Topological Entropy of Iterated Set-Valued Dynamical Systems

This paper studies topological entropy and pseudo-entropy of iterated set-valued function systems. Firstly, the notions of topological entropy defined by separating and spanning sets and by open covers are introduced respectively, and they are proved equivalent, then a formula is obtained for the topological entropy of an iterated set-valued function system concerning the corresponding skew product system, and topological entropy of iterated set-valued function systems is a topological conjugacy invariant. Finally, the notions of pseudo-entropy of set-valued function systems and iterated set-valued function systems are introduced and it is proved that the pseudo-entropy is equal to the topological entropy of iterated set-valued function systems.