## Return to Answer

1 [made Community Wiki]

Poincare Recurrence Theorem: http://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem

Let $(X,\Sigma,m)$ be a finite measure space and let $f:X \to X$ be a measure-preserving map. If $E \in \Sigma$, then almost every point in $E$ returns to $E$; i.e., $m ({x \in E: \exists N: \forall n>N \quad f^n(x) \not \in E })=0$

A proof can be found e.g. in Arnold's "Mechanics"; there are some on PlanetMath, too. All use basically the definition of a measure, and maybe (or not) a necessary condition for convergence of a series of real numbers.

The theorem describes behavior of certain systems in statistical mechanics or thermodynamics, but it also has many mathematical consequences. It was one of first results in ergodic theory. It can be used to prove e.g. that an orbit of an irrational rotation of a circle is dense. Relations with recent developments in ergodic theory and dynamical systems are discussed by Barreira, doi:10.1142/9789812704016_0039
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