Topological entropy on set-valued functions

TL;DR

This paper extends the concept of topological entropy to set-valued functions, showing how classical results adapt and identifying conditions for positive or infinite entropy in this broader context.

Contribution

It introduces a generalized definition of topological entropy for set-valued functions and explores which classical properties carry over or need modification.

Findings

Some classical results extend naturally to set-valued functions.

Conditions for positive or infinite topological entropy are established.

Certain properties of topological entropy require alteration in the set-valued context.

Abstract

Topological entropy is a widely studied indicator of chaos in topological dynamics. Here we give a generalized definition of topological entropy which may be applied to set-valued functions. We demonstrate that some of the well-known results concerning topological entropy of continuous (single-valued) functions extend naturally to set-valued functions while others must be altered. We also present sufficient conditions for a set-valued function to have positive or infinite topological entropy.