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International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 1, Pages 49-54
http://dx.doi.org/10.1155/S0161171299220492

Nonwandering sets of maps on the circle

1Department of Mathematics, Seoul National University of Technology, Nowon-Gu, Seoul 139-743, Korea
2Department of Mathematics, Hanseo University, Chungnam, Seosan 356-820, Korea

Received 30 April 1997; Revised 25 July 1997

Copyright © 1999 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let f be a continuous map of the circle S1 into itself. And let R(f),Λ(f),Γ(f), and Ω(f) denote the set of recurrent points, ω-limit points, γ-limit points, and nonwandering points of f, respectively. In this paper, we show that each point of Ω(f)\R(f)¯ is one-side isolated, and prove that

(1) Ω(f)\Γ(f) is countable and

(2) Λ(f)\Γ(f) and R(f)¯\Γ(f) are either empty or countably infinite.