On the structure of the ω-limit sets for continuous maps of the interval

Lluís Alsedà; Moira Chas; Jaroslav Smítal

doi:10.1142/s0218127499001206

On the structure of the ω-limit sets for continuous maps of the interval

Lluís Alsedà
, Moira Chas
, Jaroslav Smítal

Mathematics

Autonomous University of Barcelona
Silesian University in Opava

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point x_α ∈ I such that its center has depth at least α. This improves a result by [Sharkovskii, 1966].

Original language	English
Pages (from-to)	1719-1729
Number of pages	11
Journal	International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume	9
Issue number	9
DOIs	https://doi.org/10.1142/s0218127499001206
State	Published - Sep 1999

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abstract = "We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point xα ∈ I such that its center has depth at least α. This improves a result by [Sharkovskii, 1966].",

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AU - Chas, Moira

AU - Smítal, Jaroslav

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N2 - We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point xα ∈ I such that its center has depth at least α. This improves a result by [Sharkovskii, 1966].

AB - We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point xα ∈ I such that its center has depth at least α. This improves a result by [Sharkovskii, 1966].

UR - https://www.scopus.com/pages/publications/0038853480

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DO - 10.1142/s0218127499001206

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ER -

On the structure of the ω-limit sets for continuous maps of the interval

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