Large deviations,averaging and periodic orbits of dynamical systems

The paper deals with large deviation bounds for the proportion of periodic orbits with irregular behavior for expansive dynamical systems with specification, in particular, we obtain estimates for large deviations from the equidistribution for closed geodesics on negatively curved manifolds. We derive also large deviation bounds in the averaging principle when the fast motion is the shift along periodic orbits.Partially supported by US-Israel BSFPartially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation, (Germany)