On scrambled sets and a theorem of Kuratowski on independent sets

Abstract:

The measure of scrambled sets of interval self-maps $f:I=[0,1] \to I$ was studied by many authors, including Smítal, Misiurewicz, Bruckner and Hu, and Xiong and Yang. In this note, first we introduce the notion of “$\ast$-chaos" which is related to chaos in the sense of Li-Yorke, and we prove a general theorem which is an improvement of a theorem of Kuratowski on independent sets. Second, we apply the result to scrambled sets of higher dimensional cases. In particular, we show that if a map $f: I^{k} \to I^{k}~(k\geq 1)$ of the unit $k$-cube $I^k$ is $\ast$-chaotic on $I^{k}$, then for any $\epsilon > 0$ there is a map $g: I^{k} \to I^{k}$ such that $f$ and $g$ are topologically conjugate, $d(f,g) < \epsilon$ and $g$ has a scrambled set which has Lebesgue measure 1, and hence if $k \geq 2$, then there is a homeomorphism $f: I^{k} \to I^{k}$ with a scrambled set $S$ satisfying that $S$ is an $F_{\sigma }$-set in $I^k$ and $S$ has Lebesgue measure 1.