$\omega$-chaos without infinite LY-scrambled set on Gehman dendrite

TL;DR

This paper constructs a dendrite map that exhibits $\omega$-chaos without possessing an infinite LY-scrambled set, thus answering an open question about their relationship.

Contribution

It provides a counterexample showing that $\omega$-chaos can occur without an infinite LY-scrambled set on dendrites, clarifying the connection between these chaos notions.

Findings

Constructed a dendrite map with $\omega$-chaos but no infinite LY-scrambled set.

Answered an open question from previous research.

Demonstrated the independence of $\omega$-chaos and LY-scrambled sets on dendrites.

Abstract

We answer the last question left open in [Z.~Ko\v{c}an, \emph{Chaos on one-dimensional compact metric spaces}, Internat. J. Bifur. Chaos Appl. Sci. Engrg. \textbf{22}, article id: 1250259 (2012)] which asks whether there is a relation between an infinite LY-scrambled set and $\omega$-chaos for dendrite maps. We construct a continuous self-map of a dendrite without an infinite LY-scrambled set but containing an uncountable $\omega$-scrambled set.