$ω$-chaos without infinite LY-scrambled set on Gehman dendrite

Abstract

We answer the last question left open in [Z.~Kočan, \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 $ω$-chaos for dendrite maps. We construct a continuous self-map of a dendrite without an infinite LY-scrambled set but containing an uncountable $ω$-scrambled set.