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The attracting centre of a continuous self-map of the interval

Published online by Cambridge University Press:  19 September 2008

Xiong Jincheng
Xiong Jincheng
Affiliation:
University of Science and Technology of China, Hefei, Anhui, People's Republic of China and International Centre for Theoretical Physics, Trieste, Italy
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Abstract

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Let ƒ denote a continuous map of a compact interval I to itself. A point x ∈ I is called a γ-limit point of ƒ if it is both an ω-limit point and an α-limit point of some point y ∈ I. Let Γ denote the set of γ-limit points. In the present paper, we show that (1) −Γ is either empty or countably infinite, where denotes the closure of the set P of periodic points, (2) x ∈ I is a γ-limit point if and only if there exist y1 and y2 in I such that x is an ω-limit point of y1, and y1 is an ω-limit point of y2, and if and only if there exists a sequence y1, y2,…of points in I such that x is an ω-limit point of y1, and yi is an ω-limit point of yi+1 for every i ≥ 1, and (3) the period of each periodic point of ƒ is a power of 2 if and only if every γ-limit point is recurrent.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 2 , June 1988, pp. 205 - 213
Copyright
Copyright © Cambridge University Press 1988

References

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