We give a complete characterization of the relationship between the shape of a Euclidean polygon and the symbolic dynamics of its billiard flow. We prove that the only pairs of tables that can have the same bounce spectrum are right-angled tables that differ by an affine map. The main tool is a new theorem that establishes that a flat cone metric is completely determined by the support of its Liouville current.
|Number of pages||38|
|Journal||Commentarii Mathematici Helvetici|
|Publication status||Accepted/In press - 26 Feb 2021|