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]).

References

[1]	H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , 3 , Gauthier-Villars (1899) pp. Chapt. 26
[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)

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

[a1]	W.H. Gottschalk, G.A. Hedlund, "Topological dynamics" , Amer. Math. Soc. (1955)