Furstenberg-Kesten theorem

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 invertiblePlanetmathPlanetmath square matricesMathworldPlanetmath of size d. Consider the multiplicative cocycle (ϕn(x))n defined by the transformation A.

If log+||A|| is integrable, where log+||A||=max{log||A||,0}, then:


exists almost everywhere, and λmax+ is integrable and


If log+||A-1|| is integrable, then:


exists almost everywhere, and λmin+ is integrable and


Furthermore, both λmin and λmax are invariant for the tranformation f, that is, λminf(x)=λmin(x) and λmaxf(x)=λmax(x), for μ almost everywhere.

This theorem is a direct consequence of Kingman’s subadditive ergodic theorem, by observing that both




are subadditive sequences.

The results in this theorem are strongly improved by Oseledet’s multiplicative ergodic theorem, or Oseledet’s decomposition.

Title Furstenberg-Kesten theorem
Canonical name FurstenbergKestenTheorem
Date of creation 2014-03-19 22:14:18
Last modified on 2014-03-19 22:14:18
Owner Filipe (28191)
Last modified by Filipe (28191)
Numerical id 3
Author Filipe (28191)
Entry type Theorem
Related topic Oseledet’s decomposition
Related topic multiplicative cocycle