Topological entropy of closed sets in $[0,1]^2$

TL;DR

This paper extends the concept of topological entropy to set-valued functions on closed subsets of [0,1], exploring properties, computations, and examples that reveal complex dynamical behaviors.

Contribution

It introduces a generalized definition of topological entropy for set-valued functions on closed subsets of [0,1], analyzing its properties and providing illustrative examples.

Findings

Many properties of classical topological entropy carry over to the new setting

Some examples show entropy can change dramatically with small modifications

A specific example demonstrates zero entropy set can become infinite entropy when augmented

Abstract

We generalize the definition of topological entropy due to Adler, Konheim, and McAndrew \cite{AKM} to set-valued functions from a closed subset $A$ of the interval to closed subsets of the interval. We view these set-valued functions, via their graphs, as closed subsets of $[0,1]^{2}$. We show that many of the topological entropy properties of continuous functions of a compact topological space to itself hold in our new setting, but not all. We also compute the topological entropy of some examples, relate the entropy to other dynamical and topological properties of the examples, and we give an example of a closed subset $G$ of $[0,1]^2$ that has $0$ entropy but $G\cup \{ (p,q) \},$ where $(p,q) \in [0,1]^2\setminus G,$ has infinite entropy.