On two notions of expansiveness for continuous semiflows

Sebastián Herrero; Nelda Jaque

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arXiv · 2109.06003

On two notions of expansiveness for continuous semiflows

Sebastián Herrero·Nelda Jaque

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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.

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Sebastián Herrero, Nelda Jaque. 2021-09-13. On two notions of expansiveness for continuous semiflows. https://arxiv.org/abs/2109.06003

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