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$G_{\unicode[STIX]{x1D6FF}}$in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space.