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Proceedings of the American Mathematical Society

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

 
 

 

Homeomorphisms with many recurrent points


Author: Benjamin Halpern
Journal: Proc. Amer. Math. Soc. 59 (1976), 159-160
MSC: Primary 58F10; Secondary 58F20, 58D99
DOI: https://doi.org/10.1090/S0002-9939-1976-0415678-2
MathSciNet review: 0415678
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Abstract: Let $ X$ be a topological space and $ H(X)$ the space of all homeomorphisms of $ X$ onto itself with the compact open topology. If $ f \in H(X)$ and $ p \in X$, then $ p$ is a recurrent point of $ f$ provided $ p$ is in the closure of $ \{ {f^n}(p)\vert n \geqslant 1\} $. It is shown that if $ X$ is Hausdorff and $ V$ is a nonempty open subset of $ X$ homeomorphic to Euclidean $ n$-dimensional space with $ n \geqslant 1$, then $ \{ f \in H(X)\vert$ the recurrent points of $ f$ are dense in $ V$ is nowhere dense in $ H(X)$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0415678-2
Keywords: Homeomorphism, recurrent point, manifold, compact open topology
Article copyright: © Copyright 1976 American Mathematical Society

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