圆周连续自映射周期点集为闭集的充要条件

THE NECESSARY AND SUFFICIENT CONDITION OF A CLOSED SET PERIODIC POINTS SET OF THE CONTINUOUS SELF—MAPS OF THE CIRCLE FORM

本文对圆周连续自映射作了些讨论,证明了如下定理,设f∈c^o(s',s'),则下列条件等价. (1)P(f)=P(f),且P(f)≠φ. (2)对于_x∈R(f),总有P∈P(f),■P∈W(x,f),■=(P)中所有点都是W(x,f)的孤立点。 (3)对于■_x∈R(f),W(x,f)是有限集。 (4)对于■∈R(f),W(x,f)的导集W(x,f)'是有限集。

Let f∈c^o(s',s'), P(f),R(f),W(x,f)denote the set of periodic points of f and the set of recurrent points off and the set of W—limit points set of X,respectively,in this paper, we show that the following conditions are equivalent. (1).P(f)=P(?), P(f)≠φ; (2)(?) x∈R(f) there exists a p∈P(f) such that P∈W(x,f) and all pointoof the periodic orbit at pare isolated points of W(x,f); (3)(?) x∈R(f),W(x,f) is finite; (4)(?) x∈R(f),W(x,f) is finite.