Centralizers of rank-1 homeomorphisms

TL;DR

This paper defines rank-1 homeomorphisms on zero-dimensional Polish spaces and proves that, under a non-degeneracy condition, such homeomorphisms have trivial centralizers, meaning they only commute with their powers.

Contribution

It introduces a formal definition of rank-1 homeomorphisms and establishes a condition ensuring their centralizer is trivial, advancing understanding of their symmetry properties.

Findings

Rank-1 homeomorphisms are defined on zero-dimensional Polish spaces.

Under a non-degeneracy condition, these homeomorphisms have trivial centralizers.

They only commute with their own powers.

Abstract

We give a definition for a rank-1 homeomorphism of a zero-dimensional Polish space X. We show that if a rank-1 homeomorphism of X satisfies a certain non-degeneracy condition, then it has trivial centralizer in the group of all homeomorphisms of X, i.e., it commutes only with its integral powers.