# 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.

#### 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

A recurrent function is nothing but a point in a compact minimal set in a dynamical system of the form $( C _ {c} ^ \star ( \mathbf R , S), \{ \rho ^ {t} \} )$, where $C _ {c} ^ \star ( \mathbf R , S )$ is the space of continuous functions $f : \mathbf R \rightarrow S$ with $f ( \mathbf R )$ a pre-compact set in $S$( $S$ a metric space), endowed with the compact-open topology, and $( \rho ^ {t} f ) ( s) = f( s+ t)$ for $f \in C _ {c} ^ \star ( \mathbf R , S)$ and $s, t \in \mathbf R$. In the case $S = \mathbf R$, this system is called the Bebutov system. In [a1], the recurrent functions (according to the above definition) are called minimal functions.