On scrambled sets and a theorem of Kuratowski on independent sets期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On scrambled sets and a theorem of Kuratowski on independent sets

Authors:	Hisao Kato

Institution:	Institute of Mathematics, University of Tsukuba, Ibaraki, 305 Japan

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 -cube is $\ast$ -chaotic on $I^{k}$ , then for any $\epsilon > 0$ there is a map $g: I^{k} \to I^{k}$ such that and are topologically conjugate, $d(f,g) < \epsilon$ and 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 satisfying that is an $F_{\sigma}$ -set in and has Lebesgue measure 1.

Keywords:	Scrambled set independent set Cantor set flat Lebesgue measure

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