Positive expansiveness versus network dimension in symbolic dynamical systems

A ‘symbolic dynamical system’ is a continuous transformation Φ : X−→X of closed perfect subset X ⊆ A, 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 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 the digraph as a function of their radius. We show: if X has positive entropy and dim(V) > 1, and the system (A,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 [Kůr03, Corollary 3.9, p.106] or [BS89, p.1241]. Second, the Jewet-Krieger Theorem says that any ergodic measurepreserving system can be represented as a uniquely ergodic, minimal Cantor system [Pet89, §4.4, p.188]. If A is a finite set, and V is a countably infinite set, then the product space A is a Cantor space. Thus, any Cantor dynamical system can be represented as a self-map Φ : A−→A, or more generally, as a self-map Φ : X−→X , where X ⊂ A is a pattern space (a closed perfect subset of A). We refer to the structure (A,X ,Φ) as a symbolic dynamical system. At an abstract topological level, any pattern space X is homeomorphic to Cantor space, so a 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 (so that an element of X corresponds to some V-indexed ‘pattern’ of ‘symbols’ in the alphabet A).