40-00-FurstenbergKestenTheorem.tex

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\pmcreated{2014-03-19 22:14:18}
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\pmowner{Filipe}{28191}
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\pmtitle{Furstenberg-Kesten theorem}
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\pmtype{Theorem}
\pmrelated{Oseledet's decomposition}
\pmrelated{multiplicative cocycle}

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\begin{document}
Consider $\mu$ a probability measure, and $f:M\rightarrow M$ a measure preserving dynamical system. Consider $A:M\rightarrow GL(d,\textbf{R})$, a measurable transformation, where GL(d,\textbf{R}) is the space of invertible square matrices of size $d$.
Consider the multiplicative cocycle $(\phi^n(x))_n$ defined by the transformation $A$.

If $\log^+||A||$ is integrable, where $\log^+||A||=\max\{ \log ||A||,0\}$, then:
$$\lambda_{\max}(x)=\lim_n \frac{1}{n} \log ||\phi^n(x)||$$
exists almost everywhere, and $\lambda^+_{\max}$ is integrable and
$$\int \lambda_{\max} d\mu = \lim_n \frac{1}{n} \int \log ||\phi^n|| d\mu = \inf_n \frac{1}{n} \int \log ||\phi^n||d\mu$$

If $\log^+||A^{-1}||$ is integrable, then:
$$\lambda_{\min}(x)=\lim_n -\frac{1}{n} \log ||\phi^{-n}(x)||$$
exists almost everywhere, and $\lambda^+_{\min}$ is integrable and
$$\int \lambda_{\min} d\mu = \lim_n -\frac{1}{n} \int \log ||\phi^{-n}|| d\mu = \sup_n -\frac{1}{n} \int \log ||\phi^{-n}||d\mu$$

Furthermore, both $\lambda_{\min}$ and $\lambda_{\max}$ are invariant for the tranformation $f$, that is, $\lambda_{\min}\circ f(x)=\lambda_{\min}(x)$ and $\lambda_{\max}\circ f(x)=\lambda_{\max}(x)$, for $\mu$ almost everywhere.

This theorem is a direct consequence of Kingman's subadditive ergodic theorem, by observing that both
$$\log ||\phi^n(x)||$$ and $$\log ||\phi^{-n}(x)||$$ are subadditive sequences.

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