# On tameness of almost automorphic dynamical systems for general groups

## TL;DR

This paper investigates the tameness properties of almost automorphic dynamical systems for general groups, showing they can be tame or non-tame, and provides examples through semicocycle representations.

## Contribution

It introduces a new framework for representing almost automorphic systems via semicocycles and constructs examples of tame and non-tame systems within well-known group actions.

## Key findings

- Existence of tame but non-null almost automorphic extensions.
- Existence of non-tame almost automorphic extensions.
- Representation of systems via semicocycles for general groups.

## Abstract

Let $(X,G)$ be a minimal equicontinuous dynamical system, where $X$ is a compact metric space and $G$ some topological group acting on $X$. Under very mild assumptions, we show that the class of regular almost automorphic extensions of $(X,G)$ contains examples of tame but non-null systems as well as non-tame ones. To do that, we first study the representation of almost automorphic systems by means of semicocycles for general groups. Based on this representation, we obtain examples of the above kind in well-studied families of group actions. These include Toeplitz flows over $G$-odometers where $G$ is countable and residually finite as well as symbolic extensions of irrational rotations.