Analysis and Mathematical Physics

The Hopf-Tsuji-Sullivan theorem states that the geodesic flow on (an infinite) Riemann surface is ergodic iff the Poincare series is divergent iff the Brownian motion is recurrent. Infinite Riemann surfaces can be built by gluing infinitely many hyperbolic pairs of pants, and the Riemann surface is determined by the corresponding Fenchel-Nielsen (FN) parameters. We provide various sufficient conditions on FN parameters to guarantee that the surface belongs to this class (joint with Hakobyan and Basmajian). Then we prove the geodesic flow is ergodic iff a.e. horizontal leaf of every finite-area holomorphic quadratic differential is recurrent. We use this criterion to decide when the geodesic flow is not ergodic in terms of the FN parameters.