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Shear-shape cocycles for
measured laminations and ergodic theory of the earthquake
flow
Aaron Calderon and James Farre |
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Geometry & Topology 28 (2024) 1995–2124
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Abstract |
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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 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. |
Keywords
Teichmüller theory, shear coordinates, measured lamination,
earthquake flow, horocycle flow, quadratic differentials
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Mathematical Subject Classification
Primary: 30F30, 30F60, 32G15
Secondary: 37D40
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References |
Publication
Received: 12 July 2021
Revised: 21 January 2023
Accepted: 18 February 2023
Published: 24 August 2024
Proposed: Mladen Bestvina
Seconded: David Fisher, Anna Wienhard
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Authors |
| Aaron Calderon | |
| Department of Mathematics University of Chicago Chicago, IL United States |
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| James Farre | |
| Faculty of Mathematics and Computer
Science Universität Heidelberg Heidelberg Germany |
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| © 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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