Multiplicative ergodic theorem

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

Consider a linear homogeneous system of differential equations

(a1)

The Lyapunov exponent of a solution 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 be a measurable flow on a measure space . For all , let be an -dimensional vector space. (Think, for example, of a vector bundle .) A cocycle associated with the flow is a measurable function on that assigns to an invertible linear mapping such that

(a2)

I.e. if the collection of vector spaces is viewed as an -dimensional vector bundle over , then defines an isomorphism of vector bundles over ,

and condition (a2) simply says that . So is a flow on that lifts . is sometimes called the skew product flow defined by and . 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 , , the classical situation (a1) reappears. Let be a differential equation on a manifold . Take , the tangent bundle over . Let be the flow on defined by . The associated cocycle is defined by the differential of ,

For a skew product flow on the Lyapunov exponent at in the direction is defined by

The multiplicative ergodic theorem of V.I. Oseledets [a1] now is as follows. Let be a skew product flow and assume that there is an invariant probability measure on for , i.e. for all . Suppose, moreover, that

Then there exists a measurable -invariant set of -measure 1 such that for all there are numbers , , and corresponding subspaces of dimensions such that for all ,

If moreover is ergodic for , i.e. all -invariant subsets have -measure or , then the , , are constants independent of (or ). However, the spaces may still depend on (if the bundle is a trivial bundle so that all the can be identified). The set 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)