Parametric topological entropy of possibly discontinuous maps in compact Hausdorff spaces and hyperspaces

TL;DR

This paper explores various definitions of topological entropy for multivalued nonautonomous dynamical systems in compact Hausdorff spaces, focusing on upper semicontinuous maps and their induced systems in hyperspaces, providing tools for easier entropy estimation.

Contribution

It introduces and compares multiple entropy definitions for multivalued systems, especially relating upper semicontinuous maps to hyperspace dynamics, aiding in entropy estimation.

Findings

Different entropy definitions can be compared to facilitate estimates.

Relationships between multivalued maps and hyperspace induced systems are established.

Examples illustrate the applicability of the theoretical results.

Abstract

We will consider various definitions of topological entropy for multivalued nonautonomous dynamical systems in compact Hausdorff spaces. Some of them can deal with arbitrary multivalued maps, i.e. when no restrictions are imposed on them. For upper semicontinuous multivalued maps, we still investigate especially their relationship to the parametric topological entropy of the induced (possibly discontinuous) dynamical systems in hyperspaces. A comparison of various sorts of definitions can allow us, rather than direct calculations, to make easier the entropy estimates. Several illustrative examples are supplied.