  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.
