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arXiv · 2102.13124

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

Aaron Calderon·James Farre
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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.

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Aaron Calderon, James Farre. 2021-02-25. Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow. https://doi.org/10.2140/gt.2024.28.1995

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