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

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.

Figures (21)

• Figure 1: The orthogeodesic foliation on pairs of pants. Note that the weight of each bolded arc is a linear combination of the boundary lengths, hence the correspondence between shear-shape and Fenchel--Nielsen/Dehn--Thurston coordinates. If any of the weights is zero, the orthogeodesic foliation only picks out the two seams with non-zero weights.
• Figure 2: Truncating an (oriented) crown to compute its metric residue.
• Figure 3: The spine of a hyperbolic surface with crowned boundary. Note that the finite core $\mathop{\mathrm{\mathsf{Sp}}}\nolimits^0$ (represented in bold) contains a spine for the convex core of the surface.
• Figure 4: Inflating a lamination and deflating its complementary components.
• Figure 5: The weighted arc complex of an ideal pentagon rel its boundary.

Theorems & Definitions (149)

• Theorem A
• Theorem B
• Remark 1.1
• Theorem C
• proof
• Corollary 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5: Bonahon, Thurston
• Theorem D