Conjugacy and Dynamics in Almost Automorphism Groups of Trees

TL;DR

This paper characterizes conjugacy of almost automorphisms of regular trees by combining existing classifications and explores their dynamic properties, advancing understanding of their structure and behavior.

Contribution

It provides a criterion for conjugacy of almost automorphisms of regular trees and analyzes their dynamics, integrating prior classifications and solutions to the conjugacy problem.

Findings

Conjugacy criteria for almost automorphisms of regular trees.

Connection between automorphism classification and Thompson's V.

Analysis of the dynamic behavior of tree almost automorphisms.

Abstract

We determine when two almost automorphisms of a regular tree are conjugate. This is done by combining the classification of conjugacy classes in the automorphism group of a level-homogeneous tree by Gawron, Nekrashevych and Sushchansky and the solution of the conjugacy problem in Thompson's $V$ by Belk and Matucci. We also analyze dynamics of tree almost automorphisms.