Topological Entropy for Arbitrary Subsets of Infinite Product Spaces

Maysam Maysami Sadr, Mina Shahrestani

Published 2020-05-26Version 1

In this note a notion of topological entropy for arbitrary subsets of the space of all sequences in a compact topological space is introduced. It is shown that for a continuous map on a compact space the topological entropy of the set of all orbits of the map coincides with the classical topological entropy of the map. Some basic properties of this new notion of entropy are considered; among them are: behavior of the entropy with respect to disjoint union, cartesian product, component restriction and dilation, shift mapping, and some continuity properties with respect to Vietoris topology. As an example it is shown that any self-similar structure of a fractal given by a finite family of contractions gives rise to a notion of intrinsic topological entropy for subsets of the fractal. A notion of Bowen entropy for subsets of sequences in a metric space is also introduced and it is shown that the two new notions of entropy coincide for compact metric spaces.