Hyperbolic cone metrics and billiards

Viveka  Erlandsson; Christopher J.  Leininger; Chandrika Sadanand

doi:10.1016/j.aim.2022.108662

Hyperbolic cone metrics and billiards

Viveka Erlandsson, Christopher J. Leininger , Chandrika Sadanand

School of Mathematics

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Abstract

A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an application, we parameterize the space of hyperbolic polygons with the same symbolic coding for their billiard dynamics, and prove that generically this parameter space is a point.

Original language	English
Article number	108662
Number of pages	58
Journal	Advances in Mathematics
Volume	409
Early online date	31 Aug 2022
DOIs	https://doi.org/10.1016/j.aim.2022.108662
Publication status	E-pub ahead of print - 31 Aug 2022

Bibliographical note

Funding Information:
Acknowledgments. We would like to thank Ben Barrett, Moon Duchin, Hugo Parlier, Alan Reid, and Juan Souto for useful conversations throughout the course of this work. In particular, the argument involving triangle groups in Lemma 7.6 is due to Alan Reid. The authors would like to thank the referee for helpful comments on the initial version of the paper. Leininger was partially supported by NSF grant DMS-1811518 and DMS-2106419 . Erlandsson was partially supported by EPSRC grant EP/T015926/1 .

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abstract = "A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an application, we parameterize the space of hyperbolic polygons with the same symbolic coding for their billiard dynamics, and prove that generically this parameter space is a point.",

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N1 - Funding Information: Acknowledgments. We would like to thank Ben Barrett, Moon Duchin, Hugo Parlier, Alan Reid, and Juan Souto for useful conversations throughout the course of this work. In particular, the argument involving triangle groups in Lemma 7.6 is due to Alan Reid. The authors would like to thank the referee for helpful comments on the initial version of the paper. Leininger was partially supported by NSF grant DMS-1811518 and DMS-2106419 . Erlandsson was partially supported by EPSRC grant EP/T015926/1 . Publisher Copyright: © 2022 The Authors

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N2 - A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an application, we parameterize the space of hyperbolic polygons with the same symbolic coding for their billiard dynamics, and prove that generically this parameter space is a point.

AB - A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an application, we parameterize the space of hyperbolic polygons with the same symbolic coding for their billiard dynamics, and prove that generically this parameter space is a point.

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DO - 10.1016/j.aim.2022.108662

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VL - 409

JO - Advances in Mathematics

JF - Advances in Mathematics

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Hyperbolic cone metrics and billiards

Abstract

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