View source for Poisson stability

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The property of a point  $  x $(
a trajectory  $  f ^ { t } x $) 
of a [[Dynamical system|dynamical system]]  $  f ^ { t } $(
or  $  f ( t , \cdot ) $, 
cf. [[#References|[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|Limit point of a trajectory]]) of the trajectory  $  f ^ { t } x $. 
The concept of Poisson stability was introduced by H. Poincaré [[#References|[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|Wandering point]]). Every fixed and every periodic point, more generally, every [[Recurrent point|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–Bendixson theory]]).

Poincaré's recurrence theorem (cf. [[Poincaré return theorem|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|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. [[#References|[1]]], [[#References|[3]]]). A generalization of this theorem to dynamical systems given on a space of infinite measure is the Hopf recurrence theorem (cf. [[#References|[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. [[#References|[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 [[#References|[a1]]]. For further comments, see [[Recurrent point|Recurrent point]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,   "Les méthodes nouvelles de la mécanique céleste" , '''3''' , Gauthier-Villars  (1899)  pp. Chapt. 26</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  J.C. Oxtoby,   "Measure and category" , Springer  (1971)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W.H. Gottschalk,   G.A. Hedlund,   "Topological dynamics" , Amer. Math. Soc.  (1955) {{ZBL|0067.15204}}</TD></TR>
</table>