Poisson stability

The property of a point $x$( a trajectory $f ^ { t } x$) of a dynamical system $f ^ { t }$( or $f ( t , \cdot )$, cf. [2]), 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 [1] 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. [1], [3]). A generalization of this theorem to dynamical systems given on a space of infinite measure is the Hopf recurrence theorem (cf. [2]): 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. [2]).

Comments

In Western literature on (abstract) topological dynamics (as opposed to the qualitative theory of differential equations) often the term "recurrent" is used for Poisson stability; see [a1]. For further comments, see Recurrent point.

References

[1]	H. Poincaré, "Les méthodes nouvelles de la mécanique céleste", 3 , Gauthier-Villars (1899) pp. Chapt. 26 Zbl 30.0834.08
[2]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations", Princeton Univ. Press (1960) (Translated from Russian)
[3]	J.C. Oxtoby, "Measure and category", Springer (1971) Zbl 0217.09201
[a1]	W.H. Gottschalk, G.A. Hedlund, "Topological dynamics", Amer. Math. Soc. (1955) Zbl 0067.15204