Forum of Mathematics, Sigma (Jan 2024)

Minimal subdynamics and minimal flows without characteristic measures

  • Joshua Frisch,
  • Brandon Seward,
  • Andy Zucker

DOI
https://doi.org/10.1017/fms.2024.41
Journal volume & issue
Vol. 12

Abstract

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Given a countable group G and a G-flow X, a probability measure $\mu $ on 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?

Keywords