Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow

TL;DR

This paper extends Mirzakhani's conjugacy between earthquake and horocycle flows to all strata, introduces shear-shape cocycles for measured laminations, and classifies ergodic measures for the extended earthquake flow.

Contribution

It generalizes shear coordinates to arbitrary measured laminations and classifies ergodic measures for the extended earthquake flow action.

Findings

Establishes conjugacy between earthquake and horocycle flows on all strata.

Classifies ergodic measures as pullbacks of affine measures on quadratic differentials.

Introduces shear-shape cocycles for measured laminations.

Abstract

We extend Mirzakhani's conjugacy between the earthquake and horocycle flows to a bijection, demonstrating conjugacies between these flows on all strata and exhibiting an abundance of new ergodic measures for the earthquake flow. The structure of our map indicates a natural extension of the earthquake flow to an action of the the upper-triangular subgroup P < SL(2,R) and we classify the ergodic measures for this action as pullbacks of affine measures on the bundle of quadratic differentials. Our main tool is a generalization of the shear coordinates of Bonahon and Thurston to arbitrary measured laminations.