View source for Recurrent point

<!--
r0801901.png
$#A+1 = 43 n = 0
$#C+1 = 43 : ~/encyclopedia/old_files/data/R080/R.0800190 Recurrent point
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''of a dynamical system''

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

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

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|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  $  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====
<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====
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|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:

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r080190a.gif" />

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|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>

</td></tr> </table>

(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====
<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>