The following problem is presented in the paper Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye :

"If a dynamical system [$(X,f)$, $X$ metric space, $f$ continuous] is Li-Yorke chaotic, does there exist a Cantor scrambled set?"

I would like to know the current status of the problem. Some information is given in the paper, but it's a bit old (2015). Thanks


1 Answer 1


This question was answered recently by Geschke, Grebík, and Miller:

S. Geschke, J. Grebík, and B. D. Miller, "Scrambled Cantor sets," Proceedings of the AMS 149 (link to the arxiv version)

They show that if $X$ is analytic (i.e., the continuous image of a Polish space), and if $(X,f)$ is a dynamical system with Li-Yorke chaos, then $(X,f)$ contains a scrambled Cantor set.

Let me point out that some assumption on $X$ (such as "analytic") is necessary here. For example, this is easiest to see if CH fails. Why? Let $(Y,f)$ be any dynamical system with Li-Yorke chaos. This means that it contains a scrambled set of size $\aleph_1$. Let $X$ denote the closure of this set under $f$, and observe that $(X,f)$ is then a dynamical system with Li-Yorke chaos, but its cardinality is smaller than that of the Cantor space! If CH holds, one can construct such things via transfinite recursion.

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    $\begingroup$ Upvoted. Btw, I don’t think that’s the only reason why assumptions on $X$ are needed. When Li-Yorke chaos was introduced (to describe the behavior exhibited for instance by interval maps with a 3-cycle) they said “uncountable” scrambled set but I guess they really meant “with $\mathfrak{c}$ many points”. I think even using this definition some assumptions on the space are needed. $\endgroup$ Commented Sep 11, 2022 at 13:10

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