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 ] → I 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 → I k ( k ≥ 1 ) f: I^{k} \to I^{k}~(k\geq 1) of the unit k k -cube I k I^k is ∗ \ast -chaotic on I k I^{k} , then for any ϵ > 0 \epsilon > 0 there is a map g : I k → I k g: I^{k} \to I^{k} such that f f and g g are topologically conjugate, d ( f , g ) > ϵ d(f,g) > \epsilon and g g has a scrambled set which has Lebesgue measure 1, and hence if k ≥ 2 k \geq 2 , then there is a homeomorphism f : I k → I k f: I^{k} \to I^{k} with a scrambled set S S satisfying that S S is an F σ F_{\sigma } -set in I k I^k and S S has Lebesgue measure 1.