On various entropies of set-valued maps

Abstract

In this paper, the complexity of an upper semi-continuous set-valued map F on a compact metric space is considered via entropy-like invariants from various perspectives. Several entropies of topological version, including pointwise entropies hp(F) and hm(F), branch entropy hi(F) and tree entropy ht(F), are introduced and investigated. Some basic properties about them are given and the relations among them and the classical entropy htop(F) are considered. The calculation or estimation of these entropies for certain finitely-generated set-valued maps on intervals or graphs are discussed. As the counterparts of ht(F), the measure-theoretic lower and upper tree entropies h_tμ(F) and h‾tμ(F) are introduced for any Borel probability measure μ in a way resembling Hausdorff dimension. A variational inequality ht(F)≥sup⁡{h‾tμ(F)|μ∈P(X)}, where P(X) is the set of Borel probability measures on X, is obtained, and as its applications, the lower bounds of ht(F) are given for certain set-valued maps respectively.