# Conjugacy in the Cantor Set Automorphism Group

## TL;DR

This paper surveys and extends results on the conjugacy classes within the automorphism group of the Cantor set, focusing on the dynamics of surjective maps and their generic properties under conjugation.

## Contribution

It provides new insights into the structure of conjugacy classes in the Cantor set automorphism group and explores the dynamics of various subclasses of surjective maps.

## Key findings

- Existence of elements with dense conjugacy classes in certain subsets.
- Characterization of generic elements in these subsets.
- Extension of previous results on the automorphism group's dynamics.

## Abstract

We survey, and extend, results on the adjoint action of the homeomorphism group $H(X)$ on the space of surjective continuous maps, $C_s(X)$, where $X$ is a Cantor set. We look also at the restriction of the action to various dynamically defined subsets of $C_s(X)$, e. g. the sets of topologically transitive maps, chain transitive maps, chain mixing maps, etc. In each case, we consider whether there exist elements with a dense conjugacy class and if so, what the generic elements look like.