Poincaré return theorem

One of the basic theorems in the general theory of dynamical systems with an invariant measure (cf. also Ergodic theory).

Let the motion of a system be described by the differential equations

(1)

where the single-valued functions satisfy the condition

so that equations (1) admit a positive integral invariant

(2)

It is also assumed that if there exists a certain domain of finite volume such that if a moving point with coordinates is found inside at the initial moment of time , then it will remain inside this domain for an arbitrary long time and

The Poincaré return theorem: If one considers a domain contained in , then there is an infinite choice of initial positions of the point such that the trajectory of intersects the domain an infinite number of times. If this choice of the initial position is made at random inside , then the probability that the point does not intersect the domain an infinite number of times will be infinitely small.

In other words, if the initial conditions are not exceptional in the sense indicated, then the point passes infinitely often arbitrarily near to its initial position.

H. Poincaré called a motion in which the system returns an infinite number of times to a neighbourhood of the initial state stable in the sense of Poisson (see Poisson stability). The Poincaré return theorem was first established by Poincaré (see [1] and [2]) and its proof was improved by C. Carathéodory [3].

Carathéodory used four axioms to introduce the abstract concept of the measure of any set of a metric space , and considered a dynamical system ( for ) in ; he then called the measure invariant with respect to the system if for any -measurable set ,

An invariant measure is the natural generalization of the integral invariants (2) for the differential equations (1). Assuming the measure of the whole space to be finite, Carathéodory proved that:

1) if , then values can be found, , such that , where is the set of points belonging simultaneously to the sets and ;

2) if in a space with a countable base, for the invariant measure , then almost-all points (in the sense of the measure ) are stable in the sense of Poisson.

A.Ya. Khinchin [5] made part 1) of this theorem more precise by proving that for each measurable set , , and for any , , the inequality

is satisfied for a relatively-dense set of values of on the axis (for any ).

N.G. Chetaev (see [6], [7]) generalized Poincaré's theorem for the case when the functions in (1) depend also periodically on the time . Namely, let a) only real values of variables correspond to the real states of the system; b) the functions in the differential equations (1) of the motion be periodic with respect to with a single period common to them all; c) throughout its motion, the point does not leave a certain closed domain if its initial position is somewhere inside a given domain ; d) , where denotes the measure of the set (volume in the sense of Lebesgue) which consists of those moving points at time which started at time from ; is a certain integer, and it is assumed that the constant is not infinitesimally small. Then almost-everywhere in the domain (apart perhaps on a set of measure zero) the trajectories are stable in the sense of Poisson.

N.M. Krylov and N.N. Bogolyubov [8] described the structure of the invariant measure with respect to the given dynamical system for a very wide class of dynamical systems (see also [4]).

References

[1]	H. Poincaré, "Sur le problème des trois corps et les équations de la dynamique" Acta. Math. , 13 (1890) pp. 1–270
[2]	H. Poincaré, "Sur le problème des trois corps et les équations de la dynamique" , Oeuvres , XII , Gauthier-Villars (1952) pp. 262–479 (in particular, p. 314)
[3]	C. Carathéodory, "Ueber den Wiederkehrsatz von Poincaré" Sitz. Ber. Preuss. Akad. Wiss. Berlin (1919) pp. 580–584
[4]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[5]	A.Ya. Khinchin, "Eine Verschärfung des Poincaréschen Wiederkehrsatzes" Comp. Math. , 1 (1934) pp. 177–179
[6]	N.G. Chetaev, "Sur la stabilité à la Poisson" C.R. Acad. Sci. Paris , 187 (1928) pp. 637–638
[7]	N.G. Chetaev, Uchen. Zap. Kazan. Univ. , 89 : 2 (1929) pp. 199–201
[8]	N.N. Krylov, N.N. Bogolyubov, "La théorie de la mesure dans son application à l'etude des systèmes dynamiques de la mécanique non-linéaire" Ann. of Math. , 38 : 1 (1937) pp. 65–113

Comments

In the literature the result discussed above is also often called the Poincaré recurrence theorem.

The set in the theorem need not be open: the theorem is true provided only that . The recurrence theorem is valid for volume-preserving flows on Riemannian manifolds of finite volume. The recurrence theorem is also true for a discrete-time dynamical system, e.g. for a mapping of a bounded domain in Euclidean space to itself that preserves Lebesgue measure. See [a1] for another generalization.

There seems to be an incompatibility of the prediction by Poincaré's recurrence theorem (namely, almost surely a system will recur arbitrarily close to its original state) with the conclusions of thermodynamics as the second law and the Boltzmann -theorem (increasing entropy). In this respect the following estimation of the expected recurrence time is of interest: it is , where denotes the "event" that recurs ; for practical situations this time is much larger than the lifetime of the universe (by factors like ); see [a2].

The Poincaré recurrence theorem was used by S. Kakutani as the basis for an important construction: that of the induced or derivative transformation of a measure-preserving transformation (with as a reverse construction that of a primitive transformation). See [a3] or [a4], pp. 39, 40.

References

[a1]	P.R. Halmos, "Invariant measures" Ann. of Math. , 48 (1947) pp. 735–754
[a2]	M. Kac, "On the notion of recurrence in discrete stochastic processes" Bull. Amer. Math. Soc. , 53 (1947) pp. 1002–1010
[a3]	S. Kakutani, "Induced measure preserving transformations" Proc. Japan. Acad. , 19 (1943) pp. 635–641
[a4]	K. Petersen, "Ergodic theory" , Cambridge Univ. Press (1983) pp. 39