Automorphism groups of measures on the Cantor space. Part I: Good measures and Rokhlin properties

TL;DR

This paper investigates the automorphism groups of measures on the Cantor space, focusing on 'good' measures and their properties related to dense and comeager conjugacy classes, using Fraïssé theory.

Contribution

It characterizes good measures with rational values on clopen sets whose automorphism groups have comeager conjugacy classes, linking measure properties with group dynamics.

Findings

Identifies conditions for dense conjugacy classes in automorphism groups.

Characterizes good measures with rational values on clopen sets.

Uses Fraïssé theory to analyze measure automorphisms.

Abstract

We study criteria for the existence of a dense or comeager conjugacy class in the automorphism group of a given measure on the Cantor space. We concentrate on good measures, defined by Akin [\emph{Trans.\ Amer.\ Math.\ Soc.} \textbf{357} (2005), no. 7, 2681--2722], which we characterize as a particular subclass of ultrahomogeneous measures. We determine good measures with rational values on clopen sets whose automorphism group admits a comeager conjugacy class. Our approach uses the Fra\"{i}ss\'{e} theory.