Oseledets theorem


In mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by Valery Oseledets in 1965 and reported at the International Mathematical Congress in Moscow in 1966. A conceptually different proof of the multiplicative ergodic theorem was found by M. S. Raghunathan. The theorem has been extended to semisimple Lie groups by V. A. Kaimanovich and further generalized in the works of David Ruelle, Grigory Margulis, Anders Karlsson, and François Ledrappier.

Cocycles

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 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

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 = 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.

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
is the accumulation of f values weighted by μ.
Since addition is commutative, the accumulation of the fμ values may be done in arbitrary order.
In contrast, the time average suggests a specific ordering
of the f values along the trajectory.
Since matrix multiplication is, in general, not commutative,
accumulation of multiplied cocycle values according to
C = C... C
— for tk large and
the steps ti − ti−1 small — makes sense only for a prescribed ordering. Thus, the time average may exist , but there is no space average counterpart. In other words, the Oseledets theorem differs from additive ergodic theorems in that it guarantees the existence of the time average, but makes no claim about the space average.