat d(T ix, T iy) > ε. This means that, wherever you start, a small change can cause a significantdifference in the future. Sensitive and equicontinuous systems are related.A TDS is minimal if it has no proper subsystems (Y ⊂ X s.t. T (Y ) ⊂ Y and Y is closed).2Figure 1.2: Doubling mapTheorem 1.1.1 (Auslander-Yorke ’80). A minimal TDS is equicontinuous if and only if it is notsensitiveNote that there are non-minimal systems that are neither sensitive nor equicontinuous.This notion of equicontinuity is very strong. For example every non-periodic shift space is notequicontinuous. A weaker form of equicontinuity was introduced by Fomin in ’51. A TDS (X,T )is mean equicontinuous if for every ε > 0 there exists δ > 0 such that if d(x, y) ≤ δ thenlim sup1nn∑i=1d(T ix, T iy) ≤ ε.Every equicontinuous TDS is mean equicontinuous. There are many interesting mean equicontin-uous systems that are not equicontinuous. We will see examples later.We say (X,T ) is mean sensitive if there exists ε > 0 such that for every open set U there existsx, y ∈ U such thatlim sup1nn∑i=1d(T ix, T iy) > ε.Theorem 1.1.2 (G. see Theorem 2.3.17). A minimal TDS is mean sensitive if and only if it isnot mean equicontinuous.We say (X,µ, T ) is a measure preserving topological dynamical system (MP-TDS) if X is acompact metric space, (X,µ) is a Borel probability space, and T : X → X is a continuous measurepreserving transformation (µ(T−1(A)) = µ(A)). We say (X,µ, T ) and (X ′,µ′, T ′) are isomorphic(measure theoretically) if there exists a measure preserving function f : X → X ′ that is 1-1 almost3everywhere, f−1 is measure preserving and satisfies T ′◦f = f◦T. To every MP-TDS we can associatean operator (known as a Koopman operator) in L2(X,µ) (the range is C) given by UT f := f ◦ T.We say (X,µ, T ) has discrete spectrum if UT has discrete spectrum (i.e. basis of eigenfunctions). AMP-TDS has discrete spectrum if and only if it is isomorphic to a group rotation (thus isomorphicto an equicontinuous system). We may ask how much equicontinuity is preserved by the measureisomorphisms. In purely topological terms the answer is none. There exist topologically chaoticdynamical systems that have discrete spectrum when equipped with any ergodic Borel measure.In hybrid terms (topological and measure theoretical) we can say something. We say (X,µ, T ) isµ−mean equicontinuous if for every τ > 0 there exists a set M such that µ(M) > 1− τ and Trestricted to M is mean equicontinuous.Theorem 1.1.3 (G. see Theorem 2.3.17). Let (X,µ, T ) be an ergodic MP-TDS. We have that(X,µ, T ) has discrete spectrum if and only if it is µ−mean equicontinuous.Some characterizations of dynamical systems with discrete spectrum are the following:Functional analysis: Associated Koopman operator on L2(X,µ) has discrete spectrum (Halmos/von-Neumann).Algebraic: Isomorphic to a rotation on a compact abelian group. (Halmos/von-Neumann).Information theory: Zero sequence entropy for any sequence (Kushnirenko).Purely dynamical: µ−mean equicontinuous (G.)We also have a dichotomy with sensitivity in the hybrid case. We say (X,µ, T ) is µ−mean sensi-tive if there exists ε > 0 such that for every Borel set of positive measure U there exists x, y ∈ Usuch thatlim sup1nn∑i=1d(T ix, T iy) > εTheorem 1.1.4 (G. see Theorem 2.3.17). Let (X,µ, T ) be an ergodic MP-TDS. We have that(X,µ, T ) is µ−mean sensitive if and only if it is not µ−mean equicontinuous.1.2 Chapter 3So far we have only discussed invariant measures. When studying non-invariant measures one mayask what happens in the long term; in particular we may ask if Tnµ or the Cesa`ro average convergesin the weak topology (where Tµ is the push-forward of the measure). We will study this problemfor a particular kind of TDS, cellular automata on subshifts (also known as shift endomorphisms).Cellular automata (CA) are discrete systems that depend on local rules. Hedlund [31] charac-terized CA as dynamical systems: φ : X → X is a cellular automaton on the subshift X if andonly if φ is continuous (with respect to the Cantor product topology) and shift-commuting. Thismeans that if φ is a CA then (X,φ) is a topological dynamical system (TDS).4In [8] Blanchard-Tisseur showed that if µ is a shift-ergodic measure that gives full measure toequicontinuity points and φ is a CA then the weak limit of the Cesa`ro average of φnµ exists; alsothey asked questions about the dynamical behaviour of the limit measure. In particular they askedwhen the limit measure is shift-ergodic, a measure of maximal entropy or φ−ergodic (p. 581 in[8]). In Chapter 3 we show that these three conditions are very strong.Theorem (see Sections 3.2 and 3.3). Let X be an irreducible SFT, (X,φ) CA, and µ a shift-ergodic Borel probability measure that gives full measure to equicontinuity points. Let µ∞ be theweak limit of the Cesa`ro average of φnµ. We have that·µ∞ is the measure of maximal entropy if and only if µ is the measure of maximal entropy andφµ = µ = µ∞ (Theorem 3.2.18).·µ∞ is φ−ergodic if and only if (X,µ∞, φ) is isomorphic (measurably) to a cyclic permutationon a finite set (Corollary 3.3.4).Furthermore if we assume φ is surjective then·µ∞ is σ−ergodic if and only if µ is φ−invariant (Theorem 3.2.17).1.3 DefinitionsLet {G,+} be a locally compact semigroup. A G− measure preserving system (MPS)is a quadruple (X,Σ, µ, T ) where (X,Σ, µ) is a Lebesgue probability measure space and T :={T j : j ∈ G}is a G− measure preserving action on X. When it is not useful we will omit writingΣ. A Z+-measure preserving system can be represented with a Lebesgue probability measure space(X,µ) and a measure preserving transformation T : (X,µ)→ (X,µ). For simplicity in this the-sis we will assume G = Zd+ even though many results hold for more general actions. Nonethelessthe results are new even for d = 1.We say a subset is invariant if it is invariant under every T i;we say (X,µ, T ) is an ergodic system if it is a measure preserving system and µ is an ergodicmeasure (i.e. every invariant set has measure 0 or 1). The measurable sets with positive measureare denoted with Σ+.A measure preserving system (X,µ, T ) generates a family of unitary linear operators (knownas the Koopman operators) on the complex valued Hilbert space L2(X,µ), U jT : f 7→ f ◦ Tj . Wedenote the inner product of L2(X,µ) with 〈·, ·〉.A G−topological dynamical system (TDS) is a pair (X,T ), where X is a compact metricspace and T :={T i : i ∈ G}is a G− continuous action on X. Given a compact metric space X anda Borel probability measure µ we denote the Borel sets with BX and the Borel sets with positivemeasure with B+X .An important class of TDS are the subshifts. Let A be a finite set. For x ∈ AG and j ∈ G weuse xj to denote the ith coordinate of x andσ :={σi : AG → AG | xi+j = (σix)j for all x ∈ AGand j ∈ G}5to denote the shift maps. Using the product topology of discrete spaces, we have that AG is acompact metrizable space. A subset X ⊂ AG is a subshift (or shift space) if it is closed andσ−invariant; in this case (X,σ) is a TDS.6Chapter 2Mean sensitivity, mean equicontinuityand spectral properties2.1 Measure theoretical resultsIn this section we define mean sensitive functions and we characterize almost periodic functions.This provides a characterization of discrete and continuous spectrum for measure preserving sys-tems.Definition 2.1.1. Let S ⊂ G. We denote with Fn the n−cube [0, n]d . We define lower densityof S asD(S) := lim infn→∞|S ∩ Fn||Fn|,and upper density of S asD(S) := lim supn→∞|S ∩ Fn||Fn|.The following properties are easy to prove and will be used throughout the paper.Lemma 2.1.2. Let S, S′ ⊂ G, i ∈ G, and F ⊂ G finite set. We have that·D(S) = D(i+ S) and D(S) = D(i+ S).·D(S) +D(Sc) = 1.·If D(S) +D(S′) > 1 then S ∩ S′ 6= ∅.·D(S) := lim infn→∞|S∩FnF ||FnF |·D(S) := lim supn→∞|S∩FnF ||FnF |Let (X,µ, T ) be a MPS and B ∈ Σ+. We say x ∈ X is a generic point for B iflimn→∞∣∣i ∈ Fn : T ix ∈ B∣∣|Fn|= µ(B).7The pointwise ergodic theorem states that if (X,µ, T ) is ergodic and B ∈ Σ+ then almostevery point is a generic point for B. This theorem was originally proved for Z+−systems by Birkhoff.It also holds if G = Zd+ [28]. An equivalent version of the pointwsie ergodic theorem states that if(X,µ, T ) is ergodic and f ∈ L1(X,µ) thenlimn→∞1|Fn|∑i∈Fnf(T ix) =∫fdµ for a.e x.2.1.1 Almost periodic and µ−mean sensitive functionsDefinition 2.1.3. Let (X,µ, T ) be an ergodic system and f ∈ L2(X,µ).·We say f is an almost periodic function (i.e. f ∈ Hap) if cl({U jf : j ∈ G}) ⊂ L2(X,µ) iscompact.·We say f 6= 0 is an eigenfunction of (X,µ, T ) if there exists w∈Rd such thatU j(f) = e2pii〈w,j〉f ∀j ∈ G.In this case we say w is an eigenvalue of (X,µ, T ).Remark 2.1.4. Equivalently eigenfunctions can be described with characters. Note that everycontinuous homomorphism from G to the unit circle on C is of the form X (j) = e2pii〈w,j〉.When d = 1, f is an eigenfunction of (X,µ, T ) if and only if there exists λ∈C such that |λ| = 1andUT (f) = λf.Definition 2.1.5. A subset S ⊂ G is syndetic if there exists a finite set K ⊂ G such thatS +K = G.The following results are well known (see for example [57]).Proposition 2.1.6. Let (X,µ, T ) be an ergodic system.·Hap is the closure of the set spanned by all the eigenfunctions of T.·f ∈ L2(X,µ) is almost periodic if and only if for every ε > 0 there exists a syndetic set S ⊂ Gsuch that∫ ∣∣f − U jf∣∣ dµ ≤ ε for every j ∈ S.·The product of two almost periodic functions is almost periodic.·If f is an eigenfunction then |f | is constant almost everywhere.Definition 2.1.7. Let (X,µ, T ) and (X ′, µ′, T ′) be two MPS. We say they are isomorphic (mea-surably) if there exists an a.e. bijective and measure preserving function f : (X,µ) → (X ′, µ′)such that T ′ ◦ f = f ◦ T and the inverse is also measure preserving.8Definition 2.1.8. Let (X,µ, T ) be a ergodic system.·(X,µ, T ) has discrete spectrum if there exists an orthonormal basis for L2(X,µ) which con-sists of eigenfunctions of UT .·(X,µ, T ) has continuous spectrum if the only eigenfunctions are the constant functions.·(X,µ, T ) is a measurable isometry if it is isomorphic to an isometric TDS (i.e. d(x, y) =d(T ix, T iy)).·(X,µ, T ) is weakly mixing if T × T is ergodic.The following two results are due to Halmos and Von-Neumann (see for example [59]).Theorem 2.1.9. Let (X,µ, T ) be an ergodic system. The following are equivalent:·(X,µ, T ) has discrete spectrum.·(X,µ, T ) is a measurable isometry.·L2(X,µ) = HapTheorem 2.1.10. Let (X,µ, T ) be an ergodic system. The following are equivalent:·(X,µ, T ) has continuous spectrum.·(X,µ, T ) is weakly mixing.·Hap consists only of constant functions.Definition 2.1.11. Let f ∈ L2(X,µ). We say an MPS (X,µ, T ) is µ−f−mean sensitive if thereexists ε > 0 such that for every A ∈ Σ+ there exists x, y ∈ A such thatlim sup1|Fn|∑j∈Fn∣∣f(T jx)− f(T jy)∣∣2 > ε.If this happens we say f is µ−mean sensitive (f ∈ Hms) and ε is a sensitivity constant.Remark 2.1.12. Let (X,µ, T ) be an ergodic system and B ∈ Σ+. It is not difficult to see that(X,T ) is µ− 1B−mean sensitive if and only if there exists ε > 0 such that for every A ∈ Σ+ thereexists x, y ∈ A such thatD{i ∈ G : T ix ∈ B and T iy /∈ B or T iy ∈ B and T ix /∈ B}≥ ε.Lemma 2.1.13. Let (X,µ, T ) be an ergodic MPS. We have that Hms ⊂ L2(X,µ) is an open set.Proof. Let f ∈ Hms. There exists ε > 0 such that for every A ∈ Σ+ there exists x, y ∈ A such thatlim sup1|Fn|∑j∈Fn∣∣f(T jx)− f(T jy)∣∣2 > ε.9Assume that∫|f − g|2 dµ ≤ (ε/4). By the pointwise ergodic theorem there exists Y ⊂ X withµ(Y ) = 1 andlim1|Fn|∑j∈Fn∣∣f(T jx)− g(T jx)∣∣2 ≤ (ε/4) for every x ∈ Y.We have that for every A ∈ Σ+ there exist x, y ∈ A ∩ Y such thatlim sup1|Fn|∑j∈Fn∣∣g(T jx)− g(T jy)∣∣2≥ lim sup1|Fn|∑j∈Fn∣∣f(T jx)− f(T jy)∣∣2 −∣∣f(T jx)− g(T jx)∣∣2 −∣∣f(T jy)− g(T jy)∣∣2> ε/2.This implies that Hms ⊂ L2(X,µ) is an open set.Theorem 2.1.14. Let (X,µ, T ) be an ergodic MPS and f ∈ L2(X,µ). The following are equivalent.1)f ∈ Hcap2)f ∈ Hms3)There exists ε > 0 such that for every A ∈ Σ+ there exists x, y ∈ A such thatlim1|Fn|∑j∈Fn∣∣f(T jx)− f(T jy)∣∣2 > ε.Thus we have that Hcap = Hms.Proof. 3)⇒ 2)Obvious (lim implies lim sup).1)⇒ 3)Assume that the closure of{U jf : j ∈ G}is not compact. This implies it is not totally bounded,hence there exists ε > 0 such that for every N ∈ N there exists SN ⊂ G with |SN | = N such that∫∣∣U if − U jf∣∣2 dµ ≥ ε for every i 6= j ∈ SN . (2.1.1)We want to show that (X,T ) is µ − f−mean sensitive. Let A ∈ Σ+ and N sufficiently large.There exist s 6= t ∈ SN such thatµ(T−sA ∩ T−tA) > 0.Let g(x) :=∣∣U sf(x)− U tf(x)∣∣2 . By the pointwise ergodic theorem we have that there existsz ∈ T−sA ∩ T−tA such thatlim1|Fn|∑j∈Fng(T jz) =∫gdµ ≥ ε.10Let p := T sz ∈ A and q := T tz ∈ A. Since G is commutative we have thatlim1|Fn|∑j∈Fn∣∣f(T jp)− f(T jq)∣∣2= lim1|Fn|∑j∈Fn∣∣f(T j+sz)− f(T j+tz)∣∣2= lim1|Fn|∑j∈Fng(T jz) > ε.We conclude that for every A ∈ Σ+ there exist p, q ∈ A such thatlim1|Fn|∑j∈Fn∣∣f(T jp)− f(T jq)∣∣2 > ε.2)⇒ 1)Let g be an eigenfunction. This means there exists w∈Rd such that U j(g) = e2pii〈w,j〉g. Letε > 0. There exists a set Xε ∈ Σ+ such that |g(x)− g(y)|2 ≤ ε for every x, y ∈ Xε. Since g is aneigenfunction we have that for every x, y ∈ Xε and every j ∈ G∣∣U jg(x)− U jg(y)∣∣2=∣∣∣e2pii〈w,j〉(g(x)− g(y))∣∣∣2= |g(x)− g(y)|2 ≤ ε.Thus g ∈ HcmsUsing Lemma 2.1.13 we conclude that Hap ⊂ Hcms.Proposition 2.1.15. Let (X,µ, T ) be an ergodic system. We have that1) (X,µ, T ) has discrete spectrum if and only if 1B ∈ Hap for every {B,Bc} ⊂ Σ+.2) (X,µ, T ) is weakly mixing if and only if 1B /∈ Hap for every {B,Bc} ⊂ Σ+.Proof. 1) ⇒)Use Theorem 2.1.9.1) ⇐)Since Hap is a closed subspace (Proposition 2.1.6) and indicator functions are dense in L2(X,µ)we have that Hap = L2(X,µ).2) ⇒)If (X,µ, T ) is weakly mixing then Hap contains only constant functions (Theorem 2.1.10) so1B /∈ Hap for every {B,Bc} ⊂ Σ+.2) ⇐)11Assume (X,µ, T ) does not have continuous spectrum. This implies there exists a non-constanteigenfunction f. This means there exists w∈Rd such thatU j(f) = e2pii〈w,j〉f ∀j ∈ G.By Proposition 2.1.6 we have that |f | = r is constant almost everywhere. This implies thereexists A ⊂ {x ∈ C : |x| = r} such that 0 < µ(H) < 1 where H := f−1(A). For every ε > 0 thereexists a syndetic set S ⊂ G such that µ(H4T jH) ≤ ε for every j ∈ S. This implies that∫∣∣1H − Uj1H∣∣ dµ= µ(H4T jH) ≤ εfor every j ∈ S. Using Proposition 2.1.6 we conclude 1H is almost periodic.Corollary 2.1.16. Let (X,µ, T ) be an ergodic system. We have that1) (X,µ, T ) has discrete spectrum if and only if 1B ∈ Hcms for every {B,Bc} ⊂ Σ+.2) (X,µ, T ) is weakly mixing if and only if 1B /∈ Hcms for every {B,Bc} ⊂ Σ+.2.1.2 µ− f−Mean expansivityDefinition 2.1.17. Let (X,µ, T ) be a measure preserving system and f : X → C. We define thefollowing pseudometric.df (x, y) = lim sup1|Fn|∑j∈Fn∣∣f(T jx)− f(T jy)∣∣2 .The ε−closed balls of the pseudometric will be denoted with Bfε (x) := {y : df (x, y) ≤ ε} .Definition 2.1.18. Let f ∈ L2(X,µ). A system (X,µ, T ) is µ−f−mean expansive if there existsε > 0 such that µ× µ {(x, y) : df (x, y) > ε} = 1.Lemma 2.1.19. Let (X,µ, T ) be an ergodic system. Then g(x) := µ({y : df (x, y) ≤ ε}) is constantfor almost every x ∈ X and equal to µ× µ {(x, y) : df (x, y) ≤ ε} .Proof. One can show that df (x, y) is µ× µ−measurable. This means that {(x, y) : df (x, y) ≤ ε} isµ× µ−measurable for every ε > 0. Using Fubini’s Theorem we obtain thatµ× µ {(x, y) : df (x, y) ≤ ε} =∫X∫X1{(x,y):df (x,y)≤ε}dµ(y)dµ(x)=∫Xµ {y : df (x, y) ≤ ε} dµ(x).12Since g is T -invariant we conclude that g(x) is constant for almost every x ∈ X and equal toµ× µ {(x, y) : df (x, y) < ε} .Theorem 2.1.20. Let (X,µ, T ) be an ergodic system and f ∈ L2(X,µ). The following are equiva-lent:1) (X,T ) is µ− f−mean sensitive.2) (X,T ) is µ− f−mean expansive.3) There exists ε > 0 such that for almost every x, µ(Bfε (x)) = 0.Proof. 2)⇒ 1)Let A ∈ Σ+. This means that A×A ∈ Σ+×Σ+. By hypothesis we can find (x, y) ∈ A×A suchthat df (x, y) ≥ ε.1)⇒ 3)Suppose (X,T ) is µ − f−mean sensitive (with µ − f−mean sensitivity constant ε) and that3) is not satisfied. This means there exists x ∈ X such that {x′ : df (x, x′) ≤ ε/2} ∈ Σ+. For anyy, z ∈ {x′ : df (x, x′) ≤ ε/2} we have that df (y, z) < ε. This contradicts the assumption that (X,T )is µ− f−mean sensitive.3)⇒ 2)Using Lemma 2.1.19 we obtain that µ× µ {(x, y) : df (x, y) ≤ ε} = 0.2.1.3 Sequence entropyDefinition 2.1.21 ([44]). Let (M,µ, T ) be a MPT. Given a finite measurable partition P of X andS = {sn} ⊂ G we define hSµ(P, T ) := lim supn→∞1nH(∨ni=1T−siP), and the sequence entropy of(X,µ, T ) with respect to S as hSµ(T ) := supP hSµ(P, T ).The system is said to be µ-null (or zerosequence entropy) if hSµ(T ) = 0 for every S ⊂ G.Kushnirenko’s theorem states that an ergodic system is µ−null if and only if it has discretespectrum [44].Theorem 2.1.22 (Kushnirenko ’67 [44]). Let (X,µ, T ) be an ergodic system and B ∈ Σ+. Thenthere exists infinite S ⊂ G so that hSµ({B,Bc} , T ) > 0 if and only if cl({Un1B : n ∈ G} ⊂ L2(X,µ)is not compact.Thus an ergodic system is µ−null if and only if it has discrete spectrum.2.2 Topological resultsThe metric and ε−closed balls on a compact metric space X will be denoted by d and Bε(x)respectively.13Mathematical definitions of chaos have been widely studied. Many of them require the system tobe sensitive. A TDS (X,T ) is sensitive if there exists ε > 0 such that for every open set A ⊂ X thereexists x, y ∈ A and i ∈ G such that d(T ix, T iy) > ε. On the other hand, equicontinuity representspredictable behaviour. A TDS is equicontinuous if T is an equicontinuous family. Auslander-Yorke showed that a minimal TDS is either sensitive or equicontinuous [4]. A problem with thisclassification is that equicontinuity is a strong property and not adequate for subshifts; it is notdifficult to see that a subshift is equicontinuous if and only if it is finite.The topological version of Halmos-Von Neumann Theorem states that for transitive TDS,equicontinuous maps can be characterized as those with topological discrete spectrum, i.e., theinduced operator on C(X) has discrete spectrum (e.g. see [59]). It is easy to see that any equicon-tinuous TDS has zero topological entropy. Similar to (measure) sequence entropy one can definetopological sequence entropy. A null system is a TDS that has zero topological sequence entropy.It is well known that equicontinuity implies nullness, but the converse is false [27]. Nevertheless,one can ask if there is a sense in which every null TDS is “nearly” equicontinuous. Indeed, in theminimal case there is. Any TDS has a unique maximal equicontinuous factor [3], and Huang-Li-Shao-Ye [34] showed that for any minimal null TDS (X,T ), the factor map from X to its maximalequicontinuous factor is 1-1 on a residual set (i.e., (X,T ) is almost automorphic). We strengthenthis result for subshifts in Corollary 2.2.37, by showing that the factor map is 1-1 on a set of fullHaar measure (i.e., (X,T ) is regular).In order to establish Corollary 2.2.37, we introduce another weak topological form of equicon-tinuity that we call diam-mean equicontinuity (stronger than mean equicontinuity). We show thatfor minimal subshifts, nullness implies diam-mean equicontinuity (Corollary 2.2.36) and that analmost automorphic subshift is diam-mean equicontinuous if and only if it is regular (Theorem2.2.24).In conclusion for minimal subshifts we have the following implications:Top. discrete spectrum = equicontinuity=⇒ nullness=⇒ diam-mean equicontinuity=⇒ mean equicontinuity=⇒ µ−mean equicontinuity (for every ergodic measure µ)= every ergodic measure has discrete spectrum2.2.1 Mean equicontinuity and mean sensitivityDefinition 2.2.1. We define ∆δ(x, y) :={i ∈ G : d(T ix, T iy) > δ}and the Besicovitch pseu-dometric as db(x, y) := inf{δ > 0 : D(∆δ(x, y)) < δ}. By identifying points that are at pseudo-14distance zero we obtain a metric space (X/db, db) that will be called the Besicovitch space. Theprojection fb : (X, d)→ (X/db, db) will be called the Besicovitch projection. The ε−closed ballsof the Besicovitch pseudometric will be denoted by Bbε(x).One can check that in fact this is a pseudometric using that D(S) +D(S′) ≥ D(S ∪ S′).Definition 2.2.2. Let (X,T ) be a TDS. We say x ∈ X is a mean equicontinuity point if forevery ε > 0 there exists δ > 0 such that if y ∈ Bδ(x) thendb(x, y) < ε(equivalently D(i ∈ G : d(T ix, T iy) ≤ ε) ≥ 1 − ε). We say (X,T ) is mean equicontinuous (ormean-L-stable) if every x ∈ X is a mean equicontinuity point. We say (X,T ) is almost meanequicontinuous if the set of mean equicontinuity points is residual.Mean equicontinuous systems were introduced by Fomin [17]. They have been studied in [2],[52], [55] and [45].In [7] equicontinuity with respect to the Besicovitch pseudometric (of the shift) was studied forcellular automata; this is a different property than mean equicontinuity.If x ∈ X is a mean equicontinuity point then fb is continuous at x. This implies the Besicovitchprojection is continuous if and only if (X,T ) is mean equicontinuous.Remark 2.2.3. If (X,T ) is mean equicontinuous then fb is continuous and hence fb is uniformlycontinuous; this means that (X,T ) is uniformly mean equicontinuous i.e. for every ε > 0 thereexists δ > 0 such that if d(x, y) ≤ δ then D(i ∈ G : d(T ix, T iy) ≤ ε) ≥ 1− ε.Lemma 2.2.4. Let (X,T ) be a TDS. We have db is equivalent to the following pseudometricd′b(x, y) := lim supn→∞1|Fn|∑i∈Fnd(T ix, T iy).Proof. Without loss of generality assume the diameter of X is bounded by 1.⇐Let ε > 0. Assume that δ < ε/2 and thatD{j ∈ G : d(T ix, T iy) > δ}< δ.15This implies thatlim sup1|Fn|∑j∈Fnd(T ix, T iy)≤ D{j ∈ G : d(T ix, T iy) > δ}+D{j ∈ G : d(T ix, T iy) ≤ δ}δ≤ 2δ< ε.⇒Let ε > 0. Assume that δ ≤ ε2 and thatlim sup1|Fn|∑j∈Fnd(T ix, T iy) < δ.Suppose thatD{j ∈ G : d(T ix, T iy) > ε}≥ ε.We have thatlim sup1|Fn|∑i∈Fnd(T ix, T iy)≥ εD{i ∈ G : d(T ix, T iy) > ε}≥ ε2≥ δ.A contradiction, henceD{j ∈ G : d(T ix, T iy) > ε}< ε.It is well known that transitive equicontinuous systems are minimal. We give a similar resultby weakening one hypothesis and strengthening the other.Definition 2.2.5. A TDS (X,T ) is strongly transitive if for every open set U there exists atransitive point x ∈ U that returns to U with positive lower density.Theorem 2.2.6. Every strongly transitive mean equicontinuous system is minimal.Proof. Let x, y ∈ X and ε > 0. Since the system is strongly transitive there exists a transitive pointz ∈ Bε/2(y) such that a := D{i : T iz ∈ Bε/2(y)}> 0. Since the system is mean equicontinuousthere exists δ > 0 such that if w ∈ Bδ(x) then db(x,w) ≤ min {ε/2, a} . There exists t1 ∈ G such16that T t1z ∈ Bδ(x). By Lemma 2.1.2 (third bullet) there exists t2 ∈ G such that T t2z ∈ Bε/2(y) andd(T t2x, T t2z) ≤ ε/2; thus T t2x ∈ Bε(y). This means the system is minimal.A similar result is known for null systems ( Definition 2.2.33), i.e. every Banach transitive nullsystem is minimal [34]. It is an open question whether every transitive recurrent null system isminimal [34][26].Definition 2.2.7. We denote the set of mean equicontinuity points by Em and we defineEmε :={x ∈ X : ∃δ > 0 ∀y, z ∈ Bδ(x), D{i ∈ G : d(T iy, T iz) ≤ ε}≥ 1− ε}.Note that Em = ∩ε>0Emε .Lemma 2.2.8. Let (X,T ) be a TDS. The sets Em, Emε are inversely invariant (i.e. T−j(Em) ⊆Em, T−j(Emε ) ⊆ Emε for all j ∈ G) and Emε is open.Proof. Let j ∈ G, ε > 0, and x ∈ T−jEmε . There exists η > 0 such that if d(Tjx, z) ≤ η thenD{i : d(T i+jx, T iz) ≤ ε}≥ 1− ε. There exists δ > 0 such that if d(x, y) < δ then d(T jx, T jy) < η(and hence D{i : d(T i+jx, T i+jy) ≤ ε}≥ 1 − ε). We conclude that x ∈ Emε . This implies Em isalso inversely invariant.Let x ∈ Emε and δ > 0 be a constant that satisfies the property of the definition of Emε . Ifd(x,w) < δ/2 then w ∈ Emε . Indeed if y, z ∈ Bδ/2(w) then y, z ∈ Bδ(x).Definition 2.2.9. A TDS (X,T ) is mean sensitive if there exists ε > 0 such that for every openset U there exist x, y ∈ U such thatD(i ∈ G : d(T ix, T iy) > ε) > ε.Other strong forms of sensitivity have been studied in [50] where S was taken to be cofinite(complement is finite) or syndetic (bounded gaps).Definition 2.2.10. Let (X,T ) be a TDS. We say x ∈ X is a transitive point if{T ix : i ∈ G}isdense. We say (X,T ) is transitive if X contains a transitive point. If every x ∈ X is transitivethen we say the system is minimal.It is not hard to see that mean sensitive systems have no mean equicontinuity points, as amatter of fact we have the following dichotomies.Theorem 2.2.11. A transitive system is either almost mean equicontinuous or mean sensitive. Aminimal system is either mean equicontinuous or mean sensitive.17Proof. First, we show that if (X,T ) is a transitive system then Emε is either empty or dense. AssumeEmε is non-empty and not dense. Then U = XEmε is a non-empty open set. Since the system istransitive and Emε is open (Lemma 2.2.8) there exists t ∈ G such that U ∩ T−t(Emε ) is non empty.By Lemma 2.2.8 we have that U ∩ T−t(Emε ) ⊂ U ∩ Emε = ∅; a contradiction.If Emε is non-empty for every ε > 0 then we have that Em = ∩n≥1Em1/n is a residual set; hence thesystem is almost mean equicontinuous.If there exists ε > 0 such that Emε is empty, then for any open ball U = Bδ(x) there exist y, z ∈Bδ(x) such that D{i ∈ G : d(T iy, T iz) ≤ ε}≤ 1−ε; this means that D{i ∈ G : d(T iy, T iz) > ε}>ε. It follows that (X,T ) is mean sensitive.Now suppose (X,T ) is minimal and almost mean equicontinuous. For every x ∈ X and everyε > 0 there exists t ∈ G such that T tx ∈ Emε . Since Emε is inversely invariant x ∈ Emε and hencex ∈ Em.An analogous result appeared in [45].2.2.2 Diam-mean equicontinuity and diam-mean sensitivityDefinition 2.2.12. Let A ⊂ X. We denote the diameter of A as diam(A).Definition 2.2.13. Let (X,T ) be a TDS. We say x ∈ X is a diam-mean equicontinuity pointif for every ε > 0 there exists δ > 0 such thatD{i ∈ G : diam(T iBδ(x)) > ε}< ε.We say (X,T ) is diam-mean equicontinuous if every x ∈ X is a diam-mean equicontinuitypoint. We say (X,T ) is almost diam-mean equicontinuous if the set of diam-mean equiconti-nuity points is residual.Remark 2.2.14. Equivalently x ∈ X is a diam-mean equicontinuity point if for every ε > 0 thereexists δ > 0 such that D{i ∈ G : diam(T iBδ(x)) ≤ ε}≥ 1− ε.By adapting the proof that a continuous function on a compact space is uniformly contin-uous one can show a TDS is diam-mean equicontinuous if and only if it is uniformly diam-meanequicontinuous i.e. for every ε > 0 there exists δ > 0 such that D{i ∈ G : diam(T iBδ(x)) > ε}< ε.Definition 2.2.15. We denote the set of diam-mean equicontinuity points by Ew and we defineEwε :={x ∈ X : ∃δ > 0, s.t. D{i ∈ G : diam(T iBδ(x)) ≤ ε}≥ 1− ε}.Note that Ew = ∩ε>0Ewε .Lemma 2.2.16. Let (X,T ) be a TDS. The sets Ew and Ewε are inversely invariant (i.e. T−j(Ew) ⊆Ew and T−j(Ewε ) ⊆ Ewε for all j ∈ G) and Ewε is open.18Proof. Let ε > 0 and T jx ∈ Ewε . There exists η > 0 and S ⊂ G such that D(S) ≥ 1 − ε andd(T i+jx, T iz) ≤ ε for every i ∈ S and z ∈ Bη(T jx). There exists δ > 0 such that if d(x, y) < δ thend(T jx, T jy) < η. We conclude that x ∈ Ewε . Thus T−1(Ew) ⊆ Ew.Let x ∈ Ewε and take δ > 0 a constant that satisfies the property of the definition of Ewε . Ifd(x,w) < δ/2 then w ∈ Ewε . Indeed if y, z ∈ Bδ/2(w) then y, z ∈ Bδ(x).Definition 2.2.17. A TDS (X,T ) is diam-mean sensitive if there exists ε > 0 such that forevery open set U we haveD{i ∈ G : diam(T iU) > ε}> ε.The proof of the following result is analogous to that of Theorem 2.2.11 (using Ewε instead ofEmε ).Theorem 2.2.18. A transitive system is either almost diam-mean equicontinuous or diam-meansensitive. A minimal system is either diam-mean equicontinuous or diam-mean sensitive.2.2.3 Almost automorphic systemsLet (X1, T1) and (X2, T2) be two TDSs (over the same group) and f : X1 → X2 a continuousfunction such that f ◦ T1 = T2 ◦ f.If f is surjective we say f is a factor map and (X2, T2) is a factor of (X1, T1). If f is bijectivewe say f is a conjugacy and (X1, T1) and (X2, T2) are conjugate (topologically).Any TDS (X,T ) has a unique (up to conjugacy) maximal equicontinuous factor i.e. anequicontinuous factor feq : (X,T )→ (Xeq, Teq) such that if f2 : (X,T )→ (X2, T2) is a factor mapsuch that (X2, T2) is equicontinuous then there exists a unique factor map g : (Xeq, Teq)→ (X2, T2)such that g ◦ feq = f2. The equivalence relation whose equivalence classes are the fibers of feq iscalled the equicontinuous structure relation. This relation can be characterized using theregionally proximal relation. We say that x, y ∈ X are regionally proximal if there exist sequences{tn}∞n=1 ⊂ G and {xn}∞n=1 , {yn}∞n=1 ⊂ X such thatlimn→∞xn = x, limn→∞yn = y, and limn→∞d(T tnxn, T tnyn) = 0.The equicontinuous structure relation is the smallest closed equivalence relation containing theregional proximal relation ([3] Chapter 9).For mean equicontinuous systems we can characterize the maximal equicontinuous factor mapusing the Besicovitch pseudometric (Definition 2.2.1).Proposition 2.2.19. Let (X,T ) be mean equicontinuous. Then feq = fb.Proof. We have that fb is continuous and (X/db, db, T ) is equicontinuous so if feq(x) = feq(y)then fb(x) = fb(y). If fb(x) = fb(y), then db(x, y) = 0 so there exists a sequence {tn} such thatlimn→∞ d(T tnx, T tny) = 0; hence x and y are regionally proximal. We conclude feq(x) = feq(y).19A transitive equicontinuous system is conjugate to a system where G acts as a translation on acompact metric abelian group. If (X,T ) is a transitive TDS we denote the maximal equicontinuousfactor by Geq (since it is a group). The TDS (Geq, Teq) has a unique ergodic invariant probabilitymeasure, the normalized Haar measure on Geq; this measure has full support and will be denotedby νeq.Definition 2.2.20. We say a TDS is almost automorphic if it is an almost 1-1 extension ofits maximal equicontinuous factor i.e. if f−1eq feq(x) = {x} on a residual set. A transitive almostautomorphic TDS is regular ifνeq{g ∈ Geq : f−1eq (g) is a singleton}= 1.It is not difficult to see that if there exists a transitive point x ∈ X such that f−1eq feq(x) = {x}then (X,T ) is almost automorphic. Transitive almost automorphic systems are minimal.Two well known families of almost automorphic systems are the Sturmian subshifts (maximalequicontinuous factor is an irrational circle rotation) and Toeplitz subshifts (maximal equicontin-uous factor is an odometer, see Section 4.4 for definition and examples).An important class of TDS are the shift systems. Let A be a compact metric space (with metricdA). For x ∈ AG and i ∈ G we use xi to denote the ith coordinate of x andσ :={σi : AG → AG | xi+j = (σix)j for all x ∈ AGand j ∈ G}to denote the shift maps. Using the (Cantor) product topology generated by the topology of A,we have that AG is a compact metrizable space. A subset X ⊂ AG is a general shift system if itis closed and σ−invariant; in this case (X,σ) is a TDS. Every TDS is conjugate to a general shiftsystem (by mapping every point to its orbit).A general shift system is a subshift if A is finite.Remark 2.2.21. Let X ⊂ AG be a general shift system. We have that (X,σ) is diam-meanequicontinuous if and only if for all ε > 0 there exists δ > 0 such that for all x ∈ X there existsS ⊂ G with D(S) ≥ 1− ε, such that if d(x, y) ≤ δ then dA(xi, yi) ≤ ε for all i ∈ S.Definition 2.2.22. Let (X,σ) be a transitive general shift system. We defineD := {g ∈ Geq : ∃x, y such that feq(x) = feq(y) = g and x0 6= y0} ,where x0 and y0 represent the 0th coordinates of x and y respectively.If (X,σ) is almost automorphic we have that D is first category. If A is finite then it is closedand nowhere dense. Also note that (X,σ) is regular if and only if νeq(D) = 0.20Lemma 2.2.23. Let (X,σ) be a transitive almost automorphic subshift and F ⊂ G a finite set.Then for all g ∈ Geq, there exists k ∈ G such that T k+ieq (g) /∈ D for all i ∈ F .Proof. Let D′ := ∪i∈FT−ieq (D). This means that D′ is also closed and nowhere dense. Thus thereexists k such that T keq(g) ∈ Geq −D′, hence T k+ieq (g) /∈ D for all i ∈ F.Theorem 2.2.24. Let (X,σ) be a transitive almost automorphic subshift. Then (X,σ) is regularif and only if it is diam-mean equicontinuous.Proof. Suppose that νeq(D) = 0. Hence for every ε > 0 there exists δ > 0 such that if U :={x | d(x, f−1eq (D)) ≤ δ}then ν(U) < ε. Let x ∈ X. We define S := {i ∈ G | xi /∈ U} ; henceD(S) = 1 − ν(U) ≥ 1 − ε (using the pointwise ergodic theorem). We conclude (X,σ) is diam-mean equicontinuous because if d(x, y) ≤ δ then dA(xi, yi) ≤ ε for all i ∈ S.Now suppose (X,σ) is diam-mean equicontinuous. We have that fb = feq(Proposition 2.2.19).Let ε > 0 and δ > 0 given by Remark 2.2.21.Let x ∈ X. Note that σi(x) ∈ f−1eq (D) if and only if there exists y ∈ f−1b ◦fb(x) such that yi 6= xi.Using Lemma 2.2.23 there exists k ∈ G such that if feq(x) = feq(y) then d(σky, σkx) ≤ δ.Since (X,σ) is a diam-mean equicontinuous there exists S ⊂ G, with D(S) ≥ 1 − ε, such thatif d(σkx, σky) ≤ δ then d(xi, yi) ≤ ε for all i ∈ S.This means that for every ε > 0D{i : ∃y ∈ f−1b ◦ fb(x) s.t. dA(xi, yi) > ε}≤ ε.Using thatD = ∪n∈N {g ∈ Geq : ∃x, y s.t feq(x) = feq(y) = g and dA(x0, y0) > 1/n} ,and the pointwise ergodic theorem we conclude that νeq(D) = 0.Definition 2.2.25. A TDS is diam-mean equicontinuous if for every ε > 0 there exists δ > 0such that for every x ∈ X we have D{i ∈ G : diam(T iBδ(x)) > ε}< ε.It is clear that every diam-mean equicontinuous system is diam-mean equicontinuous. We donot know if the converse is true in general. We will show that under some conditions they areequivalent.Definition 2.2.26. A TDS (X,T ) is diam-mean sensitive if there exists ε > 0 such that forevery open set U we have D{i ∈ G : diam(T iU)) > ε}> ε.The proof of the following theorem is very similar to the proof of Theorem 2.2.11.21Proposition 2.2.27. A minimal TDS is either diam-mean equicontinuous or diam-mean sensitive.Definition 2.2.28. Let (X,T ) be a TDS and x ∈ X. We denote the orbit of x with oT (x).Proposition 2.2.29. Let (X,σ) be a transitive almost automorphic subshift . The function h :Dc → A defined as h(g) = (f−1eq (g))0 is continuous and there exists y ∈ X such that·y is transitive.·f−1eq feq(y) is a singleton, in other words oσeq(y) ∩ D = ∅.Proof. This follows from Theorem 6.4 in [14].Lemma 2.2.30. Let (X,σ) be a transitive almost automorphic subshift and w = a0...an−1 ∈ Anand U = {x : x0...xn−1 = w} ⊂ X a non-empty set . There exists p ∈ U such that p′ = feq(p) isgeneric for D (with respect to νeq) and σieqp′ ∈ Dc for i = 0, ..., n− 1.Proof. Let Ua ={g ∈ Xeq : g /∈ D, (f−1eq (g))0 = a}and h : Dc → A the continuous function fromProposition 2.2.29. This implies that for every a, Ua is an open set. Hence ∩n−1i=0 σ−ieq Uai is an openset.Let y ∈ X be the point given by Proposition 2.2.29. Since U is non-empty and y transitive,there exists z ∈ oσeq(y)∩U. Considering that oσeq(y)∩D = ∅ we obtain feq(z) ∈ ∩n−1i=0 σ−ieq Uai . Thus∩n−1i=0 σ−ieq Uai is a non-empty open set. Since νeq is fully supported it contains a generic point forD.Proposition 2.2.31. Let (X,σ) be a minimal almost automorphic subshift. If (X,σ) is not regularthen it is diam-mean sensitive.Proof. Assume (X,σ) is not regular. This means that νeq(D) > 0.Let w ∈ An and U = {x : x0...xn−1 = w} ⊂ X non-empty (these sets form a base of thetopology). Let p ∈ U be the point given by the previous lemma. Let S :={i ∈ G : σieqp′ ∈ D}.Since p′ is generic for D we have that D(S) = νeq(D). Furthermore for every i ∈ S there existsq ∈ X such that feq(p) = feq(q) and pi 6= qi. Since σjeqp′ ∈ Dc for j = 0, ..., n − 1 we have thatq ∈ U. Hence (X,σ) is diam-mean sensitive.Corollary 2.2.32. Let (X,σ) be a minimal almost automorphic subshift. The following conditionsare equivalent:1)(X,σ) is diam-mean equicontinuous .2)(X,σ) is not diam-mean sensitive.3)(X,σ) is regular.4)(X,σ) is diam-mean equicontinuous .5)(X,σ) is not diam-mean sensitive.22Proof. Apply Theorem 2.2.18 to get 1)⇔ 2).Apply Theorem 2.2.24 to get 2)⇔ 3).By definition 1)⇒ 4).Proposition 2.2.31 implies 5)⇒ 3).Proposition 2.2.27 implies 4)⇔ 5).2.2.4 Topological sequence entropyLet U and V be two open covers of X. We define U ∨ V := {U ∩ V : U ∈ U , V ∈ V} and N(U) asthe minimum cardinality of a subcover of U .Definition 2.2.33 ([27]). Let (X,T ) be a TDS, S = {sm}∞m=1 ⊂ G, and U an open cover. WedefinehStop(T,U) := limn→∞sup1nlogN(∨nm=1T−sm(U)).The topological entropy along the sequence S is defined byhStop(T ) := supopen covers UhStop(T,U)A TDS is null if the topological entropy along every sequence is zero.Lemma 2.2.34. Let K be a finite set, ε > 0, and h : K → 2G such that D(h(k)) > ε for everyk ∈ F. There exist K ′ ⊂ K and i ∈ G with |K ′| ≥ ε |K| /2 such that i ∈ h(k) for every k ∈ K ′.Proof. There exists n0 ∈ N such that|h(k) ∩ Fn0 ||Fn0 |≥ ε/2 for every k ∈ K.This means that∑j∈Fn0|k ∈ K : j ⊂ h(k)| =∑k∈K|j ∈ Fn0 : j ∈ h(k)| ≥ε2|K| |Fn0 | .Hence there exists j ∈ Fn0 such that |k ∈ K : j ⊂ h(k)| ≥ε2 |K| .Theorem 2.2.35. Let (X,T ) be a TDS. If (X,T ) is diam-mean sensitive then there exists S ⊂ Gsuch that hStop(T ) >0.23Proof. Let (X,T ) be diam-mean sensitive with sensitive constant ε. Let U := {U1, ...UN} be a finiteopen cover with balls with diameter smaller than ε/2.We will define the sequence Sn = {s1, ..., sn} inductively with s1 = 1. For every n ∈ N wedefine Ln :={∩ni=1T−siUvi 6= ∅ : v ∈ {1, ..., N}Sn}(N is the size of the cover). We denote byL′n := {Ank}k≤N(Ln) a subcover of Ln of minimal cardinality. We define the function f : L′n → 2Gas follows; m ∈ f(Ak) if and only if there exists x, y ∈ Ak ∪j<k Aj such that d(Tmx, Tmy) > ε.Assume Sn is defined. Since (X,T ) is diam-mean sensitive we have that D(f(U)) > ε forevery U ∈ L′n. By Lemma 2.2.34 there exists g ∈ G such that|{U∈L′n:g∈f(U)}|N(Ln)> ε/2; we definesn+1 := g. The definition of f implies thatN(Ln+1)N(Ln)> 1 + ε/2. Let S∞ := ∪n∈NSn. Since N(Ln) =N(∨i∈SnT−si(U)), we conclude that hS∞top (T,U) > 0.Corollary 2.2.36. Let (X,T ) be a minimal TDS. If (X,T ) is null then it is diam-mean equicon-tinuous.Proof. Apply Theorem 2.2.35 and Proposition 2.2.27.Every minimal null TDS is almost automorphic [34][33]. Using the previous corollary andCorollary 2.2.32 we obtain a stronger result for subshifts.Corollary 2.2.37. Let (X,σ) be a minimal subshift. If (X,σ) is null then it is a regular almostautomorphic subshift and hence diam-mean equicontinuous.The converse of this result is not true (Example 2.2.44).We will see that an ergodic TDS is µ−null if and only if it is µ−mean equicontinuous. If (X,T )is mean equicontinuous and µ is an ergodic measure then (X,T ) is µ−mean equicontinuous, andhence it has zero entropy. This implies that mean equicontinuous and diam-mean equicontinuoussystems have zero topological entropy.Surprisingly it was shown in [45] that transitive almost mean equicontinuous subshifts can havepositive entropy.Definition 2.2.38. Let (X,T ) be a TDS and µ an invariant measure. We say (X,T ) is meandistal if db(x, y) > 0 for every x 6= y ∈ X, and we say (X,T ) is µ−tight if there exists X ′ suchthat µ(X ′) = 1 and db(x, y) > 0 for every x 6= y ∈ X ′.Mean distal systems were studied by Ornstein-Weiss in [51]. They showed that any tight measurepreserving Z−TDS has zero entropy (assuming the system has finite entropy). A Z+−TDS haszero topological entropy if and only if it is the factor of a mean distal TDS [15][51].Let (X,T ) be a mean equicontinuous TDS. Since feq = fb we have that if (X,T ) is mean distalthen feq is 1-1 hence (X,T ) is equicontinuous. So mean equicontinuity and mean distality are bothconsidered rigid properties, and a TDS satisfies both properties if and only if it is equicontinuous.24Proposition 2.2.39. A mean equicontinuous TDS is mean distal if and only if it is equicontinuous.A measure theoretic version of mean distality was also defined. Let (X,T ) be a TDS and µ anergodic measure. The system is µ−tight if there exists X ′ such that µ(X ′) = 1 and db(x, y) > 0 forevery x 6= y ∈ X ′. Let (X,T ) be a transitive almost automorphic diam-mean equicontinuous TDSand µ an invariant measure. Using Theorem 2.2.24 it is not hard to see that (X,T ) is µ−tight.2.2.5 Counter-examplesThe following example shows there are mean equicontinuous not diam-mean equicontinuous TDS.We do not have a transitive counterexample.Example 2.2.40. Let X ⊂ {0, 1}Z+ be the subshift consisting of sequences that contain at mostone 1. For every x, y ∈ X, db(x, y) = 0, so (X,σ) is mean equicontinuous. Nonetheless, for everyε > 0, D{i ∈ Z+ : ∃x ∈ Bε(0∞) s.t. xi = 1} = 1 so 0∞ is not a diam-mean equicontinuity point.In [40] the relationship between independence and entropy was studied. We will make use oftheir characterization of null systems.Definition 2.2.41. Let (X,T ) be a TDS, and A1, A2 ⊂ X. We say S ⊂ G is an independenceset for (A1, A2) if for every non-empty finite subset F ⊂ S we have∩i∈FT−iAv(i) 6= ∅for any v ∈ {1, 2}F .Theorem 2.2.42 ([40]). Let (X,T ) be TDS. The system (X,T ) is not null if and only if thereexists x, y ∈ X (with x 6= y), such that for all neighbourhoods Ux of x and Uy of y there exists anarbitrarily large finite independence set for (Ux, Uy) .The following example shows there are transitive non-null mean equicontinuous systems.Example 2.2.43. Let S = {2n}∞n=1,Y :={x ∈ {0, 1}Z+ : xi = 0 if i /∈ S},and X the shift-closure of Y. For every x, y ∈ X we have db(x, y) = 0, hence (X,σ) is meanequicontinuous. Nonetheless, since S is an infinite independence set for ({x0 = 0} , {x0 = 1}) weconclude that (X,σ) is not null.A Z+−subshift is Toeplitz if and only if it is the orbit closure of a regularly recurrentpoint, i.e. x ∈ X such that for every j > 0 there exists m > 0 such that xj = xj+im for all i ∈ Z+.Toeplitz subshifts are precisely the minimal subshifts that are almost 1-1 extensions of odometers(for Z+−actions see [14], for finitely generated discrete group actions see [11]).25Given a Toeplitz subshift and a regularly recurrent point x ∈ X, there exists a set of pairwisedisjoint arithmetic progressions {Sn}n∈N (called the periodic structure) such that ∪n∈NSn = Z+,xi is constant for every i ∈ Sn, and every Sn is maximal in the sense that there is no larger arithmeticprogression where xi is constant. Let x be a regularly recurrent point and X the orbit closure. Wehave that (X,σ) is regular if and only if∑n∈ND(Sn) = 1 (see [14]).Example 2.2.44. There exists a regular Toeplitz subshift (hence diam-mean equicontinuous) withpositive sequence entropy.Proof. For every n ∈ N let wn be a finite word that contains all binary words of size n. We denotethe concatenation of wn by wn,∞ ∈ {0, 1}Z+ .We define the sequence {jn} ⊂ N inductively with j1 = 0 and jn+1 := min {∪m≤n ∪k∈N {k2m + jm}}c .Let x ∈ {0, 1}Z+ be the point such that for every n ∈ N we have that xjn+i2n = wn,∞i for alli ∈ Z+.We define X as the orbit closure of x. Since x is regularly recurrent we obtain that X is aToeplitz subshift (hence almost automorphic). By using the condition for regularity using theperiodic structure (see comment before this proposition) we obtain that (X,σ) is regular. Henceby Theorem 2.2.24 we get that (X,σ) is diam-mean equicontinuous.On the other hand wk contains all the binary words of size k. This implies there exists arbitrarilylong independence sets for ({x0 = 0} , {x0 = 1}) . Using Theorem 2.2.42 we conclude (X,σ) is notnull.Another class of rigid TDS are the tame systems introduced in [42][25]. These systems werecharacterized in [40] similarly to Theorem 2.2.42 but with infinitely large independence sets. Thismeans that Example 2.2.43 is also not tame (note it is not minimal). The example in Section 11 in[40] is a tame non-regular Toeplitz subshift; this means there are tame minimal systems that arenot diam-mean equicontinuous.2.3 Topological and measure theoreticalWe remind the reader that given a metric space X and a Borel probability measure µ we denotethe Borel sets with BX and the Borel sets with positive measure with B+X .2.3.1 µ−Mean equicontinuityDefinition 2.3.1. Let (X,T ) be a TDS and µ a Borel probability measure. We say (X,T ) is µ−equicontinuous if for every τ > 0 there exists a compact set M ⊂ X with µ(M) ≥ 1− τ such thatfor every ε > 0 there exists δ > 0 such that whenever x, y ∈M and d(x, y) ≤ δ thend(T ix, T iy) ≤ ε ∀i ∈ G.26This definition of µ−equicontinuity was introduced in [35] and it was shown every ergodicµ−equicontinuous TDS has discrete spectrum. We introduce a weaker notion and characterizediscrete spectrum.If (X,T ) is an equicontinuous TDS and µ a Borel probability measure then (X,T ) is µ−equicontinuous.We will see that there exists a sensitive TDS on {0, 1}Z such that (X,T ) is µ−equicontinuous forevery ergodic Markov chain µ (Example 3.1.29).Definition 2.3.2. Let (X,T ) be a TDS and µ a Borel probability measure. We say (X,T ) isµ−mean equicontinuous if for every τ > 0 there exists a compact set M ⊂ X with µ(M) ≥ 1−τsuch that for every ε > 0 there exists δ > 0 such that whenever x, y ∈M and d(x, y) ≤ δ thendb(x, y) ≤ ε.Remark 2.3.3. Sincedb(x, y) = inf{δ > 0 : lim supn→∞∣∣{i ∈ G | d(T ix, T iy) > δ}∩ Fn∣∣|Fn|< δ},and µ is Borel, db(x, y) is a Borel function. This implies that for every ε > 0 and every x ∈ X wehave that Bbε(x) is µ−measurable.Measure theoretic forms of sensitivity for TDS have been studied in [24],[9], and [35]. Inparticular in [35] it was shown that ergodic TDS are either µ−equicontinuous or µ−sensitive.Definition 2.3.4. A TDS (X,T ) is µ−mean sensitive if there exists ε > 0 such that for everyA ∈ B+X there exists x, y ∈ A such that db(x, y) > ε (and hence D{i ∈ G : d(T ix, T iy) > ε}> ε).In this case we say ε is a µ−mean sensitivity constant.Definition 2.3.5. A TDS (X,T ) is µ−mean expansive if there exists ε > 0 such that µ ×µ {(x, y) : db(x, y) > ε} = 1.The following fact is well known. We give a proof for completeness.Lemma 2.3.6. Let (Y, dY ) be a metric space. Suppose that there is no uncountable set A ⊂ Y andε > 0 such that dY (x, y) > ε for every x, y ∈ A with x 6= y, then (Y, dY ) is separable.Proof. For every ε > 0 rational we define:Fε := {A ⊂ Y : dY (x, y) > ε ∀x 6= y ∈ A} .Using Zorn’s lemma we obtain that Fε admits a maximal element Mε, which by hypothesismust be countable. Then M := ∪ε∈Q+Mε is also countable. We have that for every x ∈ X andε > 0 there exists y ∈M such that dY (x, y) ≤ ε.27Theorem 2.3.7. Let (X,µ, T ) be an ergodic TDS. The following are equivalent:1) (X,T ) is µ−mean sensitive.2) (X,T ) is µ−mean expansive.3) There exists ε > 0 such that for almost every x, µ(Bbε(x)) = 0.4)(X,T ) is not µ−mean equicontinuous.Proof. 1)⇔ 2)⇔ 3)The proof is similar to that of Theorem 2.1.20 but we use db instead of df .2)⇒ 4)If (X,T ) is µ−mean expansive then there exists ε > 0 such that µ×µ {(x, y) : db(x, y) > ε} = 1.Suppose (X,T ) is µ−mean equicontinuous. This implies there exists a compact set M such thatµ(M) > 0 and fb|M is continuous (and hence uniformly continuous). This implies there existsδ > 0 such that if x, y ∈ M and d(x, y) ≤ δ then db(x, y) ≤ ε/2. We can cover M with finitelymany δ/2−balls, this implies one of them must have positive measure. Using this and µ−meanexpansiveness we would obtain that there exists p, q ∈ M such that d(p, q) ≤ δ and db(p, q) > ε; acontradiction to the continuity of fb|M .4)⇒ 3)Suppose 3) is not satisfied. Note that Lemma 2.1.19 using db instead of df can be provedanalogously. Using this we have that for every n ∈ N there exists a set of full measure Yn andan > 0 such that µ(Bb1/n(x)) = an for all x ∈ Yn. Let Y := ∩n∈NYn. If (Ydb, db) is not separablethen by Lemma 2.3.6 there exists ε > 0 and an uncountable set A such that for every x, y ∈ Asuch that x 6= y we have that Bbε(x) ∩ Bbε(y) = ∅. This is a contradiction, hence (Ydb, db) isseparable. Using Proposition 2.3.31 (which is proved later in the paper) we conclude (X,T ) isµ−mean equicontinuous.Definition 2.3.8. Let (X,µ, T ) and (X ′, µ′, T ′) be two MPS. We say (X ′, µ′, T ′) is isomorphic to(X,µ, T ) if there exist a bijective measure preserving function f : (X,µ)→ (X ′, µ′) such that f−1 ismeasure preserving and T ◦ f = f ◦ T ′.Proposition 2.3.9. Let (X,µ, T ) and (X ′, µ′, T ′) be two ergodic TDSs. If (X,µ, T ) is µ−meanequicontinuous and (X ′, µ′, T ′) isomorphic to (X,µ, T ) then (X ′, µ′, T ′) is µ−mean equicontinuous.Proof. We will denote by d and d′ the metrics of X and X ′ respectively.Suppose (X ′, µ′, T ′) is not µ−mean equicontinuous. By the previous Theorem we have that(X ′, µ′, T ′) is µ−mean expansive, i.e. there exists ε > 0 and a set Y ′ ⊂ X ′ ×X ′such that for every(x′, y′) ∈ Y ′ we have that db(x′, y′) > ε and µ× µ(Y ′) = 1.Let f : X → X ′ be the measure preserving isomorphism. By Lusin’s Theorem we knowthat there exists a compact set K ⊂ X such that µ(K) ≥ 1 − ε/4 and f |K is continuous andbijective. This implies there exists ε1 > 0 such that if f(x), f(y) ∈ f(K) and d′(f(x), f(y)) > ε28then d(x, y) > ε1. This implies D{i ∈ G : d′(T′if(x), T′if(y) > ε}> ε. We defineE1(x, y) :={i ∈ G : T ix, T iy ∈ K}, andE2(x, y) :={i ∈ G : d(T ix, T iy) > ε1}.Using that µ(K) ≥ 1 − ε/4 and the ergodic theorem we have that for almost every x, y ∈ X wehave that (f(x), f(y)) ∈ Y ′ and D(E1(x, y)) ≥ 1− ε/2. Since{i ∈ G : d′(T′if(x), T′if(y) > ε}⊂E2(x, y) we have thatD(E2(x, y)) ≥ ε. This implies that for a.e. x, y ∈ X we have that d′(T ix, T iy) >ε1 for every i ∈ E1(x, y)∩E2(x, y), and thatD(E1(x, y)∩E2(x, y)) ≥ ε/2. Hence (X,µ, T ) is µ−meanexpansive.2.3.2 µ−Mean sensitive pairsThe notion of entropy pairs was introduced in [6]. Different kinds of pairs have been studied, inparticular sequence entropy pairs in [36] and µ−sensitive pairs [35]. In [35] µ−sensitive pairs wereused to characterize µ−sensitivity; we introduce µ−mean sensitive pairs to characterize µ−meansensitivity.Definition 2.3.10 ([36]). Let (X,µ, T ) be a measure preserving TDS. We say (x, y) is a µ−sequenceentropy pair if for any finite partition P, such that there is no P ∈ P such that x, y ∈ P , thereexists S ⊂ G such that hSµ(P, T ) > 0.The following result was proven for Z+−actions in ([36] Theorem 4.3), and was proved for moregeneral actions in [41].Theorem 2.3.11. An ergodic TDS (X,µ, T ) is µ−null if and only if there are no µ−sequenceentropy pairs.Definition 2.3.12. We say (x, y) ∈ X2 is a µ−mean sensitive pair if x 6= y and for all openneighbourhoods Ux of x and Uy of y, there exists ε > 0 such that for every A ∈ B+X there existp, q ∈ A with D(i ∈ G : T ip ∈ Ux and T iq ∈ Uy) > ε. We denote the set of µ−mean sensitive pairsas Smµ (X,T ).Theorem 2.3.13. Let (X,µ, T ) be an ergodic TDS. Then Smµ (X,T ) 6= ∅ if and only if (X,T ) isµ−mean sensitive.Proof. If (x, y) ∈ Smµ (X,T ) then there exists open neighbourhoods Ux of x and Uy of y (withd(Ux, Uy) > 0) and ε > 0 such that for every A ∈ B+X there exist p, q ∈ A and S ⊂ G with D(S) > εsuch that T ip ∈ Ux and T iq ∈ Uy for every i ∈ S. This implies thatD{i ∈ G : d(T ip, T iq) ≥ d(Ux, Uy)}>ε. Thus (X,T ) is µ−mean sensitive.Let (X,T ) be µ−mean sensitive with sensitive constant ε0 and 0 < ε < ε0.29For ε > 0 we define the compact setXε :={(x, y) ∈ X2 | d(x, y) ≥ ε}.Suppose that Smµ (X,T ) = ∅. In particular this implies that for every (x, y) ∈ Xε there exist openneighbourhoods of x and y, Ux and Uy, such that for every δ > 0 there exists a set Aδ(x, y) ∈ B+Xsuch thatD{i ∈ G : (T ip, T iq) ∈ Ux × Uy}≤ δfor all p, q ∈ Aδ(x, y).There exists a finite set of points F ⊂ Xε such thatXε ⊆ ∪(x,y)∈FUx × Uy.Let δ = ε/ |F | . Since µ is ergodic for every (x, y) ∈ F there exists n(x, y) ∈ G such that A :=∩(x,y)∈FTn(x,y)Aδ(x, y) ∈ B+X . Thus for every (x, y) ∈ FD{i ∈ G : (T ip, T iq) ∈ Ux × Uy}≤ δ,for every p, q ∈ A.Since ε is smaller than a sensitive constant there exist p, q ∈ A such thatD{i ∈ G : (T ip, T iq) ∈ Xε}> ε, and henceD{i ∈ G : (T ip, T iq) ∈ ∪(x,y)∈FUx × Uy}> ε.We have a contradiction since this means there exists (x′, y′) ∈ F such thatD{i : (T ip, T iq) ∈ Ux′ × Uy′}> ε/ |F | = δ.2.3.3 µ− f−Mean equicontinuityWe will make use of df and Bfε (see Definition 2.1.17).Definition 2.3.14. Let (X,T ) be a TDS µ a Borel probability measure and f ∈ L2(X,µ). We say(X,T ) is µ− f−mean equicontinuous if for every τ > 0 there exists a compact set M ⊂ X suchthat for every ε > 0 there exists δ > 0 such that whenever x, y ∈M and d(x, y) ≤ δ thendf (x, y) ≤ ε.In this case we say f is µ−mean equicontinuous (f ∈ Hme).30Theorem 2.3.15. Let (X,T ) be a TDS, µ ergodic and f ∈ L2(X,µ). The following are equivalent:1) (X,T ) is µ− f−mean sensitive.2) (X,T ) is µ− f−mean expansive.3) There exists ε > 0 such that for almost every x, µ(Bfε (x)) = 0.4)(X,T ) is not µ− f−mean equicontinuous.Proof. 1)⇔ 2)⇔ 3)Given by Theorem 2.1.20.2)⇒ 4) and 4)⇒ 3)The proof is similar to the proof of Theorem 2.3.7 but we use df instead of db.Using this and Theorem 2.1.14.Corollary 2.3.16. Let (X,µ, T ) be an ergodic TDS. Then Hap = Hcms = Hme.Given a measure preserving system (X,µ, T ) and a non-trivial measurable partition {B,Bc} wecan associate a shift invariant measure µB on {0, 1}G as follows.We define the function ψ : X → {0, 1}G with (ψ(x))j = 0 if and only if T ix ∈ B. We defineµB = ψµ. We have that ({0, 1}G , µB, σ) is a factor of (X,µ, T ).Theorem 2.3.17. Let (X,µ, T ) be an ergodic TDS. The following conditions are equivalent:1) (X,T ) is µ−mean equicontinuous2) (X,T ) is µ− 1B−mean equicontinuous for every {B,Bc} ⊂ B+X .3) (X,T ) is µ− f−mean equicontinuous for every f ∈ L2.4) (X,µ, T ) has discrete spectrum.5) (X,T ) is not µ−mean sensitive.Proof. 2)⇔ 3)Use Proposition 2.1.15 and Corollary 2.3.163)⇔ 4)Use Theorem 2.1.14 and Corollary 2.3.16.1)⇔ 5)Theorem 2.3.7.4)⇒ 1)Any isometry is µ−mean equicontinuous which is an isomorphism invariant property (use Propo-sition 2.3.9).1)⇒ 2)Using Proposition 2.3.9 we obtain that ({0, 1}G , σ) is µB−mean equicontinuous for every{B,Bc} ⊂ B+X . This implies (X,T ) is µ− 1B−mean equicontinuous for every {B,Bc} ⊂ B+X .31Corollary 2.3.18. Let (X,T ) be a mean equicontinuous TDS and µ an ergodic probability measure.Then (X,µ, T ) has discrete spectrum.There is work in progress regarding topological f−mean equicontinuity and the relationshipwith weakly almost periodic functions.2.3.4 µ−Mean equicontinuity pointsDefinition 2.3.19. Let (X,T ) be a TDS. We define the orbit metric dT on X as dT (x, y) :=supi∈G{d(T ix, T iy)}, and the orbit balls asBTε (x) = {y | dT (x, y) ≤ ε}={y | d(T i(x), T i(y)) ≤ ε ∀i ∈ G}.Let (X,T ) be a TDS. A point x ∈ X is an equicontinuity point of T if and only if for every εthere exists δ such that Bδ(x) ⊂ BTε (x).Based on [24] we define µ−equicontinuity points.Definition 2.3.20. A point x is a µ−equicontinuity point of (X,T ) if for all ε > 0 we havelimδ→0µ(BTε (x) ∩Bδ(x))µ(Bδ(x))= 1.If x is an equicontinuity point in the support of µ then x is a µ−equicontinuity point.We will use the Besicovitch pseudometric db and the projection to the Besicovitch space fb(Definition 2.2.1).Definition 2.3.21. A point x is a µ−mean equicontinuity point of (X,T ) if for all ε > 0 wehavelimδ→0µ(Bbε(x) ∩Bδ(x))µ(Bδ(x))= 1.We will show when µ−(mean) equicontinuity can be described locally using µ−(mean) equicon-tinuity points.Definition 2.3.22. We say (X,µ) satisfies Lebesgue’s density theorem if for every Borel setA we have thatlimδ→0µ(A ∩Bδ(x))µ(Bδ(x))= 1 for a.e. x ∈ A. (2.3.1)The original Lebesgue’s density theorem applies to Rd and the Lebesgue measure. If X ⊂ Rdand µ is a Borel probability measure then (X,µ) satisfies Lebesgue’s density theorem ( see Remark2.4 in [38]).32Theorem 2.3.23 (Levy’s zero-one law [16] pg.262). Let Σ be a sigma-algebra on a set Ω and P aprobability measure. Let {Fn} ⊂ Σ be a filtration of sigma-algebras, that is, a sequence of sigma-algebras {Fn}, such that Fi ⊂ Fi+1 for all i. If D is an event in F∞ (the smallest sigma-algebrathat contains every Fn) then P (D | Fn)→ 1D almost surely.Corollary 2.3.24. Let X be a Cantor space and µ a Borel probability measure. Then (X,µ)satisfies Lebesgue’s density theorem.Proof. Let n ∈ N, and Fn the smallest sigma-algebra that contains all the balls{B1/n′(x)}withn′ ≤ n . It is easy to see that F∞ is the Borel sigma-algebra on X. The desired result is a directapplication from Theorem 2.3.23.There exist Borel probability spaces that do not satisfy Lebesgue’s density theorem (e.g. Ex-ample 5.6 in [38]).Definition 2.3.25. We say (X,µ) is Vitali (or satisfies Vitali’s covering theorem) if forevery Borel set A ⊂M and every N, ε > 0, there exists a finite subset F ⊂ A and nx ≥ N (definedfor every x ∈ F ) such that{B1/nx(x)}x∈Fare disjoint and µ(A ∪x∈F B1/nx(x)) ≤ ε.If X ⊂ Rd and µ is a Borel probability measure then (X,µ) is Vitali ([32] p.8). This result isknown as Vitali’s covering theorem.Using a clopen base it is not hard to see that if X is a Cantor space and µ a Borel probabilitymeasure then (X,µ) satisfies Vitali’s covering theorem. For more conditions see Chapter 1 in [32].The following theorems will be proved at the end of the subsection.Theorem 2.3.26. Let (X,T ) be a TDS and µ a Borel probability measure. Consider the followingproperties:1) (X,T ) is µ−equicontinuous.2) Almost every x ∈ X is a µ−equicontinuity point.3) There exists X ′ ⊂ X such that µ(X ′) = 1 and (X ′, dT ) is separableWe have that 1)⇐⇒ 3.If (X,µ) is Vitali then 2)⇒ 1).If (X,µ) satisfies Lebesgue’s density theorem then 1)⇒ 2).Theorem 2.3.27. Let (X,T ) be a TDS and µ a Borel probability measure. Consider the followingproperties:1) (X,T ) is µ−mean equicontinuous.2) Almost every x ∈ X is a µ−mean equicontinuity point.3) There exists X ′ ⊂ X such that µ(X ′) = 1 and (X ′, db) is separableWe have that 1)⇐⇒ 3.If (X,µ) is Vitali then 2)⇒ 1).If (X,µ) satisfies Lebesgue’s density theorem then 1)⇒ 2).33Definition 2.3.28. Given a TDS (X,T ) we denote the change of metric projection map as fT :(X, d)→ (X, dT ).The function f−1T is always continuous. A point x ∈ X is an equicontinuity point of (X,T ) if andonly if it is a continuity point of fT . Thus (X,T ) is equicontinuous if and only if fT is continuous.We have already seen that (X,T ) is mean equicontinuous if and only if fb is continuous.Definition 2.3.29. Let X,Y be metric spaces and µ a Borel probability measure on X. A functionf : X → Y is µ−Lusin (or Lusin measurable) if for every τ > 0 there exists a compact setM ⊂ X such that µ(M) > 1− τ and f |M is continuous.This implies that a TDS (X,T ) is µ−equicontinuous if and only if fT is µ−Lusin and (X,T ) isµ−mean equicontinuous if and only if fb is µ−Lusin.Every µ−Lusin function is µ−measurable. Lusin’s theorem states the converse is true if Y isseparable (note that (X, dT ) is not necessarily separable). This fact is generalized in the followingtheorem.Theorem 2.3.30 ([56] pg. 145). Let f : X → Y be a function and µ a Borel probability measureon X such that there exists X ′ ⊂ X such that µ(X ′) = 1 and f(X ′) is separable. The function f isµ−Lusin if and only if for every ball B, f−1(B) is µ−measurable.Proposition 2.3.31. Let (X,T ) be a TDS and µ a Borel probability measure. We have that (X,T )is µ−mean equicontinuous if and only if there exists X ′ ⊂ X such that µ(X ′) = 1 and (X ′db, db)is separable; and (X,T ) is µ− equicontinuous if and only if there exists X ′ ⊂ X such that µ(X ′) = 1and (X ′, dT ) is separableProof. Define f := fb.If there exists X ′ ⊂ X such that µ(X ′) = 1 and (X ′db, db) is separable apply Theorem 2.3.30to obtain that fb is µ−Lusin and hence (X,T ) is µ−mean equicontinuous.If fb is µ−Lusin it means that for every κ > 0 there exists a compact set Mκ ⊂ X such thatµ(Mκ) > 1 − κ and fb |Mκ is continuous. This implies that X′ = ∪n∈NM1/n satisfies the desiredconditions.The other part of the result is proved analogously with f := fT .We will define a measure theoretic notion of continuity point (µ−continuity point) that satisfiesthe following property: x ∈ X is a µ−equicontinuity point of T if and only if x is a µ−continuitypoint of fT . We will show that if (X,µ) is Vitali and almost every x ∈ X is a µ−continuity pointof f then f is µ−Lusin; we show the converse is true if (X,µ) satisfies Lebesgue’s density theorem(Theorem 2.3.35).In this subsection X will denote a compact metric space, µ a Borel probability measure on X,and Y a metric space with metric dY (and balls BYε (y): = {z ∈ Y : dY (y, z) ≤ ε}).34Definition 2.3.32. In some cases we can talk about the measure of not necessarily measurablesets. Let A ⊂ X. We say A has full measure if A contains a measurable subset with measure one.We say µ(A) < τ, if there exists a measurable set A′ ⊃ A such that µ(A′) < τ.Definition 2.3.33. A set A ⊂ X is µ−measurable if A is in the the completion of µ.The function f : X → Y is µ−measurable if for every Borel set D ⊂ Y , we have that f−1(D)is µ−measurable.A point x ∈ X is a µ−continuity point if for every ε > 0 we have that 1f−1(BYε (f(x))) isµ× µ−measurable andlimδ→0µ[f−1(BYε (f(x))) ∩Bδ(x)]µ(Bδ(x))= 1. (2.3.2)If µ−almost every x ∈ X is a µ−continuity point we say f is µ−continuous.Lemma 2.3.34. Let (X,µ) be Vitali and f :X → Y µ−continuous. For every ε, τ > 0 there existsa finite set Fε,τ and a function δ(x) such that{Bδ(x)(x)}x∈Fε,τare disjoint, andµ(∪x∈Fε,τ[f−1(BYε (f(x))) ∩Bδ(x)(x)]) > 1− τ.Proof. Let τ > 0. For every µ−equicontinuity point x and ε > 0 there exists δ(x) > 0 such thatµ(f−1(BYε (f(x)))∩Bδ(x))µ(Bδ(x))> 1− τ for every δ ≤ δ(x). LetAi := {x | x is a µ− continuity point, 1/δ(x) ≤ i} .Using the function f = 1f−1(BYε (f(x)))·1Bδ(x) and Fubini’s theorem one can check thatµ(f−1(BYε (f(x)))∩Bδ(x))µ(Bδ(x))is a measurable function and hence the sets Ai are also µ−measurable. Since ∪i∈NAi contains theset of µ−continuity points, there exists N such that µ(XAN ) < τ. Since (X,µ) is Vitali thereexists a finite set of points Fε,τ ⊂ AN and a function δ(x) ≥ 1/N such that{Bδ(x)(x)}x∈Fε,τare disjoint and µ(∪x∈Fε,τBδ(x)(x)) > µ(AN ) − τ. This implies µ(∪x∈Fε,τBδ(x)(x)) > 1 − 2τ. SinceFε,τ ⊂ AN , we obtainµ(∪x∈Fε,τ[f−1(BYε (f(x))) ∩Bδ(x)(x)]) > 1− 3τ.Theorem 2.3.35. Let f : X → Y . Consider the following properties1) f is µ−Lusin.2) f is µ−continuous.If (X,µ) is Vitali then 2) =⇒ 1).If (X,µ) satisfies Lebesgue’s density theorem then 1) =⇒ 2).Proof. 2) + V itali =⇒ 1)35Let τ, ε > 0. By Lemma 2.3.34 there exists a finite set of points Fε,τε and a function δ(x)(defined on Fε,τε) such thatµ(∪x∈Fε,τε[f−1(BYε (f(x))) ∩Bδ(x)(x)]) > 1− τεand{Bδ(x)(x)}x∈Fε,τεare disjoint. Letx1, x2 ∈ ∪x∈Fε,τεGx := ∪x∈Fε,τε[f−1(BYε (f(x))) ∩Bδ(x)(x)].If x1 and x2 are sufficiently close then they will be in the same setGx and hence f(x1) ∈ BYε/2(f(x2)).LetM := ∩ε∈{1/2n:n∈N} ∪x∈Fε,τε[f−1(BYε (f(x))) ∩Bδ(x)(x)].Then f pM is continuous and µ(M) > 1 − τ . The regularity of the Borel measure gives us theexistence of the compact set.1) + LDT ⇒ 2)Since f is µ−Lusin we have that it is µ−measurable.Let τ > 0. There exists a compact set M ⊂ X such that µ(M) > 1− τ and f |M is continuous.Using Lebesgue’s density theorem we havelimδ→0µ(M ∩Bδ(x))µ(Bδ(x))= 1 for a.e. x ∈M.Let ε > 0. Since f |M is continuous then for sufficiently small δ and almost every x ∈ M,M ∩Bδ(x) ⊂ f−1(BYε (f(x))), and so for almost every xlimδ→0µ(f−1(BYε (f(x))) ∩Bδ(x))µ(Bδ(x))= 1.This implies the set of µ−continuity points has measure larger than 1− τ . We conclude the desiredresult.Proof of Theorem 2.3.26. Take f = fT , (Y, dY ) = (X, dT ), BYm = BTm.Note that orbit balls are countable intersections of balls, hence measurable.3)⇒ 1)Apply Theorem 2.3.30.1)⇒ 3)If fT is µ−Lusin, then that for every τ > 0 there exists a compact set Mτ ⊂ X such thatµ(Mτ ) > 1 − τ and fT |Mκ is continuous. This implies that X′ = ∪n∈NM1/n satisfies µ(X′) = 1and (X ′, dT ) is separable.For the other implications apply Theorem 2.3.35.36Proof of Theorem 2.3.27. Analogous to the proof of Theorem 2.3.26 but using fbinstead of fT .37Chapter 3Limit behaviour of µ−equicontinuouscellular automataCellular automata (CA) are discrete systems that depend on local rules. Hedlund [31] characterizedCA using dynamical properties: φ : {1, 2, ...n}Z → {1, 2, ...n}Z is a cellular automaton if and onlyif it is continuous (with respect to the Cantor product topology) and shift-commuting. This meansevery CA is a Z+−topological dynamical system, i.e. a continuous transformation φ on a compactmetric space X. The dynamical behaviour of these systems can range from very predictable to verychaotic. There are different classifications of cellular automata and TDS using equicontinuity andsensitivity (see [1] and [43]).Equicontinuity is a very strong property particularly for cellular automata; different attemptshave been made to define weaker but similar properties. Using shift-ergodic probability measures,Gilman [24][23] introduced the weaker concept of µ−equicontinuity points for cellular automata.The study of long term behaviour is a main topic of interest of dynamical systems/ergodictheory. Long term behaviour can be studied for points, sets, or measures. In particular one mayask if the orbit of a measure converges weakly. Limit behaviour of measures under CA have beenstudied mainly for two subclasses, linear and µ−equicontinuous.In [46] Lind studied the limit behaviour of the CA on the binary full-shift defined by adding thevalue of two consecutive positions mod 2. He concluded that the weak limit of the Cesaro averageof every Bernoulli measure is the uniform Bernoulli measure. This result has been generalizedto other linear expansive CA, and it has been shown that the Cesaro weak limit of an ergodicMarkov measure is the uniform Bernoulli (or in a more general setting the measure of maximalentropy)[54][48].In [8] Blanchard and Tisseur studied CA and measures that give equicontinuity points full mea-sure. These CA are µ−equicontinuous but µ−equicontinuous CA may not have any equicontinuitypoint (like in Example 3.1.29). They showed that the Cesaro weak limit exists and they askedquestions about the dynamical behaviour of the limit measure. In particular they asked when38the limit measure is shift-ergodic, a measure of maximal entropy or φ−ergodic. We address thosequestion in this thesis; we show that those three conditions are very strong.We characterize µ−equicontinuity on shifts of finite type using locally periodic behaviour;µ − LEP (Proposition 3.1.21). We present a natural generalization of Blanchard-Tisseur’s re-sult (Theorem 3.2.8). Let φ be a CA and µ a σ−ergodic measure that gives equicontinuity pointsfull measure. We show the limit measure is of maximal entropy (Theorem 3.2.18) if and only ifφ is surjective and the original measure is the measure of maximal entropy. We show that if φ issurjective then the limit measure is shift-ergodic if and only µ is φ−invariant (Theorem 3.2.17).Finally we show that if the limit measure is ergodic with respect to φ then the system is isomorphic(measurably) to a cyclic permutation on a finite set (Corollary 3.3.4). Some of our results holdfor CA on multidimensional subshifts, in the cases where they don’t we present weaker analogousresults. We also present several results for µ − LEP and µ−equicontinuous systems, which maynot have any equicontinuity points (like in Example 3.1.29).3.1 Equicontinuity and local periodicity3.1.1 DefinitionsThe n−window, Wn ⊂ Zd is defined as the cube of radius n−1 centred at the origin (W1 containsonly the origin); a window is an n−window for some n. For any set W ⊂ Zd and x ∈ AZd, xW ∈AW is the restriction of x to W. We will endow AZdwith the product topology; this is the sametopology obtained by the metric given by d(x, y) = 1m , where m is the largest integer such thatxWm = yWm . We denote the balls with B1/n(x): ={z | d(x, z) ≤ 1n}.We define the full A-shift as the metric space AZd. For i ∈ Zd we will use σi : AZd→ AZdto denote the shift maps (the maps that satisfy xi+j = (σix)j for all x ∈ X and i, j ∈ Zd). Thealgebra of sets generated by balls and their shifts is called the cylinder sets. In this chapter asubset X ⊂ AZdis a subshift (or shift space) if it is closed and σi−invariant for all i ∈ Zd. Ifd = 1 we say the space is 1D. A cellular automaton (CA) is a pair (X,φ) where X is a subshiftand φ(·) : X → X is a continuous σ−commuting map, i.e. φ commutes with all the Zd shifts. Wesay (X,φ) is a 1D CA if X is a 1D subshift. In most of the literature cellular automata is studiedonly on full-shifts. In our definition a 1D subshift itself is a CA. Cellular automata of this kind arealso known as shift endomorphisms.A one sided subshift is a set X ⊂ AN that is closed and σ−invariant (i.e σ(X) ⊂ X).The following theorem was established in [31] for 1D CA on full-shifts. The same result holdsfor CA on higher dimensional subshifts.Theorem 3.1.1 (Curtis-Hedlund-Lyndon). Let X be a shift space, and φ(·) : X → X a function. The map φ is a CA if and only if there exists a non-negative integer n, and a function φ [·] :39AWn → A, such that (φ(x))i = φ [(σix)Wn ] (note the use of ( ), and [ ] to distinguish between thetwo functions that are related). The radius of the CA is the smallest possible n.Definition 3.1.2. We say X ⊂ AZdis a shift of finite type (SFT) if there exists n ∈ N and afinite list of forbidden patterns {Bi} ⊂ AWn such that x ∈ X if and only if none of the elements of{Bi} appear in x.Example 3.1.3. The 1D SFT obtained by the forbidden word {11} (i.e. the set of doubly infinitesequences that never have two 1′s together) is commonly known as the golden mean shift.For more information on subshifts and SFTs see [47].3.1.2 Topological equicontinuity and local periodicityWe remind the reader of the following definition. We introduce new notation.Definition 3.1.4. Given a CA (X,φ), we define the orbit metric dφ on X as dφ(x, y) :=supi≥0{d(φix, φiy)}, and the orbit balls asO1/m(x): ={y | dφ(x, y) ≤1m}={y | (φi(x))j = (φi(y))j ∀i ∈ N, j ∈Wm}.Note that in this chapter we use O instead of Bφ or BT . This is to make notation easier as wewill use other superscripts.A point x is an equicontinuity point of φ if for all m ∈ N there exists n ∈ N such thatB1/n(x) ⊂ O1/m(x). The transformation φ is equicontinuous if every x ∈ X is an equicontinuitypoint.We have that φ is equicontinuous if and only if the family{φi}i∈N is equicontinuous.If X is a subshift with dense periodic points, then a CA (X,φ) is equicontinuous if and only ifit is eventually periodic (i.e. there exists p and p′ such that φp+p′= φp)[43][18].A weaker notion of periodicity that is related to equicontinuity is local periodicity.Definition 3.1.5. Let (X,φ) be a CA. The set of m-locally periodic points of φ, LPm(φ), isthe set of points x such that (φix)Wm is periodic; the set of locally periodic points is defined asLP (φ):= ∩LPm(φ). Similarly, the set of m−locally eventually periodic points of φ, LEPm(φ),is the set of points x such that (φix)Wm is eventually periodic ; the set of locally eventuallyperiodic points is defined as LEP (φ) := ∩ LEPm(φ).A transformation φ is locally eventually periodic (LEP ) if LEP (φ) = X and locally pe-riodic (LP ) if LP (φ) = X.For x ∈ LEPm(φ) the smallest period will be denoted as pm(x), and then the smallest preperiodas ppm(x).40It is easy to see that x ∈ LEP (φ) if and only if (φnx)i is eventually periodic for all i ∈ Z.Proposition 3.1.6. Let (X,φ) be a CA. If (X,φ) is equicontinuous then it is LEP.Proof. Let m ∈ N. Equicontinuity implies uniform equicontinuity, hence there exists n ∈ N suchthat if y ∈ B1/n(x) then y ∈ O1/m(x).Let x ∈ X. There exists j > j′such that (φjx)i = (φj′x)i for |i| ≤ n. Thus φjx ∈ B1/n(φj′x)and hence φjx ∈ O1/m(φj′x). This implies the orbit ball is eventually periodic so x ∈ LEPm(φ);hence φ is LEP .The converse is not true.Example 3.1.7. Let X ⊂ {0, 1}Z be the subshift that contains the points that contain at most one1. We have that (X,σ) is LEP but 0∞ is not an equicontinuity point. Hence LEP does not implyequicontinuity.Nonetheless we will see that LP implies equicontinuity.The following is an unpublished result by Chandgotia [10].Proposition 3.1.8. Let (X,σ) be a one-sided subshift. If X has infinitely many periodic pointsthen X contains a non-periodic point.Proof. For x ∈ AN a periodic point, the set Br(x) denotes all the possible words of size r thatappear in x; we also define B(x) := ∪Br(x); the minimal period of the word w ∈ B(x) in x isdenoted by px(w). We say X has bounded periodic words if for all w ∈ ∪x periodicB(x) there existsnw such that px(w) ≤ nw for all periodic points x.If x ∈ AN is periodic with minimal period n, then |Bn(x)| = n and |Br(x)| > r for 1 ≤ r < n.Suppose X is not a subshift with bounded periodic words. So there exists w ∈ B(x) and xn ∈ Xa sequence of periodic points such that pxn(w) ≥ n and xn begins with w. Any limit point of thesequence xn contains w only once, hence it is not periodic.Now suppose X has bounded periodic words. Let x1,n be a sequence of periodic points suchthat p(x1,n) > n and B1(x1,n) is constant. Inductively, define xi,n to be a subsequence of xi−1,nsuch that Bi(xi,n) is constant. We can then find a sequence of points yn ∈ X such that p(yn) > n(and hence limn→∞ |Br(yn)| > r for all r) and the sequence of sets Br(yn) is eventually constantfor all r. Let y be a subsequential limit of yn and w ∈ limn→∞ Br(yn). There exists nw such thatpyn(w) ≤ nw for all n, so w ∈ Br(yn). This means that |Br(y)| ≥ limn→∞ |Br(yn)| > r for all r andhence y is not periodic.Corollary 3.1.9. Let X be a one-sided subshift that contains only σ−periodic points. Then X isfinite and hence (X,σ) is equicontinuous.Proposition 3.1.10. Let (X,φ) be a CA. If (X,φ) is LP then it is equicontinuous.41Proof. Let Xj ={y ∈ AN | yi = (φix)j for some x ∈ X}. We have that (Xj , σ) is a one-sided sub-shift that contains only periodic points, hence there are only finitely many. This means φ isequicontinuous.A point x is recurrent if for every open neighbourhood U the orbit of x under φ intersects Uinfinitely often; the set of recurrent points is denoted by R(φ). The following lemma will be usefullater.Lemma 3.1.11. Let (X,φ) be a CA. Then R(φ) ∩ LEP (φ) = LP (φ).Proof. Using the definitions it is easy to see that LP (φ) ⊂ R(φ) ∩ LEP (φ).Let x ∈ R(φ) ∩ LEP (φ), m ∈ N, and q := ppm(x) (see Definition 3.1.5). Suppose x /∈ LP (φ).This means that q > 0, (φi+qx)Wm is periodic for i ≥ 0, and (φq−1x)Wm 6= (φq+pm(x)−1x)Wm .Using the continuity of φ, we know there exists m′ ≥ m such that for every y ∈ B1/m′(φq−1x),(φi+qx)Wm = (φi+qy)Wm for 0 ≤ i ≤ pm(x). Using the fact that φq−1x is recurrent we obtain thatthere exists N > ppm(x) + pm(x) such that φNx ∈ B1/m′(φq−1x). This means that (φNx)Wm 6=(φq+pm(x)−1x)Wmand (φN+ix)Wm = (φq+pm(x)+i−1x)Wm for 0 < i ≤ pm(x). This is a contradictionsince the second condition and the fact that pm(x) is the smallest period implies that there existsj > 0 such that N = q + jpm(x)− 1, and hence(φNx)Wm = (φq+pm(x)−1x)Wm .3.1.3 Measure theoretical equicontinuity and local periodicityWe will use µ to denote Borel probability measures on X. These do not need to be invariant underφ. For the ergodic theory point of view of µ−equicontinuity see Section 4.We repeat the definition of µ−equicontinuity points.The concept of µ−equicontinuity points first appeared in [24] [23], and it was used to classifycellular automata using Bernoulli measures. We remind the reader of the definition in this chapter’snotation.Definition 3.1.12. Let (X,φ) be a CA and µ a Borel probability measure on X. A point x ∈ Xis a µ−equicontinuity point of φ if for all m ∈ N, one haslimn→∞µ(B1/n(x) ∩O1/m(x))µ(B1/n(x))= 1.The following result is a consequence of Corollary 2.3.24, Theorem 2.3.26 and Proposition 2.3.31.Theorem 3.1.13 ( [20]). Let (X,φ) be a CA and µ a Borel probability measure. The following areequivalent:1) (X,φ) is µ−equicontinuous2) Almost every x ∈ X is a µ−equicontinuity point.3) There exists X ′ ⊂ X such that X ′ is dφ−separable and µ(X ′) = 1.42Definition 3.1.14. When (X,φ) is µ−equicontinuous we say µ is φ−equicontinuous.Definition 3.1.15. Let (X,φ) be a CA. If µ(LEP (φ)) = 1, we say (X,φ) is µ−locally eventuallyperiodic (µ−LEP ) and µ is φ−locally eventually periodic (φ−LEP ). We define µ−LP andφ− LP analogously.The concept of µ−LEP was motivated by Proposition 5.2 in [24]. In [20] the property of LEPis defined and studied for general TDS.Lemma 3.1.16. Let m ∈ N and ε > 0. If (X,φ) is µ− LEP then there exist positive integers pmεand ppmε such that µ(Ymε ) > 1− ε, whereY mε := {x | x ∈ LEP (φ), with pm(x) ≤ pmε and ppm(x) ≤ ppmε } .Proof. LetY := ∪s,k∈N {x | x ∈ LEP (φ), with pm(x) ≤ s and ppm(x) ≤ k} .Since (X,φ) is µ − LEP we have that µ(Y ) = 1. Monotonicity of the measure gives the desiredresult.Definition 3.1.17. Given a µ − LEP transformation φ and m, ε > 0, we will use pmε and ppmεto denote a particular choice of integers that satisfy the conditions of the previous lemma and thatsatisfy that pmε →∞ and ppmε →∞, as ε→ 0.Definition 3.1.18. Given a subshift X, we denote the σ−periodic points with PX(σ).The following result was proved in [23] when X is a full shift, but the same result holds whenX is an SFT.Lemma 3.1.19. Let X be a 1D SFT with forbidden words of size q, (X,φ) a CA with radius r.If there is a point x and an integer m 6= 0 such that O1/i(x) ∩ σ−mO1/i(x) 6= ∅ with i ≥ q, r thenO1/i(x) ∩ PX(σ) 6= ∅.Proposition 3.1.20. Let (X,φ) be a CA. If (X,φ) is µ− LEP then it is µ−equicontinuous.Proof. Let ε > 0 and M := ∩m∈NY mε/2m (hence µ(M) > 1− ε).Let m ∈ N. There exists K such that if d(x, y) < 1/K then d(φix, φiy) < 1/m for 0 ≤ i ≤(pmε/2m)2+ ppmε/2m . Let x, y ∈ Ymε/2m , d(x, y) < 1/K , p = pm(x)pm(y) and i ≥ p + ppmε/2m . Wecan express i = ppmε/2m + j · p + k with j ∈ N and k ≤ p. Using the fact that x, y ∈ Ymε/2m andppmε/2m + k ≤(pmε/2m)2+ ppmε/2m we obtaind(φix, φiy) ≤ d(φix, φppmε +kx) + d(φppmε +kx, φppmε +ky) + d(φppmε +ky, φiy)≤ 3/m.43This means that B1/K(x) ∩ Ymε/2m ⊂ O3/m(x) and hence φ is µ−equicontinuous.We obtain the converse with an extra hypothesis.Proposition 3.1.21. Let X be a 1D SFT, (X,φ) a CA, and µ a σ−invariant probability measureon X. Then (X,φ) is µ−equicontinuous if and only if it is µ− LEP.Proof. Let X be a 1D SFT with forbidden words of size q, (X,φ) a CA with radius r and p ≥ q, r.If x is a µ−equicontinuous point thenlimn→∞µ(B1/n(x) ∩O1/p(x))µ(B1/n(x))= 1.Using µ(O1/p(x)) > 0 and Poincare’s recurrence theorem we obtain that{y | σi(y) ∈ O1/p(x) i.o.}is not empty. Using p ≥ q, r and Lemma 3.1.19 we conclude that every orbit ball with positivemeasure contains a σ−periodic point and hence (φnx)j is eventually periodic for −p ≤ j ≤ p. Thereverse implication is obtained with Proposition 3.1.20.Proposition 3.1.21 shows that µ−equicontinuity and µ − LEP are equivalent if X is a 1DSFT and µ a σ−invariant measure. We do not know if this result holds for cellular automata onmultidimensional SFTs. We can show a weaker result (Proposition 3.1.25) by strengthening theµ−equicontinuity hypothesis.Definition 3.1.22. Let (X, ρ) be a metric space, and A ⊂ X. The closure of A is denoted withclρ(A).Recall that dφ denotes the orbit metric (Definition 3.1.4).Lemma 3.1.23. Let (X,φ) be a CA. If µ(cldφ(PX(σ))) = 1, then (X,φ) is µ−LEP.Proof. Using the fact that the φ−image of a σ−periodic point is σ−periodic with at most the sameperiod one can see that any point in PX(σ) is eventually periodic for φ. Let O1/m be an orbitball of size m. This means that if O1/m ∩ PX(σ) 6= ∅ and x ∈ O1/m then x ∈ LEPm(φ). Hence ifµ(cldφ(PX(σ))) = 1 then φ is µ−LEP.We represent the change of metric identity map with fφ : (X, d) → (X, dφ). A point x ∈ X isan equicontinuity point of φ if and only if it is a continuity point of fφ. Hence φ is equicontinuousif and only if fφ is continuous.Definition 3.1.24. Let (X,φ) be a CA. We denote the set of equicontinuity points with EQ(φ).44In [8] σ−ergodic measures that give full measure to the equicontinuity points of a CA werestudied. As a consequence of Lemma 3.1 (in that paper) we can see that if X is a 1D subshift,(X,φ) a CA, and µ a σ−ergodic probability measure on X with µ(EQ(φ)) = 1 then (X,φ) isµ− LEP.If we assume the subshift has dense periodic points (or more generally are dense in a set of fullmeasure) then we obtain that result for multidimensional subshifts.Proposition 3.1.25. Let (X,φ) be a CA and µ a measure such that µ(cld(PX(σ))) = 1. Ifµ(EQ(φ)) = 1 then (X,φ) is µ− LEP.Proof. Since equicontinuity points have full measure, then fφ is continuous on a set of full measure.This means that for almost every x ∈ cld(PX(σ)), we have x ∈ cldφ(PX(σ)). So µ(cldφ(PX(σ))) =µ(cld(PX(σ))) = 1. Using Lemma 3.1.23 we conclude that (X,φ) is µ− LEP.We already noted that if X is a subshift with dense σ−periodic points, and (X,φ) is an equicon-tinuous CA then (X,φ) is eventually periodic. Proposition 3.1.25 is a measure theoretic analogousresult.Now we present some examples.Example 3.1.26. Let x be a non eventually σ−periodic point and let µ be the delta measuresupported on {x} . Then φ = σ is µ−equicontinuous but not µ-LEP.Definition 3.1.27. A 1D CA (X,φ) has right radius 0 if there exist L ∈ N and a functionφ [·] : AL → A, such that (φ(x))i = φ [x−L...xi−1xi]Example 3.1.28 ([24]). Let X = {−1, 0, 1}Z , and (X,φ) a radius 1 CA with right radius 0 definedas follows: φ [11] = 1, φ [10] = 1, φ [1− 1] = 0, φ [a1] = 0 if a 6= 1, φ [ab] = b if a, b 6= 1. Thereader can picture a 1 moving to the right until it encounters a −1 and converts into a 0. It iseasy to see that this CA does not contain equicontinuity points. For Bernoulli measures this CA isµ−equicontinuous when µ(−1) > µ(1) .We define the set Ei :={x ∈ {0, 1}Z | xj = 1 for 0 ≤ j ≤ i}.Example 3.1.29. There exists a CA on the full 2-shift with no equicontinuity points that isµ−equicontinuous for every σ−invariant measure that satisfies∑i≥0 µ(Ei) < ∞ ( in particularevery non-trivial ergodic Markov chain).Proof. On {0, 1}Z we define φ(x) = y as yi = xi−1xi−2.One can check that for every i > 0, (φix)0 =−2i∏j=−ixj . Let a ∈ {0, 1} . If there exists n > 0such that xi = a for i ≤ −n, then (φix)0 = a for i ≥ 2n. This means that for every ball Bthere exists x, y ∈ B such that O0(x) 6= O0(y), so the CA has no equicontinuity points. Let45Ei := {x | xj = 1 for − 2i ≤ j ≤ −i} . We have that∑i≥0 µ(Ei) =∑i≥0 µ(Ei) < ∞. By theBorel-Cantelli Lemma we have that µ(Ei infinitely often) = 0. This means that the probabilitythat (φix)0 has infinitely many ones is zero; since the same argument can be given for (φix)m weconclude φ is µ− LEP and hence µ−equicontinuous.For every non-trivial ergodic Markov chain µ(Ei) decreases exponentially so∑i≥0 µ(Ei) <∞.Note that in the trivial case (i.e. when µ(1) = 1 or 0), the hypothesis is not satisfied but wealso conclude φ is µ−equicontinuous.Q: Does there exists a CA with no equicontinuity points that is µ−equicontinuous for everyσ−invariant µ?For more examples of µ−equicontinuous CA see [58].The following diagrams illustrate how the different properties relate on the topological andmeasure theoretical level.TopologicalLP ⇒ Equicontinuous⇒ LEPMeasure theoreticalµ− LP ⇒ µ− LEP ⇒ µ− equicontinuousIf X is a 1D SFT and µ σ−invariant thenµ− LEP = µ− equicontinuous3.2 Weak convergenceA sequence of measures µn (on X) converges weakly to µ∞ ( denoted as µn →w µ∞) if for everycontinuous function f : X → R, ∫fdµn →∫fdµ∞.This form of convergence is called weak convergence in the Probability literature and weak*convergence in the Functional Analysis literature.One can study limit behaviour of dynamical systems by studying the long term behaviour ofφnµ (φµ is the push-forward of the measure) or of its Cesaro averages: µcn :=1n∑ni=1 φi(µ). Inparticular we may ask if φnµ or µcn converges weakly, and which are the properties of the limitmeasure.Theorem 3.2.1 (Portmanteau [5] pg. 15). We have µn →w µ∞ if and only if for every open set U,µ∞(U) ≤ lim inf µn(U) if and only if µn(E)→ µ∞(E) for every set E with zero boundary measure.46We will see that orbit balls form a weak convergence determining class when the limit measureis φ− equicontinuous (Lemma 3.2.2). Note that even when µ is φ − LEP, the measure of theboundary of an orbit ball is not necessarily zero. For example, one can check that the orbit ballsof Example 3.1.28 are each contained in their own boundary.Lemma 3.2.2. Let (X,φ) be a CA, µn be a sequence of measures, and µ∞ a φ−equicontinuousmeasure. If for every orbit ball A we have that µn(A)→ µ∞(A) then µn →w µ∞. Also, if µ and µ′are φ−equicontinuous and µ(A) = µ′(A) for every orbit ball A then µ = µ′.Proof. If we have two orbit balls O and O′, then either O∩O′ = ∅ or one is contained in the other.This implies we have convergence for finite unions of orbit balls (since they can be written as unionsof disjoint orbit balls). Let U be an open set. We have that U is the countable union of balls.From Theorem 3.1.13 we know there exists a dφ−separable set X ′ such that µ∞(X ′) = 1 . Thismeans that for every δ > 0 there exist a finite number of orbit balls O1/i such that ∪Ni=1O1/i ⊂ Uand µ∞(U) ≤ µ∞(∪Ni=1O1/i) + δ. We have thatµ∞(U)− δ ≤ µ∞(∪Ni=1O1/i) = limµn(∪Ni=1O1/i) ≤ lim inf µn(U),and henceµ∞(U) ≤ lim inf µn(U).Therefore µn →w µ∞.Now suppose µ(A) = µ′(A) for every orbit ball A. Let X ′ be the dφ−separable set withµ(X ′) = 1 This means µ(X ′) = µ′(X ′) = 1 (that is because X ′ is equal to the union of countablymany orbit balls). Thus µ and µ′ agree on a pi−system (family closed under finite intersections)that generate the Borel sigma algebra (intersected with X ′); we conclude µ = µ′.Definition 3.2.3. For x ∈ LEPm(φ) we defineO−q1/m(x) :={y | ∃i ∈ N s.t. φipm(x)+qy ∈ O1/m(x)}.Definition 3.2.4. We denote the Cesaro average of φiµ with µcn, i.e. µcn :=1n∑ni=1 φi(µ).In the following proposition we show Cesaro convergence holds for orbit balls. Note in theproof that the convergence is actually stronger than Cesaro; there is convergence along periodicsubsequences.Lemma 3.2.5. Let (X,φ) be a µ− LEP CA, and m ∈ N. If x ∈ LPm(φ) thenµcn(O1/m(x))→1pm(x)pm(x)−1∑q=0µ(O−q1/m(x)).47Furthermore if x /∈ LPm(φ) then µcn(O1/m(x))→ 0.Proof. We have∪n≥0φ−pm(x)n−q(O1/m(x)) = O−q1/m(x).For every 0 ≤ q < pm(x) and n ≥ 0φ−pm(x)n−q(O1/m(x)) ⊂ φ−pm(x)(n+1)−q(O1/m(x)).This implies µ(φ−pm(x)n−q(O1/m(x))) is non-decreasing andlimn→∞µ(φ−pm(x)n−q(O1/m(x))) = µ(O−q1/m(x)). (3.2.1)Since we have convergence along periodic subsequences we have thatlimn→∞µcn(O1/m(x)) =1pm(x)pm(x)−1∑q=0µ(O−q1/m(x)).Let ε > 0 and x /∈ LPm(φ). If np′ > ppmε , thenφ−p′n−s(O1/m(x)) ∩ Ymε = ∅,so µ(φ−p′n−s(O1/m(x)(x))) < ε.Proposition 3.2.6. Let (X,φ) be a µ− LEP CA. If φµ = µ then (X,φ) is µ− LP.Proof. Using the invariance of µ and Poincare’s recurrence theorem we obtain that the set ofrecurrent points has full measure, i.e. µ(R(φ)) = 1. By Lemma 3.1.11 we have thatµ(LP (φ)) = µ(LEP (φ) ∩R(φ)) = 1.Remark 3.2.7. Let B be a finite union of balls (thus B is compact). If B = ∪∞i=1Bi, where{Bi}is a disjoint family of balls, then there exists K such that B = ∪Ki=1Bi. From this fact we get thatany premeasure on the algebra generated by the balls can be extended to a measure on the Borelsigma-algebra.The existence of a limit measure in the following result is a natural generalization of a resultfor 1D CA in [8] (X is multidimensional and φ may not have any equicontinuity points). Here wealso show that the limit measure is φ− LP.48Theorem 3.2.8. Let (X,φ) be a µ− LEP CA. The sequence of measures µcn converges weakly toa φ− LP measure µ∞.Proof. Let B1/m be a ball. For every ε > 0 and n ∈ N we have that∣∣∣∣∣1nn∑i=1µ(φ−i(B1/m ∩ Ymε ))−1nn∑i=1µ(φ−i(B1/m))∣∣∣∣∣≤ nε/n = ε.Consequentlylimε→01nn∑i=1µ(φ−i(B1/m ∩ Ymε )) =1nn∑i=1µ(φ−i(B1/m))uniformly on n.On the other hand for every ε > 0 there exists a finite set of disjoint orbit balls{O1/mk(xk)}such that the xk are LEP and B1/m ∩ Ymε = ∪k=Kk=1 O1/mk(xk). This implies that1nn∑i=1µ(φ−i(B1/m ∩ Ymε ))=1nn∑i=1µ(φ−i(∪k=Kk=1 O1/mk(xk))).By Lemma 3.2.5 limn→∞ 1n∑ni=1 µ(φ−i(∪k=Kk=1 O1/mk(xk))) exists.LetF (n, ε) :=1nn∑i=1µ(φ−i(B1/m ∩ Ymε )).We have shown thatlimn→∞F (n, ε) exists, andlimε→0F (n, ε) =1nn∑i=1µ(φ−i(B1/m)) uniformly on n.Thus we obtain that limn→∞ limε→0 F (n, ε) exists, andlimn→∞limε→01nn∑i=1µ(φ−i(B1/m ∩ Ymε )) = limn→∞1nn∑i=1µ(φ−i(B1/m))= limn→∞µcn(B1/m).The proof of the previous statement is common in analysis (see for example Theorem 1 in [37]).We define µ∞ as the measure that satisfies µ∞(B1/m) := limn→∞ µcn(B1/m) (see Remark 3.2.7)for every ball B1/m. Every open set U can be approximated by a finite disjoint union of balls. This49implies µ∞(U) ≤ lim inf µcn(U), and hence µcn →w µ∞.Since φ−1(LEP (φ)) = LEP (φ) we have that µcn is φ − LEP. For every m ∈ N and ε > 0 wehave that Y mε is a finite union of orbit balls and hence.µ∞(Ymε ) = limµcn(Ymε ) ≥ 1− ε.Since Y mε ⊂ LEP (φ) we obtain that µ∞ is φ−LEP. Considering that φµ∞ = µ∞ and Proposition3.2.6 we obtain that (X,φ) is µ− LP.There is a more general definition of µ − LEP for topological dynamical systems (see [20]).It is possible to check that the previous result holds for topological dynamical systems on zerodimensional spaces.Definition 3.2.9. From now on we will use µ∞ to denote the weak limit of µcn.Proposition 3.2.10. Let (X,φ) be a µ − LEP CA. Then φnµ →w µ∞ if and only if φnµ(O) →µ∞(O) for all orbit balls O, and µcn →w µ∞ if and only if µcn(O)→ µ∞(O) for all orbit balls O.Proof. Assume that φnµ→w µ∞.Let m ∈ N and x ∈ LEP (φ). Take ε > 0 so that pmε > pm(x).Let O1/m(x) = ∩i∈Nφ−iGi, where every Gi is a ball. There exists k1 such that∣∣∣µ∞(∩ki=1φ−iGi)− µ∞(O1/m(x))∣∣∣ ≤ ε for k ≥ k1.Fix k ≥ 2pmε , k1. If n ≥ ppmε thenφ−n(∩ki=1φ−iGi) ∩ Ymε = φ−n(O1/m(x)) ∩ Ymε .Since µ(Y mε ) > 1− ε we obtain∣∣∣φnµ(∩ki=1Gi)− φnµ(O1/m(x))∣∣∣ ≤ ε for n ≥ ppmε .Since ∩ki=1φ−iGi has no boundary, there exists N such that∣∣∣φnµ(∩ki=1φ−iGi)− µ∞(∩ki=1φ−iGi)∣∣∣ ≤ ε for n ≥ N.Using the inequalities we obtain∣∣φnµ(O1/m(x))− µ∞(O1/m(x))∣∣ ≤ 3ε for n ≥ N, ppmε .Hence φnµ(O1/m(x))→ µ∞(O1/m(x)).50The other direction is a corollary of Lemma 3.2.2.The proof for µcn is analogous.Definition 3.2.11. For a ∈ R and E ⊂ X Borel, we defineAEa :={y : limn→∞1|Wn|∑i∈Wn1E(σi(y)) = a}.Note that if µ is σ−ergodic then by the pointwise ergodic theoremµ(AEa ) ={1 if a = µ(E)0 otherwise.Lemma 3.2.12. Let µ be a σ−ergodic measure, (X,φ) a µ− LEP CA, m ∈ N, and x ∈ LPm(φ).If for every 0 ≤ q < pm(x) there exists Nq ∈ N such thatµ(φ−pm(x)n−q(O1/m(x))) = µ(O−q1/m(x)) for all n ≥ Nqthenµcn(AO1/m(x)a )→ µ∞(AO1/m(x)a ) =1pm(x)pm(x)−1∑q=0µ(AO−q1/m(x)a ).Proof. By hypothesis we have that for n ≥ Nqφpm(x)n+qµ(O1/m(x)) = µ(O−q1/m(x)).Note that since µ is σ−ergodic then φnµ is σ−ergodic for every n ≥ 1. Let 0 ≤ q < pm(x).If n ≥ Nq thenφpm(x)n+qµ(AO1/m(x)a ) ={1 if a = φpm(x)n+qµ(O1/m(x))0 otherwise={1 if a = µ(O−q1/m(x))0 otherwise= µ(AO−q1/m(x)a ).This implies we have convergence along periodic subsequences. Thusµcn(AO1/m(x)a )→ µ∞(AO1/m(x)a ) =1pm(x)pm(x)−1∑q=0µ(AO−q1/m(x)a ).51Using the fact that µ(φ−pm(x)n−q(O1/m(x))) → µ(O−q1/m(x)) one can check that one of thehypothesis of this result is always satisfied if we assume the CA is µ− LP.Lemma 3.2.13. Let (X,φ) be a µ − LP CA, m ∈ N, and x ∈ LPm(φ). Then for every 0 ≤ q <pm(x) we haveµ(φ−pm(x)i−q(O1/m(x))) = µ(O−q1/m(x)) for all i ∈ N.Proof. This comes from the fact that LP (φ) ∩ φ−pm(x)i−q(O1/m(x)) = LP (φ) ∩ O−q1/m(x) for alli ∈ N.Theorem 3.2.14. Let µ be a σ−ergodic measure and (X,φ) a µ−LP CA. Then µ∞ is σ−ergodicif and only if µ is φ−invariant.Proof. If φµ = µ then µ = µ∞ and hence it is σ−ergodic.Suppose µ∞ is σ−ergodic and µ is not φ−invariant. By Lemma 3.2.2 we know there exists x ∈LP (φ) and m ∈ N such thatµ(O1/m(x)) 6= µ(φ−1O1/m(x)).We have thatO1/m(x) ∩ LP (φ) = O01/m(x) ∩ LP (φ), andφ−1O1/m(x) ∩ LP (φ) = O−11/m(x) ∩ LP (φ).Since (X,φ) is µ− LP we obtain thatµ(O01/m(x)) 6= µ(O−11/m(x)).Using Lemma 3.2.12 and Lemma 3.2.13 we getµ∞(AO1/m(x)a ) =1pm(x)pm(x)−1∑q=0µ(AO−q1/m(x)a ).We reach a contradiction because AO1/m(x)µ(O01/m(x))is σ−invariant, µ∞ is σ−ergodic but1pm(x)≤ µ∞(AO1/m(x)µ(O01/m(x))) ≤pm(x)− 1pm(x).Every subshift has at least one measure of maximal entropy (MME), i.e. a measure whoseentropy is the same as the topological entropy of the subshift. A 1D subshift is irreducible if52for every pair of balls U, V there exists j ∈ Z such that σjU ∩ V 6= ∅. A well known result ofShannon and Parry state that every irreducible 1D SFT admits a unique MME, and it always hasfull support [53]. The MME of a fullshift is the uniform Bernoulli measure.Note that we are only discussing measures of maximal entropy with respect to the shift not toφ.Theorem 3.2.15 (Coven-Paul [12]). Let X be a 1D irreducible SFT with a unique MME, and (X,φ)a CA. Then (X,φ) is surjective if and only if it preserves the MME. In particular if φ : AZ → AZis a CA, then (X,φ) is surjective if and only if it preserves the uniform Bernoulli measure.Lemma 3.2.16. Let (X,φ) be CA. Assume that φ preserves a measure with full support. If x isan equicontinuity point then x is recurrent (x ∈ R(φ)).Proof. Let x be an equicontinuity point and m ∈ N. There exists n ≥ 2m such that B1/n(x) ⊂O1/2m(x). We have that φ preserves a fully supported measure. Using Poincare’s recurrence theoremwe conclude there exists j ∈ N such that φjB1/n(x)∩B1/n(x) 6= ∅. This implies that φjx ∈ B1/m(x),and thus x ∈ R(φ).In [8] it was asked under which conditions the limit measure, under a 1D CA, of σ−ergodicmeasures that give full measure to equicontinuity points i.e. µ(EQ(φ) = 1, is σ−ergodic, a measureof maximal entropy or φ−ergodic. We address those questions (the first two in this section and thelast one in the next section).Theorem 3.2.17. Let X be a 1D irreducible SFT, (X,φ) a surjective CA, and µ a σ−ergodicmeasure with µ(EQ(φ)) = 1. Then µ∞ is σ−ergodic if and only if µ is φ−invariant.Proof. If µ = φµ, then µ∞ = µ is σ−ergodic.Since φ is surjective by Shannon-Parry and Coven-Paul we obtain that φ preserves a fullysupported measure. By Proposition 3.1.25 we have that µ(LEP (φ)) = 1. Using Lemma 3.2.16 andLemma 3.1.11 we get that µ(LP (φ)) = µ(LEP (φ) ∩R(φ)) = 1; and hence (X,φ) is µ− LP. UsingTheorem 3.2.14 we obtain µ = φµ.The proof of the following result is similar.Theorem 3.2.18. Let X be a 1D irreducible SFT, (X,φ) a CA, and µ a σ−ergodic measure withµ(EQ(φ)) = 1. Then µ∞ is the MME if and only if µ is the MME and (X,φ) is surjective.Proof. If µ is the MME and µ = φµ, then µ∞ = µ is the MME.Assume µ∞ is the MME; hence it is σ−ergodic and has full support. By Proposition 3.1.25we have that µ(LEP (φ)) = 1. Using Lemma 3.2.16 and Lemma 3.1.11 we get that µ(LP (φ)) =µ(LEP (φ) ∩ R(φ)) = 1; and hence (X,φ) is µ − LP. Using Theorem 3.2.14 we obtain µ = φµ =µ∞.53There is interest in dynamical systems such that the orbit of the measure will converge in somesense to a measure of maximal entropy or an equilibrium measure (for example see Section 4.4 of[39]). It has been shown that the Markov measures under 1D linear permutive CA converge inCesaro sense to the measure of maximal entropy (e.g. [46][54][48]). Theorem 3.2.18 shows that if ameasure µ is not the measure of maximal entropy and the equicontinuity points have full measurethen the limit measure will not be the measure of maximal entropy.Under some conditions these results hold for multidimensional subshifts.One of the implications of the mentioned theorem by Coven-Paul has been generalized. Let Xbe a subshift with a unique MME, and (X,φ) a CA. If (X,φ) is surjective then it preserves theMME (Theorem 3.3 in [49]). The proof of the following results are almost the same as the proofsfor Theorems 3.2.18 and 3.2.17.Theorem 3.2.19. Let X be an subshift with dense periodic points and a unique and fully supportedMME, (X,φ) a surjective CA, and µ a σ−ergodic measure with µ(EQ(φ)) = 1. Then µ∞ isσ−ergodic if and only if µ is φ−invariant.Theorem 3.2.20. Let X be an subshift with dense periodic points and a unique and fully supportedMME, (X,φ) a CA, and µ a σ−ergodic measure with µ(EQ(φ)) = 1. Then µ∞ is the MME if andonly if µ is the MME and µ = φµ.For µ− LEP systems we can show sufficient conditions for the σ−ergodicity of µ∞.Lemma 3.2.21. Let µ be a φ − LEP measure.Then µ∞ is σ−ergodic if and only if for everyx ∈ LP (φ).µ∞(AO1/m(x)a ) ={1 if a = µ∞(O1/m(x))0 otherwise.Proof. The ⇒) implication is given by the pointwise ergodic theorem.If the equation is satisfied then the pointwise ergodic theorem conclusion holds for all sets of theformO1/m(x) with x ∈ LPm(φ). By Proposition 3.2.6 µ∞(LP (φ)) = 1. Since{O1/m(x) | m ∈ N and x ∈ LPm(φ)}generates the Borel sigma algebra (intersected with LP (φ)), we conclude µ∞ is σ−ergodic (see [59]pg.41.)Theorem 3.2.22. Let µ be a φ−LEP , σ−ergodic measure. If for every orbit ball O, with µ∞(O) >0, there exists NO such thatφnµ(O) = µ∞(O) for n ≥ NO,then µ∞ is σ−ergodic.Proof. Let m ∈ N, x ∈ LPm(φ) and 0 ≤ q < pm(x). From the proof of equation (3.2.1) ofLemma 3.2.5 one can see that for every 0 ≤ q < pm(x), φpm(x)n+qµ(O1/m(x)) is non-decreasing and54converges (as n→∞). Using this and the hypothesis we have that there exists N such thatφpmn+qµ(O1/m(x)) = µ∞(O1/m(x)) for n ≥ N.This impliesµ(O−q1/m(x)) = µ∞(O1/m(x)).Using Lemma 3.2.12 we obtainµ∞(AO1/m(x)a ) =1pm(x)pm(x)−1∑r=0µ(AO−r1/m(x)a ) for every a ∈ R.This impliesµ∞(AO1/m(x)a ) = µ(AO−q1/m(x)a ) for every a ∈ R.Using the σ−ergodicity of µ we getµ(AO−q1/m(x)a ) ={1 if a = µ(O−q1/m(x))0 otherwise.Hence, we obtainµ∞(AO1/m(x)a ) ={1 if a = µ∞(O1/m(x))0 otherwise.Using the previous lemma we conclude that µ∞ is σ−ergodic.3.3 φ−ErgodicityIn this section we are interested in measure preserving topological dynamical systems.We will now remind the reader of some definitions involving measure preserving systems. We say(X,T, µ) is a measure preserving transformation if (X,µ) is Lebesgue probability space, T : X → Xis measurable and Tµ = µ. When we say µ is ergodic we also assume it is invariant under T .Two measure preserving transformations (X1, T1, µ1) and (X2, T2, µ2) are isomorphic (measur-ably) if there exists an invertible measure preserving transformation f : (X1, µ1)→ (X2, µ2), suchthat the inverse is measure preserving and T2 ◦ f = f ◦ T1.The spectral theory for dynamical systems (TDS and measure preserving transformations) isuseful for studying rigid transformations. We will give the definitions and state the most importantresults. (For more details and proofs see [59]).A measure preserving transformation T on a measure space (M,µ) generates a unitary linearoperator on the Hilbert space L2(M,µ), by UT : f 7→ f ◦ T, known as the Koopman operator. The55spectrum of the Koopman operator is called the spectrum of the measure preserving transformation.The spectrum is pure point or discrete if there exists an orthonormal basis for L2(M,µ) whichconsists of eigenfunctions of the Koopman operator. The spectrum is rational if the eigenvaluesare complex roots of unity. Classical results by Halmos and Von Neumann state that two ergodicmeasure preserving transformation with discrete spectrum have the same group of eigenvalues if andonly if they are isomorphic, and that an ergodic measure preserving transformation has pure pointspectrum if and only if it is isomorphic to a rotation on a compact metric group. The eigenfunctionsof a rotation on a compact group are generated by the characters of the group. Discrete spectrumcan be characterized for topological dynamical systems using a weak forms of µ−equicontinuity[22].Example 3.3.1. Let S = (s0, s1, ...) be a finite or infinite sequence of integers larger or equalthan 1. The S−adic odometer is the +(1, 0, ...) (with carrying) map defined on the compact setD =∏i≥0 Zsi (for a survey on odometers see [14]).These transformations are also called adding machines. An ergodic measure preserving trans-formation has discrete rational spectrum if and only if it is isomorphic to an odometer.Any odometer can be embedded in a CA [13].Theorem 3.3.2. Let (X,φ) be a CA and µ an invariant probability measure. If (X,φ) is µ−LEPthen (X,φ, µ) has discrete rational spectrum.Proof. Using Proposition 3.2.6 we have that (X,φ) is µ− LP.Let m ∈ N, y ∈ LP (φ) and i =√−1. We define λm,y := e2pii/pm(y) ∈ C andfm,y,k :=pm(y)−1∑j=0λj·km,y · 1O1/m(φjy) ∈ L2(X,µ).Using the fact that (X,φ) is µ− LP, y ∈ LP (φ) we have that O1/m(φpm(y)y) = O1/m(y) andUφ1O1/m(φjy) ={1O1/m(φj−1y) if 1 ≤ j < pm(y)− 11O1/m(φpm(y)−1y) if j = 0.This implies thatUφfm,y,k =pm(x)−1∑j=0λj·km,y · 1Bom(φj−1y)=pm(y)−1∑j=0λ(j+1)·km,y · 1Bom(φjy)= λkm,yfm,y,k56Thus fm,y,k is an eigenfunction corresponding to the eigenvalue λkm,y, which is a complex root ofunity.Considering thatpm(y)−1∑k=0λj·km,y ={0 if j > 0pm(y) if j = 0,we obtain1pm(y)pm(y)−1∑k=0fm,y,k = 1Om(y).This means 1Om(y) ∈ Span {fm,y,k}k .Let n ∈ N, and x ∈ X. Since (X,φ) is µ− LP and X is a Cantor set there exists a sequence{yi} ⊂ LP (φ) such that On(yi) are disjoint and µ(Bn(x)) = µ(∪On(yi)). This means 1Bn(x) can beapproximated in L2 by elements inSpan{1Om(y) : m ∈ N and y ∈ LP (φ)}.Since the closure of Span{1Bn(x) : n ∈ N, x ∈ X}is L2(X,µ) we conclude the closure of Span {fm,y,k}m,y,kis L2(X,µ).This implies that every ergodic µ− LEP CA is isomorphic to an odometer.We obtain a stronger result if the measure is σ−invariant.Proposition 3.3.3. Let (X,φ) be a µ−ergodic, µ − LEP CA. If µ is σ−invariant then (X,φ, µ)is isomorphic to a cyclic permutation on a finite set.Proof. By Proposition 3.2.6 µ(LP (φ)) = 1. Since µ−LEP CA are µ−equicontinuous, there existsx ∈ LP (φ) such that µ(O1(x)) > 0. Let O∞1 be the φ−orbit of O1(x) (the orbit ball centred at theorigin). We have that µ(O∞1 ) > 0. Since φ(O∞1 ) = O∞1 and µ is φ−ergodic, then µ(O∞1 ) = 1.We have that p1(O∞1 ) = {p1(x)} . This implies the 0th column of almost every point is periodicwith period p1(x). Since µ is σ−invariant we have that almost every point is periodic (with periodp1(x)).Using this and other assumptions we can characterize when a limit measure of a µ−equicontinuousCA is φ−ergodic.Corollary 3.3.4. Let X be a 1D SFT, µ be a σ−invariant measure, (X,φ) be a µ−equicontinuousCA. We have that (X,φ, µ∞) is isomorphic to a cyclic permutation on a finite set if and only ifµ∞ is φ−ergodic.57Bibliography[1] E. Akin, J. Auslander, and K. Berg. When is a transitive map chaotic? Ohio State Univ.Math. Res. Inst. Publ., 5(2):25–40, 1996.[2] J. Auslander. Mean-l-stable systems. Illinois Journal of Mathematics, 3(4):566–579, 1959.[3] J. Auslander. Minimal Flows and Their Extensions. North-Holland Mathematics Studies.Elsevier Science, 1988.[4] J. Auslander and J. A. Yorke. Interval maps, factors of maps, and chaos. TohokuMathematical Journal, 32(2):177–188, 1980.[5] P. Billingsley. Convergence of Probability Measures. Wiley Series in Probability andStatistics. Wiley, 2009.[6] F. 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