# State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"

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

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.
• 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. Sep 11, 2022 at 13:10