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 → Rn×n satisfying
where X and T (with T = Z⁺ or T = R⁺) are the phase space
and the time range, respectively, of the dynamical system,
and In is the n-dimensional unit matrix.
The dimension n of the matrices C is not related to the phase space X.
Examples
- A prominent example of a cocycle is given by the matrix Jt in the theory of Lyapunov exponents. In this special case, the dimension n of the matrices is the same as the dimension of the manifold X.
- For any cocycle C, the determinant det C(x, t) is a one-dimensional cocycle.
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 and are L1-integrable with respect to μ. Then for μ-almost all x and each non-zero vector u ∈ Rn the limit
exists and assumes, depending on u but not on x, up to n different values.
These are the Lyapunov exponents.
Further, if λ1 > ... > λm
are the different limits then there are subspaces Rn = R1 ⊃ ... ⊃ Rm ⊃ Rm+1 = {0}, depending on x, such that the limit is λi for u ∈ Ri \ Ri+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 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:
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),tk) = C(x(tk−1),tk − tk−1) ... C(x(t0),t1 − t0)
— for tk large and
the steps ti − ti−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.