On two notions of expansiveness for continuous semiflows

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.