JournalscmhVol. 96, No. 3pp. 421–463

You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces

  • Moon Duchin

    Tufts University, Medford, USA

  • Viveka Erlandsson

    University of Bristol, UK; UiT The Arctic University of Norway, Tromsø, Norway

  • Christopher J. Leininger

    Rice University, Houston, USA

  • Chandrika Sadanand

    Bowdoin College, Brunswick, USA
You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces cover
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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.

Cite this article

Moon Duchin, Viveka Erlandsson, Christopher J. Leininger, Chandrika Sadanand, You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces. Comment. Math. Helv. 96 (2021), no. 3, pp. 421–463

DOI 10.4171/CMH/516