On Hudetz entropy localization

Abstract

In Rahimi and Riazi (2014) [22], a type of entropy of continuous maps on compact metric spaces is defined via integrating a local entropy map. This average entropy is represented in terms of the entropy in the sense of Dumitrescu, but the equality of these two types of entropy is left as an open problem in [22]. In this paper, we modify the problem stated in [22], by replacing the Dumitrescu entropy by the Hudetz entropy, since in our case, the two previous types of entropy take an infinite value and the problem stated in [22] reduces to a trivial equality. Then we conclude the paper by proving the equality of the modified average entropy and the Hudetz entropy. The key tool is a version of Jaccob's theorem for the Hudetz entropy.