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

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.