View source for Recurrent point

''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

<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>

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.

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]]).

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]]]).

====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 <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

<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>

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]].)

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

<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>

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:

<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 <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]]].

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., <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.)

====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>