Multiplicative ergodic theorem

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].

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