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Every chaotic interval map has a scrambled set in the recurrent set

Published online by Cambridge University Press:  17 April 2009

Bau-Sen Du
Bau-Sen Du
Affiliation:
Institute of Mathematics, Academia Sinica, Taipei, Taiwan 11529, Republic of China
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Abstract

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Let I denote a compact real interval and let f ∈ C0(I, I). In this note we show that if f chaotic in the sense of Li and Yorke, then there is an uncountable perfect δ-scrambled set S for f in the recurrent set of f. Furthermore, the ω-limit set of every x ∈ S under f contains S and contains infinitely many periodic points of f with arbitrarily large periods.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 39 , Issue 2 , April 1989 , pp. 259 - 264
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Auslauder, J. and Yorke, J.A., ‘Interval maps, factors of maps, and chaos’, Tôhoku Math. J. 32 (1980), 177–188.Google Scholar
[2]Bruckner, A.M. and Hu, Thakyin, ‘On scrambled sets for chaotic functions’, Trans. Amer. Math. Soc. 301 (1987), 289–297.CrossRefGoogle Scholar
[3]Butler, G.J. and Pianigiani, G., ‘Periodic points and chaotic functions in the unit interval’, Bull. Austral. Math. Soc. 18 (1978), 255–265.CrossRefGoogle Scholar
[4]Gedeon, T., ‘There are no chaotic mappings with residual scrambled sets’, Bull. Austral. Math. Soc. 36 (1987), 411–416.CrossRefGoogle Scholar
[5]Kan, I., ‘A chaotic function possessing a scrambled set with positive Lebesgue measure’, Proc. Amer. Math. Soc. 92 (1984), 45–49.CrossRefGoogle Scholar
[6]Kloeden, P.E., ‘Chaotic difference equations are dense’, Bull. Austral. Math. Soc. 15 (1976), 371–379.CrossRefGoogle Scholar
[7]Kuratowski, K., Topologie I (PWN - Polish Scientific Publishers, Warsaw, 1968). p. 387.Google Scholar
[8]Li, T.-Y. and Yorke, J.A., ‘Period three implies chaos’, Amer. Math. Monthly 82 (1975), 985–992.CrossRefGoogle Scholar
[9]Misiurewicz, M., ‘Chaos almost everywhere’, Lect. Notes in Math. 1163. pp. 125–130 (Springer-Verlag).Google Scholar
[10]Mizera, I., ‘Continuous chaotic functions of an interval have generically small scrambled sets’, Bull. Austral. Math. Soc. 37 (1988), 89–92.CrossRefGoogle Scholar
[11]Osikawa, M. and Oono, Y., ‘Chaos in C 0-endomorphism of interval’, Publ. Res. Inst. Math. Sci. 17 (1981), 165–177.CrossRefGoogle Scholar
[12]Smital, J., ‘A chaotic function with some extremal properties’, Proc. Amer. Math. Soc. 87 (1983), 54–56.CrossRefGoogle Scholar
[13]Smital, J., ‘A chaotic function with scrambled set of positive Lebesgue measure’, Proc. Amer. Math. Soc. 92 (1984), 50–54.CrossRefGoogle Scholar
[14]Smital, J., ‘Chaotic functions with zero topological entropy’, Trans. Amer. Math. Soc. 297 (1986), 269–282.CrossRefGoogle Scholar
[15]Smital, J. and Jankova, K., ‘A characterization of chaos’, Bull. Austral. Math. Soc. 34 (1986), 283–292.Google Scholar
[16]Sharkovskii, A.N., ‘Coexistence of cycles of a continuous mapping of the real line into itself’, (Russian), Ukrain. Mat. Zh. 16 (1964), 61–71.Google Scholar
[17]Sharkovskii, A.N., ‘Behavior of a mapping in the neighbourhood of an attracting set’, (Russian), Ukrain. Mat. Zh. 18 (1966), 60–83. (English), AMS Trans. (2) 97 (1970), 227–258.Google Scholar
[18]Xiong, J.-C., ‘A chaotic map with topological entropy 0’, Acta Math. Sci. 6 (1986), 439–443.CrossRefGoogle Scholar
[19]Zhou, Z.-L., ‘A note on the Li-Yorke theorem’, Kexue Tongbao (English Ed.) 31 (1986), 649–651.Google Scholar