Oseledets theorem

The multiplicative ergodic theorem is stated in terms of matrix cocycles of a dynamical system. The theorem states conditions for the existence of the defining limits and describes the Lyapunov exponents. It does not address the rate of convergence.

A cocycle of an autonomous dynamical system X is a map C : X×T → R^n×n satisfying

${\displaystyle C(x,0)=I_{n}{\rm {~for~all~}}x\in X}$

${\displaystyle C(x,t+s)=C(x(t),s)\,C(x,t){\rm {~for~all~}}x\in X{\rm {~and~}}t,s\in T}$

where X and T (with T = Z⁺ or T = R⁺) are the phase space and the time range, respectively, of the dynamical system, and I_n is the n-dimensional unit matrix. The dimension n of the matrices C is not related to the phase space X.

Let μ be an ergodic invariant measure on X and C a cocycle of the dynamical system such that for each t ∈ T, the maps ${\displaystyle x\rightarrow \log \|C(x,t)\|}$ and ${\displaystyle x\rightarrow \log \|C(x,t)^{-1}\|}$ are L¹-integrable with respect to μ. Then for μ-almost all x and each non-zero vector u ∈ Rⁿ the limit

${\displaystyle \lambda =\lim _{t\to \infty }{1 \over t}\log {\|C(x,t)u\| \over \|u\|}}$

exists and assumes, depending on u but not on x, up to n different values. These are the Lyapunov exponents.

Further, if λ₁ > ... > λ_m are the different limits then there are subspaces Rⁿ = R₁ ⊃ ... ⊃ R_m ⊃ R_m+1 = {0}, depending on x, such that the limit is λ_i for u ∈ R_i \ R_i+1 and i = 1, ..., m.

The values of the Lyapunov exponents are invariant with respect to a wide range of coordinate transformations. Suppose that g : X → X is a one-to-one map such that ${\displaystyle \partial g/\partial x}$ and its inverse exist; then the values of the Lyapunov exponents do not change.

Verbally, ergodicity means that time and space averages are equal, formally:

${\displaystyle \lim _{t\to \infty }{1 \over t}\int _{0}^{t}f(x(s))\,ds={1 \over \mu (X)}\int _{X}f(x)\,\mu (dx)}$

where the integrals and the limit exist. Space average (right hand side, μ is an ergodic measure on X) is the accumulation of f(x) values weighted by μ(dx). Since addition is commutative, the accumulation of the f(x)μ(dx) values may be done in arbitrary order. In contrast, the time average (left hand side) suggests a specific ordering of the f(x(s)) values along the trajectory.

Since matrix multiplication is, in general, not commutative, accumulation of multiplied cocycle values (and limits thereof) according to C(x(t₀),t_k) = C(x(t_k−1),t_k − t_k−1) ... C(x(t₀),t₁ − t₀) — for t_k large and the steps t_i − t_i−1 small — makes sense only for a prescribed ordering. Thus, the time average may exist (and the theorem states that it actually exists), but there is no space average counterpart. In other words, the Oseledets theorem differs from additive ergodic theorems (such as G. D. Birkhoff's and J. von Neumann's) in that it guarantees the existence of the time average, but makes no claim about the space average.

• Oseledets, V. I. (1968). "Мультипликативная эргодическая теорема. Характеристические показатели Ляпунова динамических систем" [Multiplicative ergodic theorem: Characteristic Lyapunov exponents of dynamical systems]. Trudy MMO (in Russian). 19: 179–210.
• Ruelle, D. (1979). "Ergodic theory of differentiable dynamic systems" (PDF). IHES Publ. Math. 50 (1): 27–58. doi:10.1007/BF02684768. S2CID 56389695.

• V. I. Oseledets, Oseledets theorem at Scholarpedia