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A function that is a [[Recurrent point|recurrent point]] of the [[Shift dynamical system|shift dynamical system]]. An equivalent definition is: A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r0801801.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r0801802.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r0801803.png" /> such that the limit
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r0801804.png" /></td> </tr></table>
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exists (the limit in the compact-open topology, i.e. uniformly on each segment) a sequence of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r0801805.png" /> can be found such that
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A function that is a [[Recurrent point|recurrent point]] of the [[Shift dynamical system|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 $
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such that the limit
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r0801806.png" /></td> </tr></table>
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$$
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\widetilde \phi  ( t)  = \lim\limits _ {k \rightarrow \infty }  \phi ( t _ {k} + t )
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$$
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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
 +
 
 +
$$
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\phi ( t)  =  \lim\limits _ {k \rightarrow \infty }  \widetilde \phi  ( \tau _ {k} + t )
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$$
  
 
in the compact-open topology.
 
in the compact-open topology.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r0801807.png" /> is a bounded uniformly-continuous function, then numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r0801808.png" /> can be found such that the limit (in the compact-open topology)
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If $  \phi : \mathbf R \rightarrow \mathbf R  ^ {n} $
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is a bounded uniformly-continuous function, then numbers $  t _ {k} \in \mathbf R $
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can be found such that the limit (in the compact-open topology)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r0801809.png" /></td> </tr></table>
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$$
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\widetilde \phi  ( t)  = \lim\limits _ {k \rightarrow \infty }  \phi ( t _ {k} + t)
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$$
  
 
exists and is a recurrent function. Every [[Almost-periodic function|almost-periodic function]], and, in particular, every periodic function, is recurrent.
 
exists and is a recurrent function. Every [[Almost-periodic function|almost-periodic function]], and, in particular, every periodic function, is recurrent.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A recurrent function is nothing but a point in a compact [[Minimal set|minimal set]] in a dynamical system of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r08018010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r08018011.png" /> is the space of continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r08018012.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r08018013.png" /> a pre-compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r08018014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r08018015.png" /> a metric space), endowed with the compact-open topology, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r08018016.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r08018017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r08018018.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080180/r08018019.png" />, this system is called the Bebutov system. In [[#References|[a1]]], the recurrent functions (according to the above definition) are called minimal functions.
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A recurrent function is nothing but a point in a compact [[Minimal set|minimal set]] in a dynamical system of the form $  ( C _ {c}  ^  \star  ( \mathbf R , S), \{ \rho  ^ {t} \} ) $,  
 +
where $  C _ {c}  ^  \star  ( \mathbf R , S ) $
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is the space of continuous functions $  f : \mathbf R \rightarrow S $
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with $  f ( \mathbf R ) $
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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 [[#References|[a1]]], the recurrent functions (according to the above definition) are called minimal functions.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Auslander,  F. Hahn,  "Point transitive flows, algebras of functions and the Bebutov system"  ''Fund. Math.'' , '''60'''  (1967)  pp. 117–137</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Auslander,  F. Hahn,  "Point transitive flows, algebras of functions and the Bebutov system"  ''Fund. Math.'' , '''60'''  (1967)  pp. 117–137</TD></TR></table>

Latest revision as of 08:10, 6 June 2020


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
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