On minimal flows. definably amenable groups, and o-minimality

TL;DR

This paper investigates definably amenable groups within o-minimal structures, providing a counterexample to a conjecture about generic types and minimal flows, and presenting additional positive results in this setting.

Contribution

It offers the first example in o-minimal structures where weak generic types differ from almost periodic types, addressing a question in model theory.

Findings

Counterexample showing weak generic types do not coincide with almost periodic types in o-minimal context

The union of minimal subflows is not always closed in this setting

Additional positive results on definably amenable groups in o-minimal theories

Abstract

We study definably amenable groups in NIP theories, and answer a question of Newelski (and also of Chernikov-Simon), by giving an example in the o-minimal context where weak generic types do not coincide with almost periodic types, equivalently where the union of the minimal subflows of suitable type spaces is not closed. We give other positive results in this o-minimal context.