View source for Poincaré return theorem

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

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p0731001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where the single-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p0731002.png" /> satisfy the condition

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so that equations (1) admit a positive [[Integral invariant|integral invariant]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p0731004.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

It is also assumed that if there exists a certain domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p0731005.png" /> of finite volume such that if a moving point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p0731006.png" /> with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p0731007.png" /> is found inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p0731008.png" /> at the initial moment of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p0731009.png" />, then it will remain inside this domain for an arbitrary long time and

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The Poincaré return theorem: If one considers a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310011.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310012.png" />, then there is an infinite choice of initial positions of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310013.png" /> such that the trajectory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310014.png" /> intersects the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310015.png" /> an infinite number of times. If this choice of the initial position is made at random inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310016.png" />, then the probability that the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310017.png" /> does not intersect the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310018.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310019.png" /> 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|Poisson stability]]). The Poincaré return theorem was first established by Poincaré (see [[#References|[1]]] and [[#References|[2]]]) and its proof was improved by C. Carathéodory [[#References|[3]]].

Carathéodory used four axioms to introduce the abstract concept of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310020.png" /> of any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310021.png" /> of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310022.png" />, and considered a dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310023.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310024.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310025.png" />) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310026.png" />; he then called the measure invariant with respect to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310027.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310028.png" />-measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310029.png" />,

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An invariant measure is the natural generalization of the integral invariants (2) for the differential equations (1). Assuming the measure of the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310031.png" /> to be finite, Carathéodory proved that:

1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310032.png" />, then values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310033.png" /> can be found, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310034.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310036.png" /> is the set of points belonging simultaneously to the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310038.png" />;

2) if in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310039.png" /> with a countable base, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310040.png" /> for the invariant measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310041.png" />, then almost-all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310042.png" /> (in the sense of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310043.png" />) are stable in the sense of Poisson.

A.Ya. Khinchin [[#References|[5]]] made part 1) of this theorem more precise by proving that for each measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310045.png" />, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310047.png" />, the inequality

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is satisfied for a relatively-dense set of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310049.png" /> on the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310050.png" /> (for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310051.png" />).

N.G. Chetaev (see [[#References|[6]]], [[#References|[7]]]) generalized Poincaré's theorem for the case when the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310052.png" /> in (1) depend also periodically on the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310053.png" />. Namely, let a) only real values of variables correspond to the real states of the system; b) the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310054.png" /> in the differential equations (1) of the motion be periodic with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310055.png" /> with a single period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310056.png" /> common to them all; c) throughout its motion, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310057.png" /> does not leave a certain closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310058.png" /> if its initial position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310059.png" /> is somewhere inside a given domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310060.png" />; d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310061.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310062.png" /> denotes the measure of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310063.png" /> (volume in the sense of Lebesgue) which consists of those moving points at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310064.png" /> which started at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310065.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310066.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310067.png" /> is a certain integer, and it is assumed that the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310068.png" /> is not infinitesimally small. Then almost-everywhere in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310069.png" /> (apart perhaps on a set of measure zero) the trajectories are stable in the sense of Poisson.

N.M. Krylov and N.N. Bogolyubov [[#References|[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 [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,   "Sur le problème des trois corps et les équations de la dynamique"  ''Acta. Math.'' , '''13'''  (1890)  pp. 1–270</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Carathéodory,   "Ueber den Wiederkehrsatz von Poincaré"  ''Sitz. Ber. Preuss. Akad. Wiss. Berlin''  (1919)  pp. 580–584</TD></TR><TR><TD valign="top">[4]</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">[5]</TD> <TD valign="top">  A.Ya. Khinchin,   "Eine Verschärfung des Poincaréschen Wiederkehrsatzes"  ''Comp. Math.'' , '''1'''  (1934)  pp. 177–179</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N.G. Chetaev,   "Sur la stabilité à la Poisson"  ''C.R. Acad. Sci. Paris'' , '''187'''  (1928)  pp. 637–638</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  N.G. Chetaev,   ''Uchen. Zap. Kazan. Univ.'' , '''89''' :  2  (1929)  pp. 199–201</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  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</TD></TR></table>



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

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310070.png" /> in the theorem need not be open: the theorem is true provided only that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310071.png" />. The recurrence theorem is valid for volume-preserving flows on Riemannian manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310072.png" /> of finite volume. The recurrence theorem is also true for a discrete-time dynamical system, e.g. for a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310073.png" /> of a bounded domain in Euclidean space to itself that preserves Lebesgue measure. See [[#References|[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|Boltzmann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310074.png" />-theorem]] (increasing entropy). In this respect the following estimation of the expected recurrence time is of interest: it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310075.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310076.png" /> denotes the  "event"  that recurs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310077.png" />; for practical situations this time is much larger than the lifetime of the universe (by factors like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310078.png" />); see [[#References|[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|measure-preserving transformation]] (with as a reverse construction that of a primitive transformation). See [[#References|[a3]]] or [[#References|[a4]]], pp. 39, 40.

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