# 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  and ) and its proof was improved by C. Carathéodory .

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  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 , ) 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  described the structure of the invariant measure with respect to the given dynamical system for a very wide class of dynamical systems (see also ).

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