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?

The Halpern–Läuchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles about products of perfect Polish spaces. These principles yield straightforward proofs of the Halpern–Läuchli theorem, and the same forcing from Harrington’s proof can force their consistency. We also show that these principles are not ZFC theorems by showing that they put lower bounds on the size of the continuum.

Given a G-flow X, let $\mathrm{Aut}(G, X)$ , or simply $\mathrm{Aut}(X)$ , denote the group of homeomorphisms of X which commute with the G action. We show that for any pair of countable groups G and H with G infinite, there is a minimal, free, Cantor G-flow X so that H embeds into $\mathrm{Aut}(X)$ . This generalizes results of [2, 7].

For $G$ a Polish group, we consider $G$-flows which either contain a comeager orbit or have all orbits meager. We single out a class of flows, the maximally highly proximal (MHP) flows, for which this analysis is particularly nice. In the former case, we provide a complete structure theorem for flows containing comeager orbits, generalizing theorems of Melleray, Nguyen Van Thé, and Tsankov and of Ben Yaacov, Melleray, and Tsankov. In the latter, we show that any minimal MHP flow with all orbits meager has a metrizable factor with all orbits meager, thus ‘reflecting’ complicated dynamical behavior to metrizable flows. We then apply this to obtain a structure theorem for Polish groups whose universal minimal flow is distal.