Statement of The Theorem
Let μ be an 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} 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.
Read more about this topic: Oseledets Theorem
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