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''Oseledets's multiplicative ergodic theorem, Oseledec's multiplicative ergodic theorem''
 
''Oseledets's multiplicative ergodic theorem, Oseledec's multiplicative ergodic theorem''
  
 
Consider a linear homogeneous system of differential equations
 
Consider a linear homogeneous system of 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/m/m065/m065440/m0654401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
\dot{x}  = A ( t) x ,\ \
 +
x ( 0 ; x _ {0} ) = x _ {0} \in \mathbf R  ^ {n} ,\ \
 +
t \geq  0 .
 +
$$
 +
 
 +
The Lyapunov exponent of a solution  $  x ( t ; x _ {0} ) $
 +
of (a1) is defined as
 +
 
 +
$$
 +
\lambda ( x _ {0} )  = {\lim\limits  \sup } _ {t \rightarrow
 +
\infty } \
 +
t  ^ {-} 1  \mathop{\rm log}  \| x ( t ; x _ {0} ) \| .
 +
$$
  
The Lyapunov exponent of a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m0654402.png" /> of (a1) is defined as
+
A more general setting (Lyapunov exponents for families of system of differential equations) for discussing Lyapunov exponents and related matters is as follows. Let  $  \Phi = ( \Phi _ {t} ) _ {t \in \mathbf R }  $
 +
be a [[Measurable flow|measurable flow]] on a measure space  $  ( E , \mu ) $.  
 +
For all  $  e \in E $,
 +
let  $  V _ {e} $
 +
be an  $  n $-
 +
dimensional vector space. (Think, for example, of a vector bundle  $  T \rightarrow E $.)
 +
A cocycle  $  C ( t , e ) $
 +
associated with the flow  $  \Phi $
 +
is a measurable function on  $  \mathbf R \times E $
 +
that assigns to  $  ( t , e ) $
 +
an invertible linear mapping  $  V _ {e} \rightarrow V _ {\Phi _ {t}  ( e) } $
 +
such that
  
<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/m/m065/m065440/m0654403.png" /></td> </tr></table>
+
$$ \tag{a2 }
 +
C ( t + s , e )  = C ( t , \Phi _ {s} ( e) ) C ( s , e ) .
 +
$$
  
A more general setting (Lyapunov exponents for families of system of differential equations) for discussing Lyapunov exponents and related matters is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m0654404.png" /> be a [[Measurable flow|measurable flow]] on a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m0654405.png" />. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m0654406.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m0654407.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m0654408.png" />-dimensional vector space. (Think, for example, of a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m0654409.png" />.) A cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544010.png" /> associated with the flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544011.png" /> is a measurable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544012.png" /> that assigns to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544013.png" /> an invertible linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544014.png" /> such that
+
I.e. if the collection of vector spaces  $  V _ {e} $
 +
is viewed as an $  n $-
 +
dimensional vector bundle over  $  E $,
 +
then  $  C ( t , \cdot ) $
 +
defines an isomorphism of vector bundles  $  \widetilde \Phi  _ {t} $
 +
over  $  \Phi _ {t} $,
  
<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/m/m065/m065440/m06544015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$
  
I.e. if the collection of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544016.png" /> is viewed as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544017.png" />-dimensional vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544019.png" /> defines an isomorphism of vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544020.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544021.png" />,
+
\begin{array}{ccc}
 +
V  & \mathop \rightarrow \limits ^ { {\widetilde \Phi  _ {t} }}  & V  \\
 +
\downarrow  &{}  &\downarrow  \\
 +
E  & \mathop \rightarrow \limits _ { {\Phi _ {t} }}  & E  \\
 +
\end{array}
  
<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/m/m065/m065440/m06544022.png" /></td> </tr></table>
+
$$
  
and condition (a2) simply says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544023.png" />. So <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544024.png" /> is a flow on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544025.png" /> that lifts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544026.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544027.png" /> is sometimes called the skew product flow defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544029.png" />. This set-up is sufficiently general to discuss Lyapunov exponents for non-linear flows, and even stochastic non-linear flows and such things as products of random matrices. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544031.png" />, the classical situation (a1) reappears. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544032.png" /> be a differential equation on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544033.png" />. Take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544034.png" />, the tangent bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544035.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544036.png" /> be the flow on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544037.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544038.png" />. The associated cocycle is defined by the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544040.png" />,
+
and condition (a2) simply says that $  \widetilde \Phi  _ {t+} s = \widetilde \Phi  _ {t} \circ \widetilde \Phi  _ {s} $.  
 +
So $  \widetilde \Phi  $
 +
is a flow on $  V $
 +
that lifts $  \Phi $.  
 +
$  \widetilde \Phi  $
 +
is sometimes called the skew product flow defined by $  \Phi $
 +
and $  C $.  
 +
This set-up is sufficiently general to discuss Lyapunov exponents for non-linear flows, and even stochastic non-linear flows and such things as products of random matrices. If $  E = \{ e \} $,  
 +
$  \Phi _ {t} = \mathop{\rm id} $,  
 +
the classical situation (a1) reappears. Let $  \dot{x} = f ( x) $
 +
be a differential equation on a manifold $  M $.  
 +
Take $  V = T M $,  
 +
the tangent bundle over $  M $.  
 +
Let $  \Phi _ {t} $
 +
be the flow on $  M $
 +
defined by $  \dot{x} = f ( x) $.  
 +
The associated cocycle is defined by the differential $  d \Phi _ {t} $
 +
of $  \Phi _ {t} $,
  
<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/m/m065/m065440/m06544041.png" /></td> </tr></table>
+
$$
 +
C ( t , m )  = d \Phi _ {t} ( m) : \
 +
T _ {m} M  \rightarrow  T _ {\Phi _ {t}  ( m) } M .
 +
$$
  
For a skew product flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544042.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544043.png" /> the Lyapunov exponent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544044.png" /> in the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544045.png" /> is defined by
+
For a skew product flow $  \widetilde \Phi  $
 +
on $  V $
 +
the Lyapunov exponent at $  e \in E $
 +
in the direction $  v \in V _ {e} $
 +
is defined by
  
<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/m/m065/m065440/m06544046.png" /></td> </tr></table>
+
$$
 +
\lambda ( e , v )  = {\lim\limits  \sup } _ {t \rightarrow \infty
 +
}  t  ^ {-} 1  \mathop{\rm log}  \| C ( t , e ) v \| .
 +
$$
  
The multiplicative ergodic theorem of V.I. Oseledets [[#References|[a1]]] now is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544047.png" /> be a skew product flow and assume that there is an invariant probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544048.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544049.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544050.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544051.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544052.png" />. Suppose, moreover, that
+
The multiplicative ergodic theorem of V.I. Oseledets [[#References|[a1]]] now is as follows. Let $  \widetilde \Phi  $
 +
be a skew product flow and assume that there is an invariant probability measure $  \rho $
 +
on $  ( E , \mu ) $
 +
for $  \Phi $,  
 +
i.e. $  \Phi _ {t} \rho = \rho $
 +
for all $  t \in \mathbf R $.  
 +
Suppose, moreover, that
  
<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/m/m065/m065440/m06544053.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { E }  \sup _ {- 1 \leq  t \leq  1 }  \mathop{\rm log}
 +
^ {+}  \| C ^ {\pm  1 } ( t , e ) \|  d \rho  < \infty .
 +
$$
  
Then there exists a measurable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544054.png" />-invariant set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544055.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544056.png" />-measure 1 such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544057.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544058.png" /> numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544060.png" />, and corresponding subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544061.png" /> of dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544062.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544063.png" />,
+
Then there exists a measurable $  \Phi $-
 +
invariant set $  E _ {0} \subset  E $
 +
of $  \rho $-
 +
measure 1 such that for all $  x \in E _ {0} $
 +
there are $  l ( e) $
 +
numbers  $  \lambda _ {e}  ^ {l} < \dots < \lambda _ {1}  ^ {l} $,  
 +
$  l ( e) \leq  d $,  
 +
and corresponding subspaces 0 \subset  W _ {e}  ^ {l} \subset  \dots \subset  W _ {e}  ^ {1} = V _ {e} $
 +
of dimensions $  d _ {e}  ^ {l} < \dots < d _ {e}  ^ {1} = d $
 +
such that for all $  i = 1 \dots l ( e) $,
  
<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/m/m065/m065440/m06544064.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow \infty }  t  ^ {-} 1  \mathop{\rm log} \
 +
\| C ( t , e ) v \|  = \lambda _ {e}  ^ {i}
 +
\  \iff \  v \in W _ {e}  ^ {i} \setminus  W _ {e}  ^ {i+} 1 .
 +
$$
  
If moreover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544065.png" /> is ergodic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544066.png" />, i.e. all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544067.png" />-invariant subsets have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544068.png" />-measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544069.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544070.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544073.png" /> are constants independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544074.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544075.png" />). However, the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544076.png" /> may still depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544077.png" /> (if the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544078.png" /> is a trivial bundle so that all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544079.png" /> can be identified). The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065440/m06544080.png" /> is called the Lyapunov spectrum of the flow. For more details and applications cf. [[#References|[a2]]], [[#References|[a3]]].
+
If moreover $  \rho $
 +
is ergodic for $  \Phi _ {t} $,  
 +
i.e. all $  \Phi _ {t} $-
 +
invariant subsets have $  \rho $-
 +
measure 0 $
 +
or $  1 $,  
 +
then the $  l ( e) $,  
 +
$  \lambda _ {e}  ^ {i} $,  
 +
$  d _ {e}  ^ {i} $
 +
are constants independent of $  e $(
 +
or $  E _ {0} $).  
 +
However, the spaces $  W _ {e}  ^ {i} $
 +
may still depend on $  e \in E _ {0} $(
 +
if the bundle $  V $
 +
is a trivial bundle so that all the $  V _ {e} $
 +
can be identified). The set $  \{ \lambda _ {1} \dots \lambda _ {l} \} $
 +
is called the Lyapunov spectrum of the flow. For more details and applications cf. [[#References|[a2]]], [[#References|[a3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. [V.I. Oseledets] Oseledec,  "A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems"  ''Trans. Moscow Math. Soc.'' , '''19'''  (1968)  pp. 197–231  ''Trudy Moskov. Mat. Obshch.'' , '''19'''  (1968)  pp. 179–210</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Kliemann,  "Analysis of nonlinear stochastic systems"  W. Schiehlen (ed.)  W. Wedig (ed.) , ''Analysis and estimation of stochastic mechanical systems'' , Springer (Wien)  (1988)  pp. 43–102</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Arnold (ed.)  V. Wihstutz (ed.) , ''Lyapunov exponents'' , ''Lect. notes in math.'' , '''1186''' , Springer  (1986)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. [V.I. Oseledets] Oseledec,  "A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems"  ''Trans. Moscow Math. Soc.'' , '''19'''  (1968)  pp. 197–231  ''Trudy Moskov. Mat. Obshch.'' , '''19'''  (1968)  pp. 179–210</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Kliemann,  "Analysis of nonlinear stochastic systems"  W. Schiehlen (ed.)  W. Wedig (ed.) , ''Analysis and estimation of stochastic mechanical systems'' , Springer (Wien)  (1988)  pp. 43–102</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Arnold (ed.)  V. Wihstutz (ed.) , ''Lyapunov exponents'' , ''Lect. notes in math.'' , '''1186''' , Springer  (1986)</TD></TR></table>

Latest revision as of 14:32, 7 June 2020


Oseledets's multiplicative ergodic theorem, Oseledec's multiplicative ergodic theorem

Consider a linear homogeneous system of differential equations

$$ \tag{a1 } \dot{x} = A ( t) x ,\ \ x ( 0 ; x _ {0} ) = x _ {0} \in \mathbf R ^ {n} ,\ \ t \geq 0 . $$

The Lyapunov exponent of a solution $ x ( t ; x _ {0} ) $ of (a1) is defined as

$$ \lambda ( x _ {0} ) = {\lim\limits \sup } _ {t \rightarrow \infty } \ t ^ {-} 1 \mathop{\rm log} \| x ( t ; x _ {0} ) \| . $$

A more general setting (Lyapunov exponents for families of system of differential equations) for discussing Lyapunov exponents and related matters is as follows. Let $ \Phi = ( \Phi _ {t} ) _ {t \in \mathbf R } $ be a measurable flow on a measure space $ ( E , \mu ) $. For all $ e \in E $, let $ V _ {e} $ be an $ n $- dimensional vector space. (Think, for example, of a vector bundle $ T \rightarrow E $.) A cocycle $ C ( t , e ) $ associated with the flow $ \Phi $ is a measurable function on $ \mathbf R \times E $ that assigns to $ ( t , e ) $ an invertible linear mapping $ V _ {e} \rightarrow V _ {\Phi _ {t} ( e) } $ such that

$$ \tag{a2 } C ( t + s , e ) = C ( t , \Phi _ {s} ( e) ) C ( s , e ) . $$

I.e. if the collection of vector spaces $ V _ {e} $ is viewed as an $ n $- dimensional vector bundle over $ E $, then $ C ( t , \cdot ) $ defines an isomorphism of vector bundles $ \widetilde \Phi _ {t} $ over $ \Phi _ {t} $,

$$ \begin{array}{ccc} V & \mathop \rightarrow \limits ^ { {\widetilde \Phi _ {t} }} & V \\ \downarrow &{} &\downarrow \\ E & \mathop \rightarrow \limits _ { {\Phi _ {t} }} & E \\ \end{array} $$

and condition (a2) simply says that $ \widetilde \Phi _ {t+} s = \widetilde \Phi _ {t} \circ \widetilde \Phi _ {s} $. So $ \widetilde \Phi $ is a flow on $ V $ that lifts $ \Phi $. $ \widetilde \Phi $ is sometimes called the skew product flow defined by $ \Phi $ and $ C $. This set-up is sufficiently general to discuss Lyapunov exponents for non-linear flows, and even stochastic non-linear flows and such things as products of random matrices. If $ E = \{ e \} $, $ \Phi _ {t} = \mathop{\rm id} $, the classical situation (a1) reappears. Let $ \dot{x} = f ( x) $ be a differential equation on a manifold $ M $. Take $ V = T M $, the tangent bundle over $ M $. Let $ \Phi _ {t} $ be the flow on $ M $ defined by $ \dot{x} = f ( x) $. The associated cocycle is defined by the differential $ d \Phi _ {t} $ of $ \Phi _ {t} $,

$$ C ( t , m ) = d \Phi _ {t} ( m) : \ T _ {m} M \rightarrow T _ {\Phi _ {t} ( m) } M . $$

For a skew product flow $ \widetilde \Phi $ on $ V $ the Lyapunov exponent at $ e \in E $ in the direction $ v \in V _ {e} $ is defined by

$$ \lambda ( e , v ) = {\lim\limits \sup } _ {t \rightarrow \infty } t ^ {-} 1 \mathop{\rm log} \| C ( t , e ) v \| . $$

The multiplicative ergodic theorem of V.I. Oseledets [a1] now is as follows. Let $ \widetilde \Phi $ be a skew product flow and assume that there is an invariant probability measure $ \rho $ on $ ( E , \mu ) $ for $ \Phi $, i.e. $ \Phi _ {t} \rho = \rho $ for all $ t \in \mathbf R $. Suppose, moreover, that

$$ \int\limits _ { E } \sup _ {- 1 \leq t \leq 1 } \mathop{\rm log} ^ {+} \| C ^ {\pm 1 } ( t , e ) \| d \rho < \infty . $$

Then there exists a measurable $ \Phi $- invariant set $ E _ {0} \subset E $ of $ \rho $- measure 1 such that for all $ x \in E _ {0} $ there are $ l ( e) $ numbers $ \lambda _ {e} ^ {l} < \dots < \lambda _ {1} ^ {l} $, $ l ( e) \leq d $, and corresponding subspaces $ 0 \subset W _ {e} ^ {l} \subset \dots \subset W _ {e} ^ {1} = V _ {e} $ of dimensions $ d _ {e} ^ {l} < \dots < d _ {e} ^ {1} = d $ such that for all $ i = 1 \dots l ( e) $,

$$ \lim\limits _ {t \rightarrow \infty } t ^ {-} 1 \mathop{\rm log} \ \| C ( t , e ) v \| = \lambda _ {e} ^ {i} \ \iff \ v \in W _ {e} ^ {i} \setminus W _ {e} ^ {i+} 1 . $$

If moreover $ \rho $ is ergodic for $ \Phi _ {t} $, i.e. all $ \Phi _ {t} $- invariant subsets have $ \rho $- measure $ 0 $ or $ 1 $, then the $ l ( e) $, $ \lambda _ {e} ^ {i} $, $ d _ {e} ^ {i} $ are constants independent of $ e $( or $ E _ {0} $). However, the spaces $ W _ {e} ^ {i} $ may still depend on $ e \in E _ {0} $( if the bundle $ V $ is a trivial bundle so that all the $ V _ {e} $ can be identified). The set $ \{ \lambda _ {1} \dots \lambda _ {l} \} $ is called the Lyapunov spectrum of the flow. For more details and applications cf. [a2], [a3].

References

[a1] V.I. [V.I. Oseledets] Oseledec, "A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems" Trans. Moscow Math. Soc. , 19 (1968) pp. 197–231 Trudy Moskov. Mat. Obshch. , 19 (1968) pp. 179–210
[a2] W. Kliemann, "Analysis of nonlinear stochastic systems" W. Schiehlen (ed.) W. Wedig (ed.) , Analysis and estimation of stochastic mechanical systems , Springer (Wien) (1988) pp. 43–102
[a3] L. Arnold (ed.) V. Wihstutz (ed.) , Lyapunov exponents , Lect. notes in math. , 1186 , Springer (1986)
How to Cite This Entry:
Multiplicative ergodic theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplicative_ergodic_theorem&oldid=49317