Positive expansiveness versus network dimension in symbolic dynamical systems

Abstract

a b s t r a c t A symbolic dynamical system is a continuous transformation Φ : X −→ X of closed subset X ⊆ A V , where A is a finite set and V is countable (examples include subshifts, odometers, cellular automata, and automaton networks). The function Φ induces a directed graph ('network') structure on V, whose geometry reveals information about the dynamical system (X, Φ). The dimension dim(V) is an exponent describing the growth rate of balls in this network as a function of their radius. We show that, if X has positive entropy and dim(V) > 1, and the system (A V , X, Φ) satisfies minimal symmetry and mixing conditions, then (X, Φ) cannot be positively expansive; this generalizes a well-known result of Shereshevsky about multidimensional cellular automata. We also construct a counterexample to a version of this result without the symmetry condition. Finally, we show that network dimension is invariant under topological conjugacies which are Hölder-continuous.Let X be Cantor space (the compact, perfect, zero-dimensional metrizable topological space, which is unique up to homeomorphism). A Cantor dynamical system is a continuous self-map Φ : X−→X. In addition to its intrinsic interest, the class of Cantor systems is important because it has two universal properties. First, any topological dynamical system on a compact metric space is a factor of a Cantor system; see [1, Corollary 3.9, p.106] or [2, p.1241]. Second, the Jewett-Krieger Theorem says that any ergodic measure-preserving system can be represented as a uniquely ergodic, minimal Cantor system [3, Section 4.4, p.188].If A is a finite set, and V is a countably infinite set, then the product space A V is a Cantor space. Thus, any Cantor dynamical system can be represented as a self-map Φ : A V −→A V , or more generally, as a self-map Φ : X−→X, where X ⊂ A V is a pattern space (a closed subset of A V ). We refer to the structure (A V , X, Φ) as a symbolic dynamical system. At an abstract topological level, any perfect pattern space X is homoeomorphic to Cantor space, so a perfect symbolic dynamical system is simply a Cantor dynamical system. What distinguishes symbolic dynamical systems is a particular way of representing X as a subset of some Cartesian product A V (so that an element of X corresponds to some V-indexed 'pattern' of 'symbols' in the alphabet A).The network of Φ is the digraph structure ( • →) on V defined as follows: for all v, w ∈ V, we have v • → w if and only if the value of Φ(x) w depends nontrivially on the value of x v . We say that (V, • →) has dimension δ if the cardinality of a ball of radius r grows like r δ as r→∞. (Note that δ is not necessarily an integer.) For example, if Φ : A Z D −→A Z D is a cellular automaton, then its network is just a Cayley digraph on Z D ; the dimension of this network is D.This paper explores the relationship between network dimension and the properties of (X, Φ) as a topological dynamical system. In Section 1, we formally define the dimension of a network (V, •→) and the entropy of a pattern space on ...

Expansive automata networks

“…Expansivity (positive or not) in cellular automata has received a lot of attention [16,17,18] and is still an active direction of research [19], one of the main open problem being the decidability of the property (see [20,Problem 19] or [21,Problem 7]).…”

Expansive Automata Networks

“…Expansivity (positive or not) in cellular automata has received a lot of attention [4,20,19] and is still an active direction of research [14], one of the main open problem being the decidability of the property (see [5,Problem 19] or [15,Problem 7]).…”

Pre-expansivity in cellular automata

“…In particular, the classical notion of (positive) expansivity has been applied to CA giving both a rich theory in the one-dimensional case [9,2,10] and a general inexistence result in essentially any other setting [11,12]. Even in the one-dimensional case where positive expansivity is equivalent to being conjugated to a one-sided subshift of finite type [13], it is interesting to note that outside the linear and bi-permutative examples, few construction techniques are known to produce positively expansive CA [14].…”

“…In this paper, we introduce a new dynamical property called pre-expansivity that both generalizes expansivity and refines pre-injectivity: it is the property of being expansive on asymptotic pairs. Our motivation is to better understand surjective CA and expansive-like dynamics, in particular in the higherdimensional case or in lattices where the classical notion of (positive)expansivity cannot be satisfied by any CA [11,12]. Pre-expansivity (resp.…”