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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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