Closed relations with non-zero entropy that generate no periodic points

TL;DR

This paper introduces a new concept of entropy for closed relations on compact metric spaces, explores its properties, and provides conditions for non-zero entropy with minimal periodic points or Cantor sets, extending classical dynamical systems results.

Contribution

It defines entropy for closed relations on compact spaces, studies its invariance under conjugation, and establishes conditions for non-zero entropy with few or no periodic points.

Findings

Entropy of closed relations is preserved under topological conjugation.

Conditions for closed relations on [0,1] to have non-zero entropy are established.

Examples show relations with non-zero entropy but minimal or no periodic points.

Abstract

The paper is motivated by E. Akin's book about dynamical systems and closed relations [A], and by J. Kennedy's and G. Erceg's recent paper about the entropy of closed relations on closed intervals [EK]. In present paper, we introduce the entropy of a closed relation G on any compact metric space X and show its basic properties. We also introduce when such a relation G generates a periodic point or finitely generates a Cantor set. Then we show that periodic points, finitely generated Cantor sets, Mahavier products and the entropy of closed relations are preserved by topological conjugations. Among other things, this generalizes the well-known results about the topological conjugacy of continuous mappings. Finally, we prove a theorem, giving sufficient conditions for a closed relation G on [0,1] to have a non-zero entropy. Then we present various examples of closed relations G on [0,1] such that (1) the entropy of G is non-zero, (2) no periodic point or exactly one periodic point is generated by G, and (3) no Cantor set is finitely generated by G.