# Recurrent function

A function that is a recurrent point of the shift dynamical system. An equivalent definition is: A function $\phi : \mathbf R \rightarrow S$, where $S$ is a metric space, is called recurrent if it has a pre-compact set of values, is uniformly continuous and if for each sequence of numbers $t _ {k} \in \mathbf R$ such that the limit

$$\widetilde \phi ( t) = \lim\limits _ {k \rightarrow \infty } \phi ( t _ {k} + t )$$

exists (the limit in the compact-open topology, i.e. uniformly on each segment) a sequence of numbers $\tau _ {k} \in \mathbf R$ can be found such that

$$\phi ( t) = \lim\limits _ {k \rightarrow \infty } \widetilde \phi ( \tau _ {k} + t )$$

in the compact-open topology.

If $\phi : \mathbf R \rightarrow \mathbf R ^ {n}$ is a bounded uniformly-continuous function, then numbers $t _ {k} \in \mathbf R$ can be found such that the limit (in the compact-open topology)

$$\widetilde \phi ( t) = \lim\limits _ {k \rightarrow \infty } \phi ( t _ {k} + t)$$

exists and is a recurrent function. Every almost-periodic function, and, in particular, every periodic function, is recurrent.

How to Cite This Entry:
Recurrent function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Recurrent_function&oldid=48454
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article