Bernoulli measures for the Teichmueller flow

Ursula Hamenstaedt

Published 2006-07-17, updated 2011-12-28Version 4

Let S be a nonexceptional oriented surface of finite type. We construct an uncountable family of probability measures on the space of area on holomorphic quadratic differentials over the moduli space for S containing the usual Lebesgue measure. These measures are invariant under the Teichmueller geodesic flow, and they are mixing, absolutely continuous with respect to the stable and unstable foliation adn exponentially recurrent to a compact set. Finally we show that the critical exponent of the mapping class group equals the dimension of the Teichmueller space for S. Moreover, this critical exponent coincides with the the logarithmic asymptotic of the number of closed Teichmueller geodesics in moduli space which meet a sufficiently large compact set.