The group of homeomorphisms of the Cantor set has ample generics

TL;DR

This paper proves that the group of homeomorphisms of the Cantor set has ample generics, meaning it has comeager conjugacy classes in all finite products, answering a question by Kechris and Rosendal.

Contribution

It establishes the existence of ample generics in the homeomorphism group of the Cantor set using projective Fraisse theory, providing new insights into its topological dynamics.

Findings

The group of homeomorphisms of the Cantor set has ample generics.

The generic tuple can be obtained as a limit of a projective Fraisse family.

A proof of the existence of a generic homeomorphism in this group is provided.

Abstract

We show that the group of homeomorphisms of the Cantor set $H(K)$ has ample generics, that is, for every $m$ the diagonal conjugacy action $g\cdot(h_1,h_2,..., h_m)=(gh_1g^{-1},gh_2g^{-1},..., gh_mg^{-1})$ of $H(K)$ on $H(K)^m$ has a comeager orbit. This answers a question of Kechris and Rosendal. We show that the generic tuple in $H(K)^m$ can be taken to be the limit of a certain projective Fraisse family. We also present a proof of the existence of the generic homeomorphism of the Cantor set in the context of the projective Fraisse theory.