Hyperbolic cone metrics and billiards

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 associat...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2022-11, Vol.409, p.108662, Article 108662
Main Authors: Erlandsson, Viveka, Leininger, Christopher J., Sadanand, Chandrika
Format: Article
Language:English
Subjects:
Billiard
Cone metric
Geodesic current
Hyperbolic
Rigidity
Surface
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Summary: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.
ISSN:0001-8708
1090-2082
1090-2082
DOI:10.1016/j.aim.2022.108662