# Recurrent function

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A function that is a recurrent point of the shift dynamical system. An equivalent definition is: A function , where 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 such that the limit exists (the limit in the compact-open topology, i.e. uniformly on each segment) a sequence of numbers can be found such that in the compact-open topology.

If is a bounded uniformly-continuous function, then numbers can be found such that the limit (in the compact-open topology) 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=15003
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article