Bounds on the Hausdorff Dimension of Teichmüller Horocycle Flow Orbit Closures

Abstract

Abstract We show that the Hausdorff dimension of any proper Teichmüller horocycle flow orbit closure on any irreducible $\textrm {SL}{(2,\textbf {R})}$-invariant subvariety of Abelian or quadratic differentials is bounded away from the dimension of the subvariety in terms of the polynomial mixing rate of the Teichmüller horocycle flow on the subvariety. The proof is based on abstract methods for measurable flows adapted from work of Bourgain and Katz on sparse ergodic theorems.