# Recurrent function

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.

#### References

[1] | N.A. Izobov, "Linear systems of ordinary differential equations" J. Soviet Math. , 5 (1976) pp. 46–96 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 71–146 |

#### Comments

A recurrent function is nothing but a point in a compact minimal set in a dynamical system of the form , where is the space of continuous functions with a pre-compact set in ( a metric space), endowed with the compact-open topology, and for and . In the case , this system is called the Bebutov system. In [a1], the recurrent functions (according to the above definition) are called minimal functions.

#### References

[a1] | J. Auslander, F. Hahn, "Point transitive flows, algebras of functions and the Bebutov system" Fund. Math. , 60 (1967) pp. 117–137 |

**How to Cite This Entry:**

Recurrent function.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Recurrent_function&oldid=15003