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 (also spelled "Oseledec") 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.

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 → R^n×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 I_n is the n-dimensional unit matrix. The dimension n of the matrices C is not related to the phase space X.

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 ${\displaystyle x\rightarrow \log \|C(x,t)\|}$ and ${\displaystyle x\rightarrow \log \|C(x,t)^{-1}\|}$ are L¹-integrable with respect to μ. Then for μ-almost all x and each non-zero vector u ∈ Rⁿ the limit

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