Multiplicative ergodic theorem

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

Consider a linear homogeneous system of differential equations

$$\begin{array}{}\text{(a1)}& \dot{x}=A(t)x,x(0;x_0)=x_0\in {\mathbf{R}}^n,t\ge 0.\end{array}$$

The Lyapunov exponent of a solution $x(t;x_0)$ of (a1) is defined as

$$\lambda (x_0)={limsup}_{t\to \mathrm{\infty }}{t}^{-}1\log \Vert x(t;x_0)\Vert .$$

A more general setting (Lyapunov exponents for families of system of differential equations) for discussing Lyapunov exponents and related matters is as follows. Let $\mathrm{\Phi }=({\mathrm{\Phi }}_t{)}_{t\in \mathbf{R}}$ 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 $T\to E$.) A cocycle $C(t,e)$ associated with the flow $\mathrm{\Phi }$ is a measurable function on $\mathbf{R}\times E$ that assigns to $(t,e)$ an invertible linear mapping $V_e\to V_{{\mathrm{\Phi }}_t(e)}$ such that

$$\begin{array}{}\text{(a2)}& C(t+s,e)=C(t,{\mathrm{\Phi }}_s(e))C(s,e).\end{array}$$

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 ${\tilde{\mathrm{\Phi }}}_t$ over ${\mathrm{\Phi }}_t$,

$$\begin{array}{ccc}V& \stackrel{{\tilde{\mathrm{\Phi }}}_t}{\to }& V\\ \downarrow & & \downarrow \\ E& \underset{{\mathrm{\Phi }}_t}{\to }& E\end{array}$$

and condition (a2) simply says that ${\tilde{\mathrm{\Phi }}}_{t+}s={\tilde{\mathrm{\Phi }}}_t\circ {\tilde{\mathrm{\Phi }}}_s$. So $\tilde{\mathrm{\Phi }}$ is a flow on $V$ that lifts $\mathrm{\Phi }$. $\tilde{\mathrm{\Phi }}$ is sometimes called the skew product flow defined by $\mathrm{\Phi }$ 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 $E=\{e\}$, ${\mathrm{\Phi }}_t=\mathrm{id}$, the classical situation (a1) reappears. Let $\dot{x}=f(x)$ be a differential equation on a manifold $M$. Take $V=TM$, the tangent bundle over $M$. Let ${\mathrm{\Phi }}_t$ be the flow on $M$ defined by $\dot{x}=f(x)$. The associated cocycle is defined by the differential $d{\mathrm{\Phi }}_t$ of ${\mathrm{\Phi }}_t$,

$$C(t,m)=d{\mathrm{\Phi }}_t(m):T_{m}M\to T_{{\mathrm{\Phi }}_t(m)}M.$$

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

$$\lambda (e,v)={limsup}_{t\to \mathrm{\infty }}{t}^{-}1\log \Vert C(t,e)v\Vert .$$

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

$$\underset{E}{\int }\underset{-1\le t\le 1}{sup}\stackrel{+}{\log }\Vert C^{\pm 1}(t,e)\Vert d\rho <\mathrm{\infty }.$$

Then there exists a measurable $\mathrm{\Phi }$- invariant set $E_0\subset E$ of $\rho$- measure 1 such that for all $x\in E_0$ there are $l(e)$ numbers ${\lambda }_e^l<\cdots <{\lambda }_1^l$, $l(e)\le d$, and corresponding subspaces $0\subset W_e^l\subset \cdots \subset W_e^1=V_e$ of dimensions $d_e^l<\cdots <d_e^1=d$ such that for all $i=1\dots l(e)$,

$$\underset{t\to \mathrm{\infty }}{lim}{t}^{-}1\log \Vert C(t,e)v\Vert ={\lambda }_e^i⟺v\in W_e^i\setminus W_e^{i+}1.$$

If moreover $\rho$ is ergodic for ${\mathrm{\Phi }}_t$, i.e. all ${\mathrm{\Phi }}_t$- 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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