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

Comments

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.

References

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