Sufficient conditions for non-zero entropy and finite relations

TL;DR

This paper establishes new conditions under which closed relations on compact metric spaces exhibit non-zero entropy, and characterizes finite relations with non-zero entropy using chaos theory concepts.

Contribution

It introduces the notions of returns, dispersions, and well-aligned sets, providing novel criteria for non-zero entropy in closed relations.

Findings

Conditions for non-zero entropy in closed relations

Characterization of finite relations with non-zero entropy

Connection between entropy and chaos in finite relations

Abstract

We introduce the notions of returns, dispersions and well-aligned sets for closed relations on compact metric spaces and then we use them to obtain non-trivial sufficient conditions for such a relation to have non-zero entropy. In addition, we give a characterization of finite relations with non-zero entropy in terms of Li-Yorke and DC2-chaos.