Topological Entropy for Arbitrary Subsets of Infinite Product Spaces

TL;DR

This paper introduces a generalized topological entropy for arbitrary subsets of infinite product spaces, linking it to classical entropy and exploring its properties and applications to fractals and sequence spaces.

Contribution

It defines a new generalized entropy concept for subsets of sequence spaces and demonstrates its equivalence to classical entropy in certain cases, expanding entropy theory.

Findings

Generalized entropy coincides with classical topological entropy for continuous maps.

Properties of the new entropy under disjoint union, product, and shift are established.

Application to fractals shows intrinsic entropy from self-similar structures.

Abstract

In this note a notion of generalized 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 generalized 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: the 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 generalized notion of Bowen's entropy associated to any increasing sequence of compatible semimetrics on a topological space is introduced and some of its basic properties are considered. As a special case for $1\leq p\leq\infty$ the Bowen $p$-entropy of sets of sequences of any metric space is introduced. It is shown that the notions of generalized topological entropy and Bowen $\infty$-entropy for compact metric spaces coincide.