Minimal subdynamics and minimal flows without characteristic measures

TL;DR

This paper constructs minimal flows for any countable group that lack characteristic measures, addressing a question about the existence of such flows and exploring related minimal subdynamics questions.

Contribution

It provides the first known construction of minimal flows without characteristic measures for all countable groups, advancing understanding of group actions and invariant measures.

Findings

Constructed minimal flows without characteristic measures for all countable groups

Addressed questions on minimal subdynamics involving subgroups and minimal actions

Provided new insights into the structure of group actions without invariant measures

Abstract

Given a countable group $G$ and a $G$-flow $X$, a measure $\mu\in P(X)$ is called characteristic if it is $\mathrm{Aut}(X, G)$-invariant. Frisch and Tamuz asked about the existence of a minimal $G$-flow, for any group $G$, which does not admit a characteristic measure. We construct for every countable group $G$ such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group $G$ and a collection of infinite subgroups $\{\Delta_i: i\in I\}$, when is there a faithful $G$-flow for which every $\Delta_i$ acts minimally?