It was proved by Valery Oseledets (also spelled "Oseledec") in 1965 and reported at the International Mathematical Congress in Moscow in 1966.
[citation needed][1] 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.
[citation needed] 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.
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.
Verbally, ergodicity means that time and space averages are equal, formally: where the integrals and the limit exist.
In contrast, the time average (left hand side) suggests a specific ordering of the f(x(s)) values along the trajectory.