Positive expansiveness versus network dimension in symbolic dynamical systems

A symbolic dynamical system is a continuous transformation Φ:X⟶X of closed subset X⊆AV, 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 (AV,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.