Topological Entropy for Discontinuous Semiflows

TL;DR

This paper extends the concept of topological entropy to discontinuous semiflows on compact metric spaces, establishing its relation to existing definitions and providing bounds and equivalences under certain conditions.

Contribution

It introduces new definitions of topological entropy for discontinuous semiflows, proves their consistency with classical definitions, and relates them to existing entropy measures through semiconjugation.

Findings

New entropy definitions for discontinuous semiflows are consistent with classical Bowen entropy.

The new entropy provides a lower bound for the $ au$-entropy by Alves, Carvalho, and Vázquez.

For impulsive semiflows with regularity, a related continuous semiflow has the same topological entropy.

Abstract

We study two variations of Bowen's definitions of topological entropy based on separated and spanning sets which can be applied to the study of discontinuous semiflows on compact metric spaces. We prove that these definitions reduce to Bowen's ones in the case of continuous semiflows. As a second result, we prove that our entropies give a lower bound for the $\tau$-entropy defined by Alves, Carvalho and V\'asquez (2015). Finally, we prove that for impulsive semiflows satisfying certain regularity condition, there exists a continuous semiflow defined on another compact metric space which is related to the first one by a semiconjugation, and whose topological entropy equals our extended notion of topological entropy by using separated sets for the original semiflow.