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

Calderon, Aaron¹ ; Farre, James²

¹ Department of Mathematics, University of Chicago, Chicago, IL, United States
² Faculty of Mathematics and Computer Science, Universität Heidelberg, Heidelberg, Germany

Geometry & topology, Tome 28 (2024) no. 5, pp. 1995-2124.

Résumé

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 $P<{SL}_2ℝ$ 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.

Détail

DOI : 10.2140/gt.2024.28.1995

Keywords: Teichmüller theory, shear coordinates, measured lamination, earthquake flow, horocycle flow, quadratic differentials

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