Positive expansiveness versus network dimension in symbolic dynamical systems

TL;DR

This paper investigates how the network dimension influences positive expansiveness in symbolic dynamical systems, revealing that higher network dimensions prevent positive expansiveness under certain conditions and establishing invariance under specific conjugacies.

Contribution

It generalizes Shereshevsky's result to broader systems and introduces the invariance of network dimension under Holder-continuous conjugacies.

Findings

Positive entropy systems with network dimension > 1 are not positively expansive under minimal symmetry and mixing.

Counterexample shows the necessity of symmetry conditions.

Network dimension remains invariant under Holder-continuous topological conjugacies.

Abstract

A `symbolic dynamical system' is a continuous transformation F:X-->X of a closed perfect subset X of A^V, where A is a finite set and V is countable. (Examples include subshifts, odometers, cellular automata, and automaton networks.) The function F induces a directed graph structure on V, whose geometry reveals information about the dynamical system (X,F). 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^V,X,F) satisfies minimal symmetry and mixing conditions, then (X,F) 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 Holder-continuous.