Discretely shrinking targets in moduli space

Abstract

We consider the discrete shrinking target problem for Teichmüller geodesic flow on the moduli space of abelian or quadratic differentials and prove that the discrete geodesic trajectory of almost every differential will hit a shrinking family of targets infinitely often provided the measures of the targets are not summable. This result applies to any ergodic \(\mathrm {SL}(2,\mathbb {R})\)–invariant measure and any nested family of spherical targets. Under stronger conditions on the targets, we moreover prove that almost every differential will eventually always hit the targets. As an application, we obtain a logarithm law describing the rate at which generic discrete trajectories accumulate on a given point in moduli space. These results build on work of Kelmer (Geom Funct Anal 27:1257–1287, 2017) and generalize theorems of Aimino, Nicol, and Todd (Ann Inst Henri Poincaré Probab Stat 53:1371–1401, 2017).