With a local system on Riemann surface one can associate Lyapunov exponents. This can be done using either random walk or geodesic flow for a hyperbolic metric. About 15 years ago I suggested a topological formula for the sum of positive exponents in the case when the local system carries a variation of Hodge structures of weight one. This was related with Teichmuller geodesic flow and interval exchange maps. Recently I have made a series of computer experiments indicating that the theory generalizes to some variations of Hodge structures of higher weight, including many examples from mirror symmetry. Good cases for weight 3 correspond to so called "thin groups" in terminology of P.Sarnak. Hypothetically, the topological formula for Lyapunov exponents holds iff certain period of Calabi-Yau motive is everywhere non-vanishing on the universal cover.