# 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

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
How to Cite This Entry:
Poincaré return theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_return_theorem&oldid=48207
This article was adapted from an original article by V.V. Rumyantsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article