You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces
Geometric Topology
2019-10-17 v4 Dynamical Systems
Abstract
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.
Keywords
Cite
@article{arxiv.1804.05690,
title = {You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces},
author = {Moon Duchin and Viveka Erlandsson and Christopher J. Leininger and Chandrika Sadanand},
journal= {arXiv preprint arXiv:1804.05690},
year = {2019}
}
Comments
v2: Updated language and references, including a reference for the flat strip theorem. v3: Added a section describing how the bounce spectrum can be determined given the countable set of bounce sequences of generalized diagonals. v4: Stylistic edits