On two notions of expansiveness for continuous semiflows

TL;DR

This paper investigates two types of expansiveness in continuous semiflows, showing that under certain conditions, these properties imply the space is finite or composed of simple dynamical components.

Contribution

It establishes new results linking two notions of expansiveness to the topological and dynamical structure of the space and semiflow.

Findings

Expansive semiflows on metric spaces are trivial and spaces are discrete.

Compact positively expansive semiflows have finitely many simple components.

Results connect expansiveness notions to the structure of semiflows and spaces.

Abstract

We study two notions of expansiveness for continuous semiflows: expansiveness in the sense of Alves, Carvalho and Siqueira (2017), and an adaptation of positive expansiveness in the sense of Artigue (2014). We prove that if $X$ is a metric space and $\phi$ is an expansive semiflow on $X$ according to the first definition, then the semiflow $\phi$ is trivial and the space $X$ is uniformly discrete. In particular, if $X$ is compact then it is finite. With respect to the second definition, we prove that if $X$ is a compact metric space and $\phi$ is a positive expansive semiflow on it, then $X$ is a union of at most finitely many closed orbits, unbranched tails and isolated singularities.