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One of the basic theorems in the general theory of dynamical systems with an [[Invariant measure|invariant measure]] (cf. also [[Ergodic theory|Ergodic theory]]).
 
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
 
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>
+
$$ \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
  
where the single-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p0731002.png" /> satisfy the condition
+
$$
 +
\sum _ { i= } 1 ^ { n }
  
<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/p0731003.png" /></td> </tr></table>
+
\frac{\partial  ( M X _ {i} ) }{\partial  x _ {i} }
 +
  = 0 ,\ \
 +
> 0 ,
 +
$$
  
 
so that equations (1) admit a positive [[Integral invariant|integral invariant]]
 
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>
+
$$ \tag{2 }
 +
\int\limits _ { V } M  d x _ {1} \dots d x _ {n} .
 +
$$
  
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
+
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
  
<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/p07310010.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { V } M  d x _ {1} \dots d x _ {n}  < \infty .
 +
$$
  
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.
+
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 <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.
+
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|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]]].
 
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" />,
+
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 $,
  
<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/p07310030.png" /></td> </tr></table>
+
$$
 +
\mu f ( A , t )  = \mu A ,\  - \infty < t < + \infty .
 +
$$
  
An invariant measure is the natural generalization of the integral invariants (2) for the differential equations (1). Assuming the measure of the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073100/p07310031.png" /> to be finite, Carathéodory proved that:
+
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 <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" />;
+
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 <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.
+
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 [[#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
+
A.Ya. Khinchin [[#References|[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
  
<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/p07310048.png" /></td> </tr></table>
+
$$
 +
\mu ( t)  = \mu ( E \cdot f ( E , t ) )  > \lambda m  ^ {2}
 +
$$
  
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" />).
+
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 [[#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.G. Chetaev (see [[#References|[6]]], [[#References|[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 [[#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]]]).
 
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]]]).
Line 45: Line 136:
 
====References====
 
====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>
 
<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====
 
====Comments====
 
In the literature the result discussed above is also often called the Poincaré recurrence theorem.
 
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.
+
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 [[#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]]].
+
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 $  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 [[#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.
 
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.

Revision as of 08:06, 6 June 2020


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
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