Recurrent homeomorphisms on 𝑅² are periodic

Oversteegen, Lex G.; Tymchatyn, E. D.

doi:10.1090/S0002-9939-1990-1037216-3

Recurrent homeomorphisms on $\textbf {R}^ 2$ are periodic
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by Lex G. Oversteegen and E. D. Tymchatyn

Proc. Amer. Math. Soc. 110 (1990), 1083-1088

DOI: https://doi.org/10.1090/S0002-9939-1990-1037216-3

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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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