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

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 < 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.30F30, 30F60, 32G15; 37D40 1. Main results 1996 2.About the proof 2001 3. Outline of the paper 2009 4. Crowned hyperbolic surfaces 2012 5.The orthogeodesic foliation 2016 6. Cellulating crowned Teichmüller spaces 2022 7. Transverse and shear-shape cocycles 2030 8.The structure of shear-shape space 2041 9. Train track coordinates for shear-shape space 2047 10.Shear-shape coordinates for transverse foliations 2053 11.Flat deformations in shear-shape coordinates 2064 12. Shear-shape coordinates for hyperbolic metrics 2066 13.Measuring hyperbolic shears and shapes 2068 14.Shape