Positive expansiveness versus network dimension in symbolic dynamical systems

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 dimensiondim(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.