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



State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"
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](https://arxiv.org/abs/2006.08277) 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.


