Birkhoff recurrence theorem
= Birkhoff recurrence theorem
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= Birkhoff's recurrence theorem
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{synonym}
Every continuous self-map of a nonempty compact <metric space> has a <recurrent point>. Choose a <minimal subsystem>. The forward orbit of every point in it is dense, and the same is true of every tail of that orbit; consequently each neighborhood is revisited at arbitrarily large times. A nonempty subsystem is all that is guaranteed: the identity map on a singleton has no larger subsystem.