Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow - Calderon, Aaron; Farre, James (Geometry & topology, 2024)
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 upper-triangular subgroup P<SL2R 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.