Oseledets Theorem - Additive Versus Multiplicative Ergodic Theorems

Additive Versus Multiplicative Ergodic Theorems

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

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.