# Poisson stability

The property of a point $x$( a trajectory $f ^ { t } x$) of a dynamical system $f ^ { t }$( or $f ( t , \cdot )$, cf. ), given in a topological space $S$, consisting in the following: There are sequences $t _ {k} \rightarrow + \infty$, $\tau _ {k} \rightarrow - \infty$ such that

$$\lim\limits _ {k \rightarrow \infty } f ^ { t _ {k} } x = \ \lim\limits _ {k \rightarrow \infty } f ^ { \tau _ {k} } x = x .$$

In other words, $x$ is an $\alpha$- and $\omega$- limit point (cf. Limit point of a trajectory) of the trajectory $f ^ { t } x$. The concept of Poisson stability was introduced by H. Poincaré  on the basis of an analysis of results of Poisson on the stability of planetary orbits.

Every Poisson-stable point is non-wandering; the converse is not true (cf. Wandering point). Every fixed and every periodic point, more generally, every recurrent point, is Poisson stable. If $S = \mathbf R ^ {2}$ and the dynamical system is smooth (i.e. given by a vector field of class $C ^ {1}$), then every Poisson-stable point is either fixed or periodic (cf. Poincaré–Bendixson theory).

Poincaré's recurrence theorem (cf. Poincaré return theorem): If a dynamical system is given in a bounded domain of $\mathbf R ^ {n}$ and if Lebesgue measure is an invariant measure of the system, then all points are Poisson stable, with the exception of a certain set of the first category of measure zero (cf. , ). A generalization of this theorem to dynamical systems given on a space of infinite measure is the Hopf recurrence theorem (cf. ): If a dynamical system is given on an arbitrary domain in $\mathbf R ^ {n}$( e.g. on $\mathbf R ^ {n}$ itself) and if Lebesgue measure is an invariant measure of the system, then every point, with the exception of the points of a certain set of measure zero, is either Poisson stable or divergent, i.e.

$$| f ^ { t } x | \rightarrow \infty \ \textrm{ as } | t | \rightarrow \infty .$$

There are still more general formulations of the theorems of Poincaré and E. Hopf (cf. ).

How to Cite This Entry:
Poisson stability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_stability&oldid=48221
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article