On Mirzakhani's twist torus conjecture

Abstract

We address a conjecture of Mirzakhani about the statistical behavior of certain expanding families of ``twist tori'' in the moduli space of hyperbolic surfaces, showing that they equidistribute to a certain Lebesgue-class measure along almost all sequences. We also identify a number of other expanding families of twist tori whose limiting distributions are mutually singular to Lebesgue.

The final section of the paper was rewritten in order to incorporate referee's suggestions and fine-tune the exposition. Final version, to appear in Duke