Oseledets multiplicative ergodic theorem

Oseledets multiplicative ergodic theorem, or Oseledets decomposition, considerably extends the results of Furstenberg-Kesten theorem, under the same conditions.

Consider μ a probability measureMathworldPlanetmath, and f:MM a measure preserving dynamical systemMathworldPlanetmathPlanetmath. Consider A:MGL(d,𝐑), a measurable transformation, where GL(d,R) is the space of invertible square matrices of size d. Consider the multiplicative cocycle (ϕn(x))n defined by the transformation A, and assume log+||A|| and log+||A-1|| are integrable.

Then, μ almost everywhere xM, one can find a natural numberMathworldPlanetmath k=k(x) and real numbers λ1(x)>>λk(x) and a filtrationPlanetmathPlanetmath


such that, for μ almost everywhere and for all i{1,,k}

  1. 1.

    k(f(x))=k(x) and λi(f(x))=λi(x) and A(x)Vxi=Vf(x)i;

  2. 2.

    limn1nlog||ϕn(x)v||=λi(x) for all vVxi\Vxi+1;

  3. 3.

    limn1nlog|detϕn(x)|=i=1kdi(x)λi(x) where di(x)=dimVxi-dimVxi+1

Furthermore, the numbers ki(x) and the subspaces Vxi depend measurably on the point x.

The numbers λi(x) are called Lyapunov exponents of A relatively to f at the point x. Each number di(x) is called the multiplicity of the Lyapunov exponent λi(x). We also have that λ1=λmax and λk=λmin, where λmax and λmin are as given by Furstenberg-Kesten theorem.

Title Oseledets multiplicative ergodic theorem
Canonical name OseledetsMultiplicativeErgodicTheorem
Date of creation 2014-03-26 14:21:35
Last modified on 2014-03-26 14:21:35
Owner Filipe (28191)
Last modified by Filipe (28191)
Numerical id 6
Author Filipe (28191)
Entry type Theorem
Classification msc 37H15
Synonym Oseledets decomposition
Related topic Lyapunov exponent
Related topic Furstenberg-Kesten Theorem