Hyperbolic cone metrics and billiards

TL;DR

This paper characterizes when negatively curved hyperbolic cone metrics are uniquely determined by their Liouville current, showing rigidity is generic, and explores implications for billiard dynamics in polygons.

Contribution

It provides a complete characterization of rigidity and flexibility of hyperbolic cone metrics, and links these properties to billiard dynamics in polygons.

Findings

Rigidity is a generic property among hyperbolic cone metrics.

Flexible metrics are parameterized by a deformation space.

For most polygons, the billiard dynamics are uniquely determined by symbolic coding.

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.