On minimal flows of commutative $p$-adic groups

TL;DR

This paper investigates the relationship between weakly generic and almost periodic types in the definable topological dynamics of commutative $p$-adic groups, establishing conditions for their coincidence related to definable $f$-generics and stationarity.

Contribution

It extends previous work by characterizing when weakly generic types coincide with almost periodic types specifically for commutative $p$-adic groups, linking this to definable $f$-generics and stationarity.

Findings

Weakly generic types coincide with almost periodic types if and only if $G$ has definable $f$-generics or is stationary.

The results apply specifically to commutative groups definable over the $p$-adic numbers.

The work builds on and extends previous results in definable topological dynamics and $p$-adic groups.

Abstract

We study the definable topological dynamics $(G,S_G(M))$ of a definable group acting on its type space, where $M$ is a structure and $G$ is a group definable in $M$. In \cite{Newelski-I}, Newelski raised a question of whether weakly generic types coincide with almost periodic types in definable topological dynamics. In \cite{YZ-Sta}, we introduced the notion of stationarity, showing the answer is positive when $G$ is a stationary definably amenable group definable over the field of $p$-adic numbers or an $o$-minimal expansion of real closed field. In this paper, we continue with the work of \cite{YZ-Sta}, focusing on the case where $G$ is a commutative group definable over the field of $p$-adic numbers, and showing that weakly generic types coincide with almost periodic types if and only if either $G$ has definable $f$-generics or $G$ is stationary.