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

$$ \tag{1 } \frac{d x _ {i} }{dt} = \ X _ {i} ( x _ {1} \dots x _ {n} ) ,\ i = 1 \dots n , $$

where the single-valued functions $ X _ {i} ( x _ {1} \dots x _ {n} ) $ satisfy the condition

$$ \sum _ {i=1} ^ { n } \frac{\partial ( M X _ {i} ) }{\partial x _ {i} } = 0 ,\ \ M > 0 , $$

so that equations (1) admit a positive integral invariant

$$ \tag{2 } \int\limits _ { V } M d x _ {1} \dots d x _ {n} . $$

It is also assumed that if there exists a certain domain $ V $ of finite volume such that if a moving point $ P $ with coordinates $ x _ {1} \dots x _ {n} $ is found inside $ V $ at the initial moment of time $ t _ {0} $, then it will remain inside this domain for an arbitrary long time and

$$ \int\limits _ { V } M d x _ {1} \dots d x _ {n} < \infty . $$

The Poincaré return theorem: If one considers a domain $ U _ {0} $ contained in $ V $, then there is an infinite choice of initial positions of the point $ P $ such that the trajectory of $ P $ intersects the domain $ U _ {0} $ an infinite number of times. If this choice of the initial position is made at random inside $ U _ {0} $, then the probability that the point $ P $ does not intersect the domain $ U _ {0} $ 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 $ P $ 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 $ \mu A $ of any set $ A \subset R $ of a metric space $ R $, and considered a dynamical system $ f ( p , t ) $( $ p = P $ for $ t = 0 $) in $ R $; he then called the measure invariant with respect to the system $ f ( p , t ) $ if for any $ \mu $- measurable set $ A $,

$$ \mu f ( A , t ) = \mu A ,\ - \infty < t < + \infty . $$

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

1) if $ \mu A = m > 0 $, then values $ t $ can be found, $ | t | \geq 1 $, such that $ \mu [ A \cdot f ( A , t ) ] > 0 $, where $ A \cdot f ( A , t ) $ is the set of points belonging simultaneously to the sets $ A $ and $ f ( A , t ) $;

2) if in a space $ R $ with a countable base, $ \mu R = 1 $ for the invariant measure $ \mu $, then almost-all points $ p \in R $( in the sense of the measure $ \mu $) 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 $ E $, $ \mu E = m > 0 $, and for any $ t $, $ - \infty < t < + \infty $, the inequality

$$ \mu ( t) = \mu ( E \cdot f ( E , t ) ) > \lambda m ^ {2} $$

is satisfied for a relatively-dense set of values of $ t $ on the axis $ - \infty < t < + \infty $( for any $ \lambda < 1 $).

N.G. Chetaev (see [6], [7]) generalized Poincaré's theorem for the case when the functions $ X _ {i} $ in (1) depend also periodically on the time $ t $. Namely, let a) only real values of variables correspond to the real states of the system; b) the functions $ X _ {i} $ in the differential equations (1) of the motion be periodic with respect to $ t $ with a single period $ \tau $ common to them all; c) throughout its motion, the point $ P $ does not leave a certain closed domain $ R $ if its initial position $ P _ {0} $ is somewhere inside a given domain $ W _ {0} $; d) $ \mathop{\rm mes} W _ {k} \geq a \mathop{\rm mes} W _ {0} $, where $ \mathop{\rm mes} W _ {k} = \int _ {W _ {k} } d x _ {1} \dots d x _ {n} $ denotes the measure of the set $ W _ {k} $( volume in the sense of Lebesgue) which consists of those moving points at time $ t = t _ {0} + k \tau $ which started at time $ t _ {0} $ from $ W _ {0} $; $ k $ is a certain integer, and it is assumed that the constant $ a $ is not infinitesimally small. Then almost-everywhere in the domain $ W _ {0} $( 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 $ U _ {0} $ in the theorem need not be open: the theorem is true provided only that $ \mu ( U _ {0} ) > 0 $. The recurrence theorem is valid for volume-preserving flows on Riemannian manifolds $ V $ of finite volume. The recurrence theorem is also true for a discrete-time dynamical system, e.g. for a mapping $ f $ 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 $ H $- theorem (increasing entropy). In this respect the following estimation of the expected recurrence time is of interest: it is $ 1/ \mu ( E) $, where $ E $ denotes the "event" that recurs $ ( \mu ( E) > 0) $; for practical situations this time is much larger than the lifetime of the universe (by factors like $ 2 ^ {100} $); 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