Extremal Exponents of Random Dynamical Systems Do Not Vanish

A family of random diffeomorphisms on a manifold M is said to be a random dynamical system or RDS if it has the so-called cocycle property. The multiplicative ergodic theorem assigns d (=dim M) Lyapunov exponents to every invariant measure of the system. Take the maximum of the leading exponents associated with the various invariant measures. The resulting number is said to be the maximal exponent of the system. The minimal exponent is defined in a similar fashion. It is shown that the minimal exponent of an RDS (p on a compact manifold is negative, provided not all invariant measures are determined by the future of ~0. A similar statement relates the maximal exponent with the past ofq~. We proceed by introducing Markov systems and Markov measures. This notion covers flows of stochastic differential equations as well as products of random diffeomorphisms in Markovian dependence, in particular, products of iid diffeomorphisms. Markov measures are characterized by the fact that they are functionals of the past. Consequently, if there exists a non-Markovian invariant measure, then the maximal exponent does not vanish. Typically, Markov systems do have non-Markovian invariant measures. Finally, for linear systems we recover results of Ledrappier. In particular, these results provide another proof of Furstenberg's theorem on the positivity of the leading exponent of a product of lid unimodular matrices.

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