ω-Chaos Without Infinite LY-Scrambled Set on Gehman Dendrite

Abstract

We answer the last question left open in [Z. Kočan, Chaos on one-dimensional compact metric spaces, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 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.

Dynamics on dendrites with closed endpoint sets

“…This extends the work of Kočan, Kurková, and Málek, who use the Gehman dendrite and its subdendrites to represent every shiftinvariant subset of {0, 1} N -in this way, any subshift can be embedded into a dendrite map so that all other points are eventually fixed [8]. Shift spaces are a rich source of examples and counterexamples in dynamical systems, and this construction has led to an abundance of dendrite maps with various combinations of chaotic and non-chaotic behavior [3,4,7,9,13]. Our construction allows us to start with other zero-dimensional systems and gives us isometric embeddings rather than just topological ones.…”

“…T n ≤ T n − 10 n T n and this converges to 1 10 as n → ∞. Since both situations occur for infinitely many n, it follows that Φ x,x ′ (s) ≤ 1 10 ≤ 9 10 ≤ Φ * x,x ′ (s) on the whole interval s ∈ (1,4). Thus x, x ′ are a DC3 pair.…”