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

TL;DR

The paper develops a Fraïssé-theoretic framework to study automorphism groups of measures on the Cantor space, focusing on good measures characterized by the Subset Condition. It proves that good measures coincide with maximal ultrahomogeneous measures, and provides two equivalent characterizations via Fraïssé limits and maximal ultrahomogeneity, tying generic conjugacy properties to the algebraic structure of the clopen-values set $V_{\mu}$. A central contribution is the explicit link between the directedness of categories ${\mathcal M}_V$ (and related cofinal amalgamation) and Rokhlin-type properties for $\mathrm{Homeo}(2^{\mathbb{N}},\mu)$, enabling criteria for when dense or comeager conjugacy classes occur, especially in the ${\mathbb{Q}}$-like or ring-like cases. The results lay a rigorous foundation for Part II, which will examine transitivity and finer dynamical properties of these automorphism groups.

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 Fraïssé theory.

Theorems & Definitions (52)

• Theorem A: see Theorem \ref{['thm:fraisseisgood']}
• Theorem B: see Theorem \ref{['thm:Q-ring']}
• Lemma 2.1
• proof
• Proposition 3.1
• proof
• Definition 3.2
• Proposition 3.3
• proof
• Theorem 3.4