On minimal flows and definable amenability in some distal NIP theories

TL;DR

This paper investigates the relationship between weakly generic and almost periodic types in definable groups within certain distal NIP theories, providing conditions for their coincidence and presenting counterexamples.

Contribution

It extends the understanding of definable topological dynamics in distal NIP theories by establishing when weakly generic types match almost periodic types and providing new counterexamples.

Findings

Weakly generic types coincide with almost periodic types when G has finitely many global weakly generic types.

Counterexamples show divergence when G has infinitely many global weakly generic types.

Results extend previous work on minimal flows and definably amenable groups to broader contexts.

Abstract

We study the definable topological dynamics $(G(M), S_G(M))$ of a definable group acting on its type space, where $M$ is either an $o$-minimal structure or a $p$-adically closed field, and $G$ a definable amenable group. We focus on the problem raised by Neweslki of whether weakly generic types coincide with almost periodic types, showing that the answer is positive when $G$ has boundedly many global weakly generic types. We also give two "minimal counterexamples" where $G$ has unboundedly many global weakly generic types, extending the main results of "On minimal flows, definably amenable groups, and o-minimality" to a more general context.