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A characterization of ω-limit sets of maps of the interval with zero topological entropy†

Published online by Cambridge University Press:  19 September 2008

A. M. Bruckner  and
J. Smítal
A. M. Bruckner
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 931-06, USA
J. Smítal
Affiliation:
Institute of Mathematics, Comenius University, 84215 Bratislava, and Institute of Mathematics, Silesian University, 74601 Opava, Czechoslovakia
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Abstract

We prove that an infinite W ⊂ (0, 1) is an ω-limit set for a continuous map ƒ of [0,1] with zero topological entropy iff W = Q ∪ P where Q is a Cantor set, and P is countable, disjoint from Q, dense in W if non-empty, and such that for any interval J contiguous to Q, card (J ∩ P) ≤ 1 if 0 or 1 is in J, and card (J ∩ P) ≤ 2 otherwise. Moreover, we prove a conjecture by A. N. Šarkovskii from 1967 that P can contain points from infinitely many orbits, and consequently, that the system of ω-limit sets containing Q and contained in W, can be uncountable.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 13 , Issue 1 , March 1993 , pp. 7 - 19
Copyright
Copyright © Cambridge University Press 1993

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References

REFERENCES

[ABCP]Agronsky, S. J., Bruckner, A. M., Ceder, J. G. & Pearson, T. L.. The structure of ω-limit sets for continous functions. Real Analysis Exchange 15 (1989–1990), 483–510.CrossRefGoogle Scholar
[B]Block, L. S.. Simple periodic orbits of mappings of the interval. Trans. Amer. Math. Soc. 254 (1979), 391–398.Google Scholar
[BC]Bruckner, A.M. & Ceder, J.. Chaos in terms of the map x→ω(x, f). Pacific J. Math. 156 (1992) 63–96.CrossRefGoogle Scholar
[BIC]Block, L. S. & Coven, E. M., ω-limit sets for maps of the interval. Ergod. Th. & Dynam. Sys. 6 (1985), 335–344.CrossRefGoogle Scholar
[BC]Bruckner, A. M. & Smital, J.. The structure of ω-limit sets for continuous maps of the interval. Math. Bohemica 117 (1992), 42–47.CrossRefGoogle Scholar
[CX]Chu, H. & Xiong, J. C.. A counterexample in dynamical systems. Proc. Amer. Math. Soc. 97 (1986), 361–366.CrossRefGoogle Scholar
[D]Denjoy, A.. Sur les courbes définies par équations différentielles à la surface du tore. J. Math. Pures Appl. (9), 11 (1932), 333–375.Google Scholar
[FSS]Fedorenko, V. V., Šarkovskii, A. N. & Smítal, J.. Characterization of weakly chaotic maps of the interval. Proc. Amer. Math. Soc. 110 (1990), 141–148.CrossRefGoogle Scholar
[H]Harrison, J.. Wandering intervals. Dynamical systems and turbulence (Warwick 1980). Springer Lecture Notes in Mathematics 898. Springer, Berlin, 1981, 154–163.Google Scholar
[K]Kirchheim, B.. A chotic function with zero topological entropy having non-perfect attractors. Math. Slovaca 40 (1990), 267–272.Google Scholar
[KŠ]Kenžegulov, Ch. K. and Šarkovskii, A. N.. On properties of the set of limit points of an iterated sequence of a continuous function. Volž Mat. Sb. 3 (1965), 343–348 (Russian).Google Scholar
[LY]Li, T. Y. & Yorke, J.. Period three implies chaos. Amer. Math. Monthly 82 (1975), 985–992.CrossRefGoogle Scholar
[M]Misiurewicz, M.. Invariant measures for continuous transformations of [0, 1] with zero topological entropy, Ergodic Theory (Oberwolfach 1978), Springer Lecture Notes in Mathematics 729, Springer, Berlin, 1979, 144–152.CrossRefGoogle Scholar
[S]Smital, J.. Chaotic maps with zero topological entropy. Trans. Amer. Math. Soc. 297 (1986), 269–282.CrossRefGoogle Scholar
[Ša1]Šarkovskii, A. N.. The partially ordered system of attracting sets. Dokl. Akad. Nauk. USSR 170 (1966), 1276–1278.Google Scholar
[Ša2]Šarkovskii, A. N.. Attracting sets containing no cycles. Ukrain. Mat. Ž. 20 (1968), 136–142. (Russian)Google Scholar
[Ša3]Šarkovskii, A. N.. A mapping of zero topological entropy possessing continuum of Cantor minimal sets. Dynamical Systems and Turbulence. Akad. Nauk USSR, Inst. of Math., Kiev, 1989, 109–117 (Russian).Google Scholar
[VŠ]Verejkina, M. B. & Šarkovskii, A. N.. Recurrence in one-dimensional dynamical systems. Approx. and Qualitative Methods of the Theory of Differential-Functional Equations. Inst. Mat. Akad. Nauk USSR, Kiev 1983, 35–46 (Russian).Google Scholar