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Suppose $(X,A,\mu)$ a probability space, where $X$ is a compact Riemann manifold, $T:X\to X$ a diffeomorphism and $T$ is a measure-preserving transformation( over the borel $\sigma$ algebra). Prove that $\mu$- a.e $x\in X$ there is the Lyapunov exponent,

$$\lambda(x):=\lim_{n\to \infty} \frac{1}{n}\log\|DT^n(x)\|$$

where the operator norm is defined as $\displaystyle\|L\|=\sup_{\|x\|=1}\|L(x)\|$

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Hey you're doing dynamical systems as well! Where is this problem taken from? – anegligibleperson Jan 2 '13 at 19:49
@anegligibleperson Take from of my teacher. I don't know the book. – user52188 Jan 2 '13 at 20:18
up vote 3 down vote accepted

You can use the Subadditive Ergodic Theorem. Note that the sequence $\log{\|DT^n(x)\|}$ is subadditive. To see this apply the chain rule to $\|DT^{n+m}(x)\|$.

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