Factoring strongly irreducible group shift actions onto full shifts of lower entropy

TL;DR

This paper proves that certain strongly irreducible group shift actions can be factored onto lower-entropy full shifts for amenable groups with the comparison property, under specific aperiodicity conditions.

Contribution

It establishes a factorization result for strongly irreducible group shifts onto full shifts of lower entropy in the context of amenable groups with the comparison property.

Findings

Strong irreducibility and aperiodicity imply factorization onto lower-entropy full shifts.

Factorization holds when the log of the number of symbols is less than the topological entropy.

Results extend the understanding of symbolic dynamics for amenable group actions.

Abstract

We show that if $G$ is a a countable amenable group with the comparison property, and $X$ is a strongly irreducible $G$-shift satisfying certain aperiodicity conditions, then $X$ factors onto the full $G$-shift over $N$ symbols, so long as the logarithm of $N$ is less than the topological entropy of $G$.