Transitive sensitive subsystems for interval maps

TL;DR

This paper explores the relationships between transitive sensitive subsystems, topological entropy, and chaos in continuous interval maps, providing examples that distinguish these concepts.

Contribution

It establishes implications among transitive sensitive subsystems, topological entropy, and chaos, and presents examples showing their distinctions in interval maps.

Findings

Positive topological entropy implies the existence of transitive sensitive subsystems.

Transitive sensitive subsystems imply chaos in the sense of Li-Yorke.

Examples demonstrate that these notions are not equivalent in continuous interval maps.

Abstract

We state that for continuous interval maps the existence of a non empty closed invariant subset which is transitive and sensitive to initial conditions is implied by positive topological entropy and implies chaos in the sense of Li-Yorke, and we exhibit examples showing that these three notions are distinct.