Topological entropy of multivalued maps in topological spaces and hyperspaces

Abstract

The main aim of this article is two-fold: (i) to correct some discrepancies in our recent paper entitled “Chaos for multivalued maps and induced hyperspace maps”[Chaos, Solitons & Fractals 138(2):109898, 1–8, 2020], (ii) to generalize the investigation analysis to multivalued maps in compact Hausdorff topological spaces. We will introduce various (some newly) definitions of topological entropy for multivalued maps and test whether or not their positive entropy implies the same for induced hyperspace maps. All of them reduce to the standard definition for single-valued maps. On the other hand, they exhibit different properties. In particular, only some definitions share the above implication (forcing property) with single-valued maps. Several illustrative examples are supplied.