Geometry Seminar

Given a nested decreasing family of targets B_n in a measure space X equipped with a flow phi_t (or transformation), the shrinking target problem asks to characterize when there is a full measure set of points x that hit the targets infinitely often in the sense that {n \in N : phi_n(x)\in B_n} is unbounded. This talk will examine the discrete shrinking target problem for the Teichmüller flow on the moduli space of unit-area quadratic differentials and show that for any ergodic probability measure, almost every differential will hit a nested spherical targets infinitely often provided the measures of the targets are not summable. Our key tool is an effective mean ergodic theorem stating that the time-average of any L^2 function converges to its space-average at a uniform rate in L^2. As an application, we obtain a logarithm law describing how quickly generic discrete geodesic trajectories accumulate on a given point. Joint with Grace Work. Speaker(s): Spencer Dowdall (Vanderbilt University)