Special ergodic theorems and dynamical large deviations
Résumé
Let f : M -> M be a self-map of a compact Riemannian manifold M, admitting a global SRB measure mu. For a continuous test function phi: M -> R and a constant alpha > 0, consider the set K-phi,K-alpha of the initial points for which the Birkhoff time averages of the function phi differ from its mu-space average by at least alpha. As the measure mu is a global SRB one, the set K-phi,K-alpha should have zero Lebesgue measure. The special ergodic theorem, whenever it holds, claims that, moreover, this set has a Hausdorff dimension less than the dimension of M. We prove that for Lipschitz maps, the special ergodic theorem follows from the dynamical large deviations principle. We also define and prove analogous result for flows. Applying the theorems of Young and of Araujo and Pacifico, we conclude that the special ergodic theorem holds for transitive hyperbolic attractors of C-2-diffeomorphisms, as well as for some other known classes of maps (including the one of partially hyperbolic non-uniformly expanding maps) and flows.