Discrete and Continuous Topological Dynamics: Fields of Cross Sections and Expansive Flows

TL;DR

This paper develops methods to translate discrete dynamical system concepts to flows, constructs cross sections for flows on Peano continua, and shows that expansive flows have specific stability properties, excluding their existence on certain surfaces.

Contribution

It introduces continuous fields of cross sections for flows and applies them to analyze expansive flows on Peano continua, establishing new non-existence results.

Findings

Expansive flows have no stable points.

Every point in an expansive flow has a non-trivial stable set.

No expansive flow exists on certain surfaces like the plane or compact surfaces.

Abstract

In this article we consider the general problem of translating definitions and results from the category of discrete-time dynamical systems to the category of flows. We consider the dynamics of homeomorphisms and flows on compact metric spaces, in particular Peano continua. As a translating tool, we construct continuous, symmetric and monotonous fields of local cross sections for an arbitrary flow without singular points. Next, we use this structure in the study of expansive flows on Peano continua. We show that expansive flows admit no stable point and that every point contains a non-trivial continuum in its stable set. As a corollary we obtain that no Peano continuum with an open set homeomorphic with the plane admits an expansive flow. In particular compact surface admits no expansive flow without singular points.