Direct limits of large orbits and the Knaster continuum homeomorphism group

TL;DR

This paper investigates the structure of the homeomorphism group of the universal Knaster continuum, showing it contains a large open subgroup with a dense conjugacy class, and develops a general framework for comeager orbits in Polish group actions.

Contribution

It establishes the existence of a comeager conjugacy class within an open subgroup of the homeomorphism group of the Knaster continuum and introduces a general method for identifying comeager orbits in Polish groups.

Findings

The homeomorphism group of the Knaster continuum has an open subgroup with a comeager conjugacy class.

A general criterion for comeager orbits in Polish group actions is developed.

Analysis of the conjugacy action of Homeo_+[0,1] supports the main results.

Abstract

The main result is that the group $\textrm{Homeo} (K)$ of homeomorphisms of the universal Knaster continuum contains an open subgroup with a comeager conjugacy class. Actually, this open subgroup is the very natural subgroup consisting of degree-one homeomorphisms. We give a general fact about finding comeager orbits in Polish group actions which are approximated densely by direct limits of actions with comeager orbits. The main theorem comes as a result of this fact and some finer analysis of the conjugacy action of the group $\textrm{Homeo}_+[0,1]$.