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The structure of equicontinuous maps

Author: Jie-Hua Mai
Journal: Trans. Amer. Math. Soc. 355 (2003), 4125-4136
MSC (2000): Primary 54E40, 54H20; Secondary 37B20, 37E25
Published electronically: June 18, 2003
MathSciNet review: 1990578
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Abstract: Let $(X,d)$ be a metric space, and $f:X\rightarrow X$ be a continuous map. In this paper we prove that if $R(f)$ is compact, and $\omega (x,f)\not =\emptyset $ for all $x\in X$, then $f$ is equicontinuous if and only if there exist a pointwise recurrent isometric homeomorphism $h$ and a non-expanding map $g$ that is pointwise convergent to a fixed point $v_{0}$ such that $f$ is uniformly conjugate to a subsystem $(h\times g)\vert S$ of the product map $h\times g$. In addition, we give some still simpler necessary and sufficient conditions of equicontinuous graph maps.

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Additional Information

Jie-Hua Mai
Affiliation: Institute of Mathematics, Shantou University, Shantou, Guangdong, 515063, People’s Republic of China

Keywords: Metric space, equicontinuous map, recurrent point, uniform conjugacy, graph map
Received by editor(s): March 4, 2002
Received by editor(s) in revised form: November 1, 2002
Published electronically: June 18, 2003
Additional Notes: Project supported by the Special Foundation of National Prior Basic Researches of China (Grant No. G1999075108)
Article copyright: © Copyright 2003 American Mathematical Society

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