Symbolic dynamics on amenable groups: the entropy of generic shifts

TL;DR

This paper investigates the generic properties of shifts over finitely generated amenable groups, revealing that zero entropy shifts are typical and that shifts with any given entropy are also generic within certain classes.

Contribution

It establishes the genericity of zero entropy and specified entropy shifts in the space of all shifts over amenable groups, extending understanding of their structural complexity.

Findings

Zero entropy shifts are generic in the space of shifts.

Shifts with any fixed entropy are generic among shifts with at least that entropy.

The set of strongly irreducible shifts is not a $G_\delta$ and these shifts are non-isolated.

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.