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

Moon Duchin; Viveka Erlandsson; Christopher J. Leininger; Chandrika Sadanand

doi:10.4171/cmh/516

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
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- zbMATH Open
Viveka Erlandsson
University of Bristol, UK; UiT The Arctic University of Norway, Tromsø, Norway
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- zbMATH Open
Christopher J. Leininger
Rice University, Houston, USA
Chandrika Sadanand
Bowdoin College, Brunswick, USA
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- zbMATH Open

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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