Mean Equicontinuity Research Articles

The research of mean equicontinuity, mean sensitivity, and shadowing property

We study the dynamical properties of mean equicontinuity, mean sensitivity, tracking property, and periodic points set in the hyperspace of uniform space. Let (Y,ξ) be uniform space, T:Y→Y be uniformly continuous and (C(Y),Cξ) be hyperspace of (Y,ξ).Then, we can get some conclusions: (a) T is mean equicontinous if and only if the induced map CT is mean equicontinous; (b) if the induced map CT is mean sensitive, then T is mean sensitive; (c) the induced map CT has tracking property implying that T has tracking property; (d) P(T) is dense in Y implying that P(CT) is dense in C(Y). In addition, we also study dynamical property of (G,h)− tracking property in metric G− space, and prove that if the map T has (G,h)− tracking property, then, for any k>1, the map Tk has (G,h)− tracking property.

Weak Mean Equicontinuity for a Countable Discrete Amenable Group Action

The weak mean equicontinuity for a countable discrete amenable group G acting continuously on a compact metrizable space X is studied. It is shown that weak mean equicontinuity, continuously pointwise ergodicity and uniformity are coincided. Moreover, we prove that (X, G) is mean equicontinuous if and only if the product system \((X \times X,G)\) is weak mean equicontinuous.

The structure of mean equicontinuous group actions

We study mean equicontinuous actions of locally compact σ-compact amenable groups on compact metric spaces. In this setting, we establish the equivalence of mean equicontinuity and topo-isomorphy to the maximal equicontinuous factor and provide a characterization of mean equicontinuity of an action via properties of its product. This characterization enables us to show the equivalence of mean equicontinuity and the weaker notion of Besicovitch-mean equicontinuity in fairly high generality, including actions of abelian groups as well as minimal actions of general groups. In the minimal case, we further conclude that mean equicontinuity is equivalent to discrete spectrum with continuous eigenfunctions. Applications of our results yield a new class of non-abelian mean equicontinuous examples as well as a characterization of those extensions of mean equicontinuous actions which are still mean equicontinuous.

Local non-periodic order and diam-mean equicontinuity on cellular automata

Diam-mean equicontinuity is a dynamical property that has been of use in the study of non-periodic order. Using some type of “local” skew product between a shift and an odometer looking cellular automaton (CA), we will show there exists an almost diam-mean equicontinuous CA that is not almost equicontinuous, (and hence not almost locally periodic). Previously, we constructed a CA that is almost mean equicontinuous [4] but not almost diam-mean equicontinuous [3].

Mean equicontinuity, almost automorphy and regularity

The aim of this article is to obtain a better understanding and classification of strictly ergodic topological dynamical systems with (measurable) discrete spectrum. To that end, we first determine when an isomorphic maximal equicontinuous factor map of a minimal topological dynamical system has trivial (one point) fibers. In other words, we characterize when minimal mean equicontinuous systems are almost automorphic. Furthermore, we investigate another natural subclass of mean equicontinuous systems, so-called diam-mean equicontinuous systems, and show that a minimal system is diam-mean equicontinuous if and only if the maximal equicontinuous factor is regular (the points with trivial fibers have full Haar measure). Combined with previous results in the field, this provides a natural characterization for every step of a natural hierarchy for strictly ergodic topological models of ergodic systems with discrete spectrum. We also construct an example of a transitive almost diam-mean equicontinuous system with positive topological entropy, and we give a partial answer to a question of Furstenberg related to multiple recurrence.

Discrete spectrum for amenable group actions

<p style='text-indent:20px;'>In this paper, we study discrete spectrum of invariant measures for countable discrete amenable group actions.</p><p style='text-indent:20px;'>We show that an invariant measure has discrete spectrum if and only if it has bounded measure complexity. We also prove that, discrete spectrum can be characterized via measure-theoretic complexity using names of a partition and the Hamming distance, and it turns out to be equivalent to both mean equicontinuity and equicontinuity in the mean.</p>

Mean equicontinuity and mean sensitivity on cellular automata

AbstractWe show that a cellular automaton (or shift-endomorphism) on a transitive subshift is either almost equicontinuous or sensitive. On the other hand, we construct a cellular automaton on a full shift (hence a transitive subshift) that is neither almost mean equicontinuous nor mean sensitive.

The Mean Sensitivity and Mean Equicontinuity in Uniform Spaces

Some characteristics of mean sensitivity and Banach mean sensitivity using Furstenberg families and inverse limit dynamical systems are obtained. The iterated invariance of mean sensitivity and Banach mean sensitivity are proved. Applying these results, the notion of mean sensitivity and Banach mean sensitivity is extended to uniform spaces. It is proved that a point-transitive dynamical system in a Hausdorff uniform space is either almost (Banach) mean equicontinuous or (Banach) mean sensitive.

Formula omitted]-equicontinuity and an analogue of Auslander-Yorke dichotomy theorem

In this paper, we introduce a new type of F-equicontinuity, which is the opposite of the existence of kF-sensitive pairs almost everywhere and show an analogue of Auslander-Yorke dichotomy theorem. Precisely, under the condition that kF is a translation invariant family, we prove that a transitive system either has F-sensitive pairs almost everywhere or is almost kF-equicontinuous of new type. Also we show that the new type of F-equicontinuity is preserved by an open factor map and consider the implication between the new type of F-equicontinuity and mean equicontinuity.

Mean equicontinuity, complexity and applications

We will review the recent development of the research related to mean equicontinuity, focusing on its characterizations, its relationship with discrete spectrum, topo-isomorphy, and bounded complexity. Particularly, the application of the complexity function in the mean metric to the Sarnak and the logarithmic Sarnak Mobius disjointness conjecture will be addressed.

A new metric for statistical properties of long time behaviors

Let (X,T) be a topological dynamical system with metric d. We define a new function F‾(x,y)=limsupn→+∞infσ∈Sn⁡1n∑k=1nd(Tkx,Tσ(k)y) by using permutation group Sn. It's shown F(x,y)=limn→+∞⁡infσ∈Sn⁡1n∑k=1nd(Tkx,Tσ(k)y) exists when x,y∈X are generic points. Applying this function, we prove (X,T) is uniquely ergodic if and only if F‾(x,y)=0 for any x,y∈X. The characterizations of ergodic measures and physical measures by F‾(x,y) are given. We introduce the notion of weak mean equicontinuity and prove that (X,T) is weak mean equicontinuous if and only if the time averages f⁎(x)=limn→+∞⁡1n∑k=1nf(Tkx) are continuous for all f∈C(X).

Bounded complexity, mean equicontinuity and discrete spectrum

We study dynamical systems that have bounded complexity with respect to three kinds metrics: the Bowen metric $d_{n}$, the max-mean metric $\hat{d}_{n}$ and the mean metric $\bar{d}_{n}$, both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_{n}$ (respectively $\hat{d}_{n}$) if and only if it is equicontinuous (respectively equicontinuous in the mean). However, we construct minimal systems that have bounded complexity with respect to $\bar{d}_{n}$ but that are not equicontinuous in the mean. It turns out that an invariant measure $\unicode[STIX]{x1D707}$ on $(X,T)$ has bounded complexity with respect to $d_{n}$ if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-equicontinuous. Meanwhile, it is shown that $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\hat{d}_{n}$ if and only if $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\bar{d}_{n}$, if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-mean equicontinuous and if and only if it has discrete spectrum.

Measure-theoretic mean equicontinuity and bounded complexity

Let (X,B,μ,T) be a measure preserving system. We say that a function f∈L2(X,μ) is μ-mean equicontinuous if for any ϵ>0 there is k∈N and measurable sets A1,A2,⋯,Ak with μ(⋃i=1kAi)>1−ϵ such that whenever x,y∈Ai for some 1≤i≤k, one haslimsupn→∞1n∑j=0n−1|f(Tjx)−f(Tjy)|<ϵ. Measure-theoretic complexity with respect to f is also introduced. It is shown that f is an almost periodic function if and only if f is μ-mean equicontinuous if and only if μ has bounded complexity with respect to f.Ferenczi studied measure-theoretic complexity using α-names of a partition and the Hamming distance. He proved that if a measure preserving system is ergodic, then the complexity function is bounded if and only if the system has discrete spectrum. We show that this result holds without the assumption of ergodicity.

Null systems in the non-minimal case

In this paper, it is shown that if a dynamical system is null and distal, then it is equicontinuous. It turns out that a null system with closed proximal relation is mean equicontinuous. As a direct application, it follows that a null dynamical system with dense minimal points is also mean equicontinuous. Meanwhile, a distal system with trivial $\text{Ind}_{\text{fip}}$-pairs and a non-trivial regionally proximal relation of order $\infty$ are constructed.

Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems

We show that an $R^d$-topological dynamical system equipped with an invariant ergodic measure has discrete spectrum if and only it is $\mu$-mean equicontinuous (proven for $Z^d$ before). In order to do this we introduce mean equicontinuity and mean sensitivity with respect to a function. We study this notion in the topological and measure theoretic setting. In the measure theoretic case we characterize almost periodic functions and in the topological case we show that weakly almost periodic functions are mean equicontinuous (the converse does not hold).

A Note on Mean Equicontinuity

In this note, it is shown that several results concerning mean equicontinuity proved before for minimal systems are actually held for general topological dynamical systems. Particularly, it turns out that a dynamical system is mean equicontinuous if and only if it is equicontinuous in the mean if and only if it is Banach (or Weyl) mean equicontinuous if and only if its regionally proximal relation is equal to the Banach proximal relation. Meanwhile, a relation is introduced such that the smallest closed invariant equivalence relation containing this relation induces the maximal mean equicontinuous factor for any system.

On dynamics of graph maps with zero topological entropy

We explore the dynamics of graph maps with zero topological entropy. It is shown that a continuous map f on a topological graph G has zero topological entropy if and only if it is locally mean equicontinuous, that is the dynamics on each orbit closure is mean equicontinuous. As an application, we show that Sarnak’s Möbius disjointness conjecture is true for graph maps with zero topological entropy. We also extend several results known in interval dynamics to graph maps. We show that a graph map has zero topological entropy if and only if there is no 3-scrambled tuple if and only if the proximal relation is an equivalence relation; a graph map has no scrambled pairs if and only if it is null if and only if it is tame.

Isomorphic extensions and applications

If $\pi\colon (X,T)\to(Z,S)$ is a topological factor map between uniquely ergodic topological dynamical systems, then $(X,T)$ is called an isomorphic extension of $(Z,S)$ if $\pi$ is also a measure-theoretic isomorphism. We consider the case when the systems are minimal and we pay special attention to equicontinuous $(Z,S)$. We first establish a characterization of this type of isomorphic extensions in terms of mean equicontinuity, and then show that an isomorphic extension need not be almost one-to-one, answering questions of Li, Tu and Ye.

Weak forms of topological and measure-theoretical equicontinuity: relationships with discrete spectrum and sequence entropy

We define weaker forms of topological and measure-theoretical equicontinuity for topological dynamical systems, and we study their relationships with sequence entropy and systems with discrete spectrum. We show that for topological systems equipped with ergodic measures having discrete spectrum is equivalent to$\unicode[STIX]{x1D707}$-mean equicontinuity. In the purely topological category we show that minimal subshifts with zero topological sequence entropy are strictly contained in diam-mean equicontinuous systems; and that transitive almost automorphic subshifts are diam-mean equicontinuous if and only if they are regular (i.e. the maximal equicontinuous factor map is one–one on a set of full Haar measure). For both categories we find characterizations using stronger versions of the classical notion of sensitivity. As a consequence, we obtain a dichotomy between discrete spectrum and a strong form of measure-theoretical sensitivity.

Mean Equicontinuity, Sensitivity, Expansiveness and Distality of Product Dynamical Systems

The mean equicontinuity, mean sensitivity, mean expansiveness and mean distality of the product dynamical system (Π Xi, Π fi) have been studied when each of the constituent dynamical system (Xi, fi) has these properties and vice versa.