Local $\mathcal{P}$ entropy and stabilized automorphism groups of subshifts

TL;DR

This paper introduces local $ ext{P}$ entropy to analyze stabilized automorphism groups of subshifts, revealing their structure is closely tied to the topological entropy of the shift, especially for mixing shifts of finite type.

Contribution

It defines local $ ext{P}$ entropy for groups and applies it to classify stabilized automorphism groups of full shifts based on topological entropy.

Findings

Local $ ext{P}$ entropy is determined by topological entropy for mixing shifts of finite type.

Complete classification of stabilized automorphism groups of full shifts.

Establishes a link between group entropy and dynamical complexity.

Abstract

For a homeomorphism $T \colon X \to X$ of a compact metric space $X$, the stabilized automorphism group $\text{Aut}^{(\infty)}(T)$ consists of all self-homeomorphisms of $X$ which commute with some power of $T$. Motivated by the study of these groups in the context of shifts of finite type, we introduce a certain entropy for groups called local $\mathcal{P}$ entropy. We show that when $(X,T)$ is a non-trivial mixing shift of finite type, the local $\mathcal{P}$ entropy of the group $\text{Aut}^{(\infty)}(T)$ is determined by the topological entropy of $(X,T)$. We use this to give a complete classification of the isomorphism type of the stabilized automorphism groups of full shifts.