Bounds on the Hausdorff dimension of Teichm\"uller horocycle flow orbit closures

TL;DR

This paper establishes bounds on the Hausdorff dimension of Teichmüller horocycle flow orbit closures, linking geometric properties to mixing rates using advanced ergodic theory methods.

Contribution

It introduces a novel approach connecting orbit closure dimensions with mixing rates, extending measurable flow techniques to Teichmüller dynamics.

Findings

Hausdorff dimension bounds depend on polynomial mixing rates

Orbit closures are quantitatively smaller than the ambient subvariety

Method adapts Bourgain and Katz's sparse ergodic theorem techniques

Abstract

We show that the Hausdorff dimension of any proper Teichm\"uller horocycle flow orbit closure on any $\mathrm{SL}{(2,\mathbf{R})}$-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 Teichm\"uller 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.