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

Viveka Erlandsson; Moon Duchin; Christopher Leininger; Chandrika Sadanand

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

Viveka Erlandsson, Moon Duchin, Christopher Leininger, Chandrika Sadanand

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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.

Original language	English
Number of pages	38
Journal	Commentarii Mathematici Helvetici
Publication status	Accepted/In press - 26 Feb 2021

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https://arxiv.org/abs/1804.05690

Dr Viveka Erlandsson
- v.erlandsson@bristol.ac.uk
- School of Mathematics - Lecturer in Mathematics
Person: Academic

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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.",

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AU - Duchin, Moon

AU - Leininger, Christopher

AU - Sadanand, Chandrika

PY - 2021/2/26

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N2 - 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.

AB - 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.

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JO - Commentarii Mathematici Helvetici

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

Abstract

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Dr Viveka Erlandsson

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