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On scrambled sets and a theorem
of Kuratowski on independent sets


Author: Hisao Kato
Journal: Proc. Amer. Math. Soc. 126 (1998), 2151-2157
MSC (1991): Primary 54H20, 26A18
DOI: https://doi.org/10.1090/S0002-9939-98-04344-5
MathSciNet review: 1451813
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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.


References [Enhancements On Off] (What's this?)

  • 1. B A. M. Bruckner and T. Hu, On scrambled sets for chaotic functions, Trans. Amer. Math. Soc. 301 (1987), 289-297. MR 88f:26003
  • 2. W. J. Gorman III, The homeomorphic transformation of c-sets into d-sets, Proc. Amer. Math. Soc. 17 (1966), 825-830. MR 34:7734
  • 3. H. Kato, Attractors and everywhere chaotic homeomorphisms in the sense of Li-Yorke on manifolds and $k$-dimensional Menger manifolds, Topology Appl. 72 (1996), 1-17.
  • 4. Ku K. Kuratowski, Applications of Baire-category method to the problem of independent sets, Fund. Math. 81 (1973), 65-72. MR 49:3855
  • 5. K. Kuratowski, Topology I, PWN, Warsaw, 1966. MR 36:840
  • 6. T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. MR 52:5898
  • 7. M. Misiurewicz, Chaos almost everywhere, Lecture Notes in Math. 1163 (1985), 125-130. MR 87e:58152
  • 8. E. E. Moise, Geometric topology in dimension 2 and 3, Springer, 1977. MR 58:7631
  • 9. J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. 42 (1941), 873-920. MR 3:211b
  • 10. T. B. Rushing, Topological embeddings, Academic Press, New York, 1973. MR 50:1247
  • 11. J. Smítal, A chaotic function with some extremal properties, Proc. Amer. Math. Soc. 87 (1983), 54-56. MR 84h:26008
  • 12. J. Smítal, A chaotic function with a scrambled set of positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), 50-54. MR 86b:26009b
  • 13. J. Xiong and Z. Yang, Chaos caused by a topologically mixing map, World Scientific, Advanced Series in Dynamical Systems Vol 9, 550-572. MR 93c:58153

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Additional Information

Hisao Kato
Affiliation: Institute of Mathematics, University of Tsukuba, Ibaraki, 305 Japan
Email: hisakato@sakura.cc.tsukuba.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-98-04344-5
Keywords: Scrambled set, independent set, Cantor set, flat, Lebesgue measure
Received by editor(s): August 29, 1996
Received by editor(s) in revised form: December 20, 1996
Communicated by: Mary Rees
Article copyright: © Copyright 1998 American Mathematical Society

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