Discretely shrinking targets in moduli space

TL;DR

This paper studies how typical trajectories in moduli space under Teichmüller flow repeatedly hit shrinking targets, establishing conditions for infinite recurrence and eventual hitting, and deriving a logarithm law for accumulation rates.

Contribution

It extends previous work by proving hitting properties for discrete trajectories in moduli space under general ergodic measures and nested targets, generalizing earlier theorems.

Findings

Almost every differential hits shrinking targets infinitely often if measures are not summable.

Under stronger conditions, almost every differential eventually always hits the targets.

A logarithm law describes the rate of accumulation of trajectories on points in moduli space.

Abstract

We consider the discrete shrinking target problem for Teichm\"uller 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 and generalize theorems of Aimino, Nicol, and Todd.