# 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].

#### 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)
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Multiplicative ergodic theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplicative_ergodic_theorem&oldid=49322