The Weil--Petersson (WP) metric is an incomplete Riemannian metric on the moduli space of finite type Riemann surfaces and the completion is the Deligne--Mumford compactification. Since WP geodesics that get to the boundary strata in finite time have zero Liouville measure, the WP flow (as a flow) makes measure-theoretic sense. The dynamical properties of the flow then become objects of interest.
Recently, the WP flow has been shown to be ergodic for non-exceptional moduli spaces and exponentially mixing for exceptional moduli. In the work that we will describe, we begin the first steps in leveraging these properties to derive statistics along typical geodesics. Our focus will be cusp excursion statistics along typical geodesics which can be thought of as analogues of logarithm laws/ shrinking target problems in this incomplete setting. Some of this is joint work with Carlos Matheus.