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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Recurrent homeomorphisms on $ {\bf R}\sp 2$ are periodic


Authors: Lex G. Oversteegen and E. D. Tymchatyn
Journal: Proc. Amer. Math. Soc. 110 (1990), 1083-1088
MSC: Primary 54H20
MathSciNet review: 1037216
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Abstract: A homeomorphism $ f:(X,d) \to (X,d)$ of a metric space $ (X,d)$ onto $ X$ is recurrent provided that for each $ \varepsilon > 0$ there exists a positive integer $ n$ such that $ {f^n}$ is $ \varepsilon $-close to the identity map on $ X$. The notion of a recurrent homeomorphism is weaker than that of an almost periodic homeomorphism. The result announced in the title generalizes the theorem of Brechner for almost periodic homeomorphisms and answers a question of R. D. Edwards.


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DOI: https://doi.org/10.1090/S0002-9939-1990-1037216-3
Keywords: Recurrent, periodic, homeomorphisms on $ {\mathbb{R}^2}$
Article copyright: © Copyright 1990 American Mathematical Society