# Recurrent point

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)

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=48455
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article