Dynamics Seminar

Let $X$ be an infinite Riemann surface with an upper-bounded geodesic pants decomposition and a subsequence of cuffs whose lengths converge to zero. The vertices of the corresponding dual graph $\mathcal{G}$ are pairs of pants, and edges are cuffs with conductances equal to their lengths. We prove that the geodesic flow on $X$ is ergodic if and only if the random walk on $\mathcal{G}$ is recurrent. This yields explicit criteria for deciding, in terms of cuff-length growth, whether the geodesic flow is ergodic. We provide concrete and new families of Riemann surfaces with an explicit understanding of the phase transitions from recurrent to non-recurrent geodesic flows.

The above equivalence result uses a characterization of the measured geodesic laminations on $X$ that arise as straightened horizontal foliations of finite-area holomorphic quadratic differentials. The conditions on the measured laminations are translated into the conditions on the existence of a square summable flow function on $\mathcal{G}$.

This is a joint work with C. Bordanave and X. Dong.