Topological entropy and metric entropy for regular impulsive semiflows

TL;DR

This paper extends Bowen's topological entropy to impulsive semiflows with discontinuities, showing equivalence of different entropy definitions and establishing a variational principle for these systems.

Contribution

It introduces a modified pseudosemimetric framework for impulsive semiflows, proving the equivalence of entropy definitions and a variational principle for regular impulsive semiflows.

Findings

Spanning and separated set entropies coincide for regular impulsive semiflows.

The entropy satisfies a variational principle.

The new pseudosemimetric approach generalizes Bowen's entropy to discontinuous systems.

Abstract

In arXiv:1801.01238 a variation of Bowen's topological entropy that can be applied to the study of discontinuous semiflows on compact metric spaces was introduced. The main novetly is the use of certain family of pseudosemimetrics associated to the semiflow, in a such a way that in the continuous case they coincide with the classical Bowen's entropies. It was proved that regular impulsive semiflow can be semiconjugated by uniformly continuous bijection to certain continuous semiflow defined in a compact metric space. In the cited paper, the entropy is defined by the growth rate of separated sets. The main results of the present work is show that, for regular impulsive semiflows, the entropy defined by using spanning sets agrees with the entropy using separated sets. Moreover, this notion of entropy satisfy a variational principle. This results are obtained introducing a little change on the family of pseudosemimetrics associated to.