# Recurrent point

of a dynamical system

A point $x$ of a dynamical system $f ^ { t }$( also denoted by $f( t, \cdot )$, see ) 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 ).

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