On the homeomorphisms which satisfy the Poincaré recurrence theorem

Ho, Chung Wu

doi:10.1090/S0002-9939-1976-0420670-8

On the homeomorphisms which satisfy the Poincaré recurrence theorem
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Proc. Amer. Math. Soc. 58 (1976), 272-276 Request permission

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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Über den Wiederkehrsatz von Poincaré

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