Symbolic dynamics on amenable groups: the entropy of generic shifts

Abstract

Let $G$ be a finitely generated amenable group. We study the space of shifts on $G$ over a given finite alphabet $A$. We show that the zero entropy shifts are generic in this space, and that more generally the shifts of entropy $c$ are generic in the space of shifts with entropy at least $c$. The same is shown to hold for the space of transitive shifts and for the space of weakly mixing shifts. As applications of this result, we show that for every entropy value $c\in [0,\log |A|]$ there is a weakly mixing subshift of $A^G$ with entropy $c$. We also show that the set of strongly irreducible shifts does not form a $G_{\delta }$ in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space.

References in corpus (1)

• Tilings of amenable groups

Cited by in corpus (10)

• Pseudo-Orbit Tracing and Algebraic actions of countable amenable groups
• Bernoulli disjointness
• The comparison property of amenable groups
• On the entropies of subshifts of finite type on countable amenable groups
• Strong amenability and the infinite conjugacy class property
• Subsystem entropies of shifts of finite type and sofic shifts on countable amenable groups
• An embedding theorem for subshifts over amenable groups with the comparison property
• Ubiquity of entropies of intermediate factors
• Minimal subdynamics and minimal flows without characteristic measures
• Homomorphisms from aperiodic subshifts to subshifts with the finite extension property