A geometric framework for asymptoticity and expansivity in topological dynamics

TL;DR

This paper introduces a geometric framework for analyzing asymptoticity and expansivity in topological dynamics, extending classical theorems and revealing new properties for specific group actions.

Contribution

It develops a geometric approach applicable to second countable locally compact groups, extending Schwartzman's theorem and identifying nonexpansive directions in ${f Z}^d$ actions.

Findings

Extension of Schwartzman's theorem in this context

Identification of nonexpansive directions in ${f Z}^d$

Rigidity properties of distal Cantor systems

Abstract

We develop a geometric framework to address asymptoticity and nonexpansivity in topological dynamics. Our framework can be applied when the acting group is second countable and locally compact. As an application, we show extensions of Schwartzman's theorem in this context. Also, we get new results when the acting groups is ${\mathbb Z}^d$: any half-space of ${\mathbb R}^d$ contains a vector defining a (oriented) nonexpansive direction in the sense of Boyle and Lind. Finally, we deduce rigidity properties of distal Cantor systems.