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Proceedings of the American Mathematical Society

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

 

 

On the homeomorphisms which satisfy the Poincaré recurrence theorem


Author: Chung Wu Ho
Journal: Proc. Amer. Math. Soc. 58 (1976), 272-276
MSC: Primary 54H20; Secondary 57E05
DOI: https://doi.org/10.1090/S0002-9939-1976-0420670-8
MathSciNet review: 0420670
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Abstract: It is shown that for a large class of spaces, almost all the homeomorphisms of the space do not satisfy the Poincaré Recurrence Theorem. More specifically, let $ X$ be a compact manifold with a nonzero Euler characteristic and $ H(X)$ be the space of all homeomorphisms of $ X$ onto $ X$. $ H(X)$ is given the compact open topology. Then the set of all the homeomorphisms of $ X$ which satisfy the Poincaré Recurrence Theorem is nowhere dense in $ H(X)$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0420670-8
Keywords: Poincaré Recurrence Theorem, Borel measure, recurrent homeomorphism, Euler characteristic, nowhere dense subset
Article copyright: © Copyright 1976 American Mathematical Society