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

超空间与测度动力学中的均值等连续及相关性质

Mean Equicontinuity and Related Properties in Hyperspace and Measure Dynamics

研究动力系统(X,T)诱导的测度系统(ℳ(X),T)和超空间系统(𝒦(X),T)的均值等连续性，证明(X,T)的直径均值等连续等价于(ℳ(X),T)的直径均值等连续，并给出(𝒦(X),T)的均值等连续等价条件。

非均匀元胞自动机的可逆性、平衡性与膨胀性

Reversibility, balance and expansivity of non-uniform cellular automata

本文研究一维非均匀元胞自动机，证明均匀回归规则分布下双射性蕴含可逆性，满射或双射蕴含平衡性，并给出膨胀性蕴含敏感性的结论。

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 systems with discrete spectrum and zero sequence entropy. In the topological category we show systems with zero topological sequence entropy are strictly contained in the 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 1-1 on a set of full Haar measure). In the measure category we show that for ergodic topo…

The generic limit set of cellular automata

In this article, we consider a topological dynamical system. The generic limit set is the smallest closed subset which has a comeager realm of attraction. We study some of its topological properties, and the links with equicontinuity and sensitivity. We emphasize the case of cellular automata, for which the generic limit set is included in all subshift attractors, and discuss directional dynamics, as well as the link with measure-theoretical similar notions.

A family of sand automata

We study some dynamical properties of a family of two-dimensional cellular automata: those that arise from an underlying one dimensional sand automaton whose local rule is obtained using a latin square. We identify a simple sand automaton G whose local rule is algebraic, and classify this automaton as having equicontinuity points, but not being equicontinuous. We also show it is not surjective. We generalise some of these results to a wider class of sand automata.

Non-Uniform Cellular Automata: classes, dynamics, and decidability

The dynamical behavior of non-uniform cellular automata is compared with the one of classical cellular automata. Several differences and similarities are pointed out by a series of examples. Decidability of basic properties like surjectivity and injectivity is also established. The final part studies a strong form of equicontinuity property specially suited for non-uniform cellular automata.

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 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) fibres. 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 …

Some topological properties of one dimensional cellular automata

Cellular automata, CA for short are continuous maps defined on the set of configurations over a finite alphabet A that commutes with the shift. They are characterized by the existence of local function which determine by local behavior the image of an element of the configurations space. We will study periodic factors of cellular automata. We classify periodic factors according to their periods. We show that for a surjective cellular automaton with equicontinuity points but without being equicontinuous there is an infinity of classes of equivalence of periodic factors

Traced communication complexity of cellular automata

We study cellular automata with respect to a new communication complexity problem: each of two players know half of some finite word, and must be able to tell whether the state of the central cell will follow a given evolution, by communicating as little as possible between each other. We present some links with classical dynamical concepts, especially equicontinuity, expansiveness, entropy and give the asymptotic communication complexity of most elementary cellular automata.

关于多元 diam 平均等度连续性与频繁稳定性的注记

A note on multivariate diam mean equicontinuity and frequent stability

本文证明了极小拓扑动力系统的 diam-平均等度连续性与频繁稳定性的二分性质，并在群为阿贝尔或顺从等条件下，通过弱平均敏感元组给出了 $m$-正则性的局部等价刻画。

D维元胞自动机的无理特征值

Irrational eigenvalues of D-Dimensional Cellular automata

本文研究D维元胞自动机的谱性质，从拓扑与遍历视角刻画其谱，并证明含完全阻塞模式或正测度等度连续点集的元胞自动机不存在可测无理特征值。

关于细胞自动机在群上的Kurka二元性

On K{\r u}rka's dichotomy for cellular automata on groups

本文研究了细胞自动机在群上的拓扑动力学，证明了虚Z群上初始条件敏感性与无等连续点的等价性，但自由群上不成立，并指出该性质在子群上也成立。

元胞自动机、渗流与动力学二分法

Cellular automata, percolation and dynamical dichotomies

本文建立了群Cayley图上的渗流与元胞自动机动力学多样性的联系，证明了有限生成群满足Gilman二分法当且仅当其具有平凡渗流阈值，并得出可数群满足该二分法当且仅当其局部虚循环的结论。

Cellular automata and substitutions in topological spaces defined via edit distances

The Besicovitch pseudo-metric is a shift-invariant pseudo-metric on the set of infinite sequences, that enjoys interesting properties and is suitable for studying the dynamics of cellular automata. They correspond to the asymptotic behavior of the Hamming distance on longer and longer prefixes. Though dynamics of cellular automata were already studied in the literature, we propose the first study of the dynamics of substitutions. We characterize those that yield a well-defined dynamical system as essentially the uniform ones. We also explore a variant of this pseudo-metric, the Feldman pseudo-…