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''of a dynamical system''
 
''of a dynamical system''
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r0801901.png" /> of a [[Dynamical system|dynamical system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r0801902.png" /> (also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r0801903.png" />, see [[#References|[2]]]) in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r0801904.png" /> that satisfies the following condition: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r0801905.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r0801906.png" /> such that all points of the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r0801907.png" /> are contained in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r0801908.png" />-neighbourhood of any arc of time length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r0801909.png" /> of this trajectory (in other words, with any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019010.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019011.png" />-neighbourhood of the set
+
A point $  x $
 +
of a [[Dynamical system|dynamical system]] $  f ^ { t } $(
 +
also denoted by $  f( t, \cdot ) $,  
 +
see [[#References|[2]]]) in a metric space $  S $
 +
that satisfies the following condition: For any $  \epsilon > 0 $
 +
there exists a $  T > 0 $
 +
such that all points of the trajectory $  f ^ { t } x $
 +
are contained in an $  \epsilon $-
 +
neighbourhood of any arc of time length $  T $
 +
of this trajectory (in other words, with any $  \tau \in \mathbf R $,  
 +
the $  \epsilon $-
 +
neighbourhood of the set
  
<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/r080190/r08019012.png" /></td> </tr></table>
+
$$
 +
\{ {f ^ { t } x } : {t \in [ \tau , \tau + T ] } \}
 +
$$
  
contains all of the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019013.png" />). In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019014.png" /> is called a recurrent trajectory.
+
contains all of the trajectory $  f ^ { t } x $).  
 +
In this case $  f ^ { t } x $
 +
is called a recurrent trajectory.
  
Birkhoff's theorem: If the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019015.png" /> is complete (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019016.png" />), then: 1) for a point to be recurrent it is necessary and sufficient that the closure of its trajectory be a compact [[Minimal set|minimal set]]; and 2) for the existence of a recurrent point it is sufficient that there be a point that is stable according to Lagrange (see [[Lagrange stability|Lagrange stability]]).
+
Birkhoff's theorem: If the space $  S $
 +
is complete (e.g. $  S = \mathbf R  ^ {n} $),  
 +
then: 1) for a point to be recurrent it is necessary and sufficient that the closure of its trajectory be a compact [[Minimal set|minimal set]]; and 2) for the existence of a recurrent point it is sufficient that there be a point that is stable according to Lagrange (see [[Lagrange stability|Lagrange stability]]).
  
A recurrent point is stable according to Poisson (see [[Poisson stability|Poisson stability]]), and if the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019017.png" /> is complete, also stable according to Lagrange (see [[Lagrange stability|Lagrange stability]]). An almost-periodic (in particular, a fixed or periodic) point of a dynamical system is recurrent. In general, any point of a strictly ergodic dynamical system (in a complete space) is recurrent, but the restriction of a dynamical system to the closure of a recurrent trajectory (a minimal set) need not be a strictly ergodic dynamical system (Markov's example, see [[#References|[2]]]).
+
A recurrent point is stable according to Poisson (see [[Poisson stability|Poisson stability]]), and if the space $  S $
 +
is complete, also stable according to Lagrange (see [[Lagrange stability|Lagrange stability]]). An almost-periodic (in particular, a fixed or periodic) point of a dynamical system is recurrent. In general, any point of a strictly ergodic dynamical system (in a complete space) is recurrent, but the restriction of a dynamical system to the closure of a recurrent trajectory (a minimal set) need not be a strictly ergodic dynamical system (Markov's example, see [[#References|[2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.D. Birkhoff,  "Dynamical systems" , Amer. Math. Soc.  (1927)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.D. Birkhoff,  "Dynamical systems" , Amer. Math. Soc.  (1927)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
An almost-periodic point of a dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019018.png" /> on a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019019.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019020.png" /> with the following property: For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019021.png" /> the set
+
An almost-periodic point of a dynamical system $  f ^ { t } $
 +
on a metric space $  ( S , \rho ) $
 +
is a point $  x \in G $
 +
with the following property: For every $  \epsilon > 0 $
 +
the set
  
<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/r080190/r08019022.png" /></td> </tr></table>
+
$$
 +
AP ( x, \epsilon )  = \{ {t \in \mathbf R } : {
 +
\rho ( f ^ { s+ t } ( x), f ^ { s } ( x) ) < \epsilon \
 +
\textrm{ for  all  } s \in \mathbf R } \}
 +
$$
  
is relatively dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019023.png" />, i.e., there exists a length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019024.png" /> such that every interval in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019025.png" /> with length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019026.png" /> contains a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019027.png" />. (Thus, one might say that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019028.png" /> is almost-periodic; cf. [[Almost-period|Almost-period]].)
+
is relatively dense in $  \mathbf R $,  
 +
i.e., there exists a length $  l ( \epsilon ) $
 +
such that every interval in $  \mathbf R $
 +
with length $  h \geq  l ( \epsilon ) $
 +
contains a point of $  AP( x, \epsilon ) $.  
 +
(Thus, one might say that the function $  t \mapsto f ^ { t } ( x) : \mathbf R \rightarrow S $
 +
is almost-periodic; cf. [[Almost-period|Almost-period]].)
  
Another important notion is that of an almost-recurrent point: A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019029.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019030.png" /> the set
+
Another important notion is that of an almost-recurrent point: A point $  x \in S $
 +
such that for every $  \epsilon > 0 $
 +
the set
  
<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/r080190/r08019031.png" /></td> </tr></table>
+
$$
 +
R( x , U _  \epsilon  )  = \
 +
\{ {t \in \mathbf R } : {f ^ { t } ( x) \in U _  \epsilon  } \}
 +
$$
  
is relatively dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019033.png" /> is the open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019034.png" />-ball around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019035.png" />. (This definition can easily be generalized to the case of a flow or a cascade on an arbitrary topological space.) The relationship of these notions with that of a recurrent point, Birkhoff's theorem and a number of other implications can be visualized as follows:
+
is relatively dense in $  \mathbf R $,  
 +
where $  U _  \epsilon  $
 +
is the open $  \epsilon $-
 +
ball around $  x $.  
 +
(This definition can easily be generalized to the case of a flow or a cascade on an arbitrary topological space.) The relationship of these notions with that of a recurrent point, Birkhoff's theorem and a number of other implications can be visualized as follows:
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r080190a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r080190a.gif" />
Line 33: Line 84:
 
Figure: r080190a
 
Figure: r080190a
  
Here the implication indicated by the dotted arrow holds only in a complete space, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019036.png" /> denotes the closure of the trajectory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019037.png" />. The property  "Sx is Lyapunov stable rel(ative) Sx"  means that the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019038.png" /> of functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019039.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019040.png" /> is equicontinuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019041.png" /> (see [[Lyapunov stability|Lyapunov stability]]). For a refinement of this diagram (including the notions of a pseudo-recurrent point and a uniformly Poisson-stable trajectory) see [[#References|[a3]]].
+
Here the implication indicated by the dotted arrow holds only in a complete space, and $  \Sigma _ {x} $
 +
denotes the closure of the trajectory of $  x $.  
 +
The property  "Sx is Lyapunov stable rel(ative) Sx"  means that the family $  \{ f ^ { t } \mid  _ {\Sigma _ {x}  } \} _ {t \in \mathbf R }  $
 +
of functions from $  \Sigma _ {x} $
 +
into $  \Sigma _ {x} $
 +
is equicontinuous on $  \Sigma _ {x} $(
 +
see [[Lyapunov stability|Lyapunov stability]]). For a refinement of this diagram (including the notions of a pseudo-recurrent point and a uniformly Poisson-stable trajectory) see [[#References|[a3]]].
  
In the literature on topological dynamics (in particular, in the literature directly or indirectly influenced by [[#References|[a2]]]) another terminology is in use:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">Above, [[#References|[2]]], [[#References|[a3]]]</td> <td colname="2" style="background-color:white;" colspan="1">[[#References|[a2]]]</td> <td colname="3" style="background-color:white;" colspan="1">[[#References|[a1]]]</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">almost periodic</td> <td colname="2" style="background-color:white;" colspan="1">—</td> <td colname="3" style="background-color:white;" colspan="1">—</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">recurrent</td> <td colname="2" style="background-color:white;" colspan="1">—</td> <td colname="3" style="background-color:white;" colspan="1">—</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">almost recurrent</td> <td colname="2" style="background-color:white;" colspan="1">almost periodic</td> <td colname="3" style="background-color:white;" colspan="1">uniformly recurrent</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Poisson-stable</td> <td colname="2" style="background-color:white;" colspan="1">recurrent</td> <td colname="3" style="background-color:white;" colspan="1">recurrent</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">(non-) wandering</td> <td colname="2" style="background-color:white;" colspan="1">(not) regionally recurrent</td> <td colname="3" style="background-color:white;" colspan="1">—</td> </tr> </tbody> </table>
+
In the literature on topological dynamics (in particular, in the literature directly or indirectly influenced by [[#References|[a2]]]) another terminology is in use:<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">Above, [[#References|[2]]], [[#References|[a3]]]</td> <td colname="2" style="background-color:white;" colspan="1">[[#References|[a2]]]</td> <td colname="3" style="background-color:white;" colspan="1">[[#References|[a1]]]</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">almost periodic</td> <td colname="2" style="background-color:white;" colspan="1">—</td> <td colname="3" style="background-color:white;" colspan="1">—</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">recurrent</td> <td colname="2" style="background-color:white;" colspan="1">—</td> <td colname="3" style="background-color:white;" colspan="1">—</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">almost recurrent</td> <td colname="2" style="background-color:white;" colspan="1">almost periodic</td> <td colname="3" style="background-color:white;" colspan="1">uniformly recurrent</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Poisson-stable</td> <td colname="2" style="background-color:white;" colspan="1">recurrent</td> <td colname="3" style="background-color:white;" colspan="1">recurrent</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">(non-) wandering</td> <td colname="2" style="background-color:white;" colspan="1">(not) regionally recurrent</td> <td colname="3" style="background-color:white;" colspan="1">—</td> </tr> </tbody> </table>
  
 
</td></tr> </table>
 
</td></tr> </table>
  
(To be precise, in [[#References|[a1]]]  "recurrent"  means positive Poisson stable, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019042.png" /> belongs only to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080190/r08019043.png" />-limit set of its own trajectory.)
+
(To be precise, in [[#References|[a1]]]  "recurrent"  means positive Poisson stable, i.e., $  x $
 +
belongs only to the $  \omega $-
 +
limit set of its own trajectory.)
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Furstenberg,  "Recurrence in ergodic theory and combinatorial number theory" , Princeton Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.H. Gottschalk,  G.A. Hedlund,  "Topological dynamics" , Amer. Math. Soc.  (1955)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K.S. [K.S. Sibirskii] Sibirsky,  "Introduction to topological dynamics" , Noordhoff  (1975)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Furstenberg,  "Recurrence in ergodic theory and combinatorial number theory" , Princeton Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.H. Gottschalk,  G.A. Hedlund,  "Topological dynamics" , Amer. Math. Soc.  (1955)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K.S. [K.S. Sibirskii] Sibirsky,  "Introduction to topological dynamics" , Noordhoff  (1975)  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:10, 6 June 2020


of a dynamical system

A point $ x $ of a dynamical system $ f ^ { t } $( also denoted by $ f( t, \cdot ) $, see [2]) in a metric space $ S $ that satisfies the following condition: For any $ \epsilon > 0 $ there exists a $ T > 0 $ such that all points of the trajectory $ f ^ { t } x $ are contained in an $ \epsilon $- neighbourhood of any arc of time length $ T $ of this trajectory (in other words, with any $ \tau \in \mathbf R $, the $ \epsilon $- neighbourhood of the set

$$ \{ {f ^ { t } x } : {t \in [ \tau , \tau + T ] } \} $$

contains all of the trajectory $ f ^ { t } x $). In this case $ f ^ { t } x $ is called a recurrent trajectory.

Birkhoff's theorem: If the space $ S $ is complete (e.g. $ S = \mathbf R ^ {n} $), then: 1) for a point to be recurrent it is necessary and sufficient that the closure of its trajectory be a compact minimal set; and 2) for the existence of a recurrent point it is sufficient that there be a point that is stable according to Lagrange (see Lagrange stability).

A recurrent point is stable according to Poisson (see Poisson stability), and if the space $ S $ is complete, also stable according to Lagrange (see Lagrange stability). An almost-periodic (in particular, a fixed or periodic) point of a dynamical system is recurrent. In general, any point of a strictly ergodic dynamical system (in a complete space) is recurrent, but the restriction of a dynamical system to the closure of a recurrent trajectory (a minimal set) need not be a strictly ergodic dynamical system (Markov's example, see [2]).

References

[1] G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927)
[2] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)

Comments

An almost-periodic point of a dynamical system $ f ^ { t } $ on a metric space $ ( S , \rho ) $ is a point $ x \in G $ with the following property: For every $ \epsilon > 0 $ the set

$$ AP ( x, \epsilon ) = \{ {t \in \mathbf R } : { \rho ( f ^ { s+ t } ( x), f ^ { s } ( x) ) < \epsilon \ \textrm{ for all } s \in \mathbf R } \} $$

is relatively dense in $ \mathbf R $, i.e., there exists a length $ l ( \epsilon ) $ such that every interval in $ \mathbf R $ with length $ h \geq l ( \epsilon ) $ contains a point of $ AP( x, \epsilon ) $. (Thus, one might say that the function $ t \mapsto f ^ { t } ( x) : \mathbf R \rightarrow S $ is almost-periodic; cf. Almost-period.)

Another important notion is that of an almost-recurrent point: A point $ x \in S $ such that for every $ \epsilon > 0 $ the set

$$ R( x , U _ \epsilon ) = \ \{ {t \in \mathbf R } : {f ^ { t } ( x) \in U _ \epsilon } \} $$

is relatively dense in $ \mathbf R $, where $ U _ \epsilon $ is the open $ \epsilon $- ball around $ x $. (This definition can easily be generalized to the case of a flow or a cascade on an arbitrary topological space.) The relationship of these notions with that of a recurrent point, Birkhoff's theorem and a number of other implications can be visualized as follows:

Figure: r080190a

Here the implication indicated by the dotted arrow holds only in a complete space, and $ \Sigma _ {x} $ denotes the closure of the trajectory of $ x $. The property "Sx is Lyapunov stable rel(ative) Sx" means that the family $ \{ f ^ { t } \mid _ {\Sigma _ {x} } \} _ {t \in \mathbf R } $ of functions from $ \Sigma _ {x} $ into $ \Sigma _ {x} $ is equicontinuous on $ \Sigma _ {x} $( see Lyapunov stability). For a refinement of this diagram (including the notions of a pseudo-recurrent point and a uniformly Poisson-stable trajectory) see [a3].

In the literature on topological dynamics (in particular, in the literature directly or indirectly influenced by [a2]) another terminology is in use:

<tbody> </tbody>
Above, [2], [a3] [a2] [a1]
almost periodic
recurrent
almost recurrent almost periodic uniformly recurrent
Poisson-stable recurrent recurrent
(non-) wandering (not) regionally recurrent

(To be precise, in [a1] "recurrent" means positive Poisson stable, i.e., $ x $ belongs only to the $ \omega $- limit set of its own trajectory.)

References

[a1] H. Furstenberg, "Recurrence in ergodic theory and combinatorial number theory" , Princeton Univ. Press (1981)
[a2] W.H. Gottschalk, G.A. Hedlund, "Topological dynamics" , Amer. Math. Soc. (1955)
[a3] K.S. [K.S. Sibirskii] Sibirsky, "Introduction to topological dynamics" , Noordhoff (1975) (Translated from Russian)
How to Cite This Entry:
Recurrent point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Recurrent_point&oldid=15436
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article