December  2021, 41(12): 5537-5549. doi: 10.3934/dcds.2021087

Hearing the shape of right triangle billiard tables

School of Mathematical Sciences, Peking University, Beijing 100871, China

* Corresponding author: Jiazhong Yang

Received  December 2020 Revised  April 2021 Published  December 2021 Early access  May 2021

Fund Project: The paper is supported by NSFC-12071006

In this paper, we give a positive answer to the problem that whether one can identify the shape of a right triangle billiard table by a single bounce sequence. Moreover, a convenient method to calculate the shape of polygons is given in this paper, too.

Citation: Yang Shen, Jiazhong Yang. Hearing the shape of right triangle billiard tables. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5537-5549. doi: 10.3934/dcds.2021087
References:
[1]

J. Bobok and S. Troubetzkoy, Does a billiard orbit determine its (polygonal) table?, Fundamenta Mathematicae, 212 (2011), 129-144.  doi: 10.4064/fm212-2-2.  Google Scholar

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J. Bobok and S. Troubetzkoy, Code and order in polygonal billiards, Topology and its Applications, 159 (2012), 236-247.  doi: 10.1016/j.topol.2011.09.007.  Google Scholar

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A. Calderon, S. Coles, D. Davis, J. Lanier and A. Oliveira, How to hear the shape of a billiard table, preprint, arXiv: 1806.09644, 2018. Google Scholar

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M. Duchin, V. Erlandsson, C. J. Leininger and C. Sadanand, You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces, preprint, arXiv: 1804.05690, 2019. Google Scholar

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G. Galperin, Non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons, Comm. Math. Phys., 91 (1983), 187-211.  doi: 10.1007/BF01211158.  Google Scholar

[6]

G. GalperinT. Krüger and S. Troubetzkoy, Local instability of orbits in polygonal and polyhedral billiards, Comm. Math. Phys., 169 (1995), 463-473.  doi: 10.1007/BF02099308.  Google Scholar

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P. Hooper, Periodic billiard paths in right triangles are unstable, Geom. Dedicata, 125 (2007), 39-46.  doi: 10.1007/s10711-007-9129-9.  Google Scholar

[8]

M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly, 73 (1966), 1-23.  doi: 10.1080/00029890.1966.11970915.  Google Scholar

[9]

A. Katok and B. Hasselblatt, Flows on surface and related dynmical systems, in Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, (1995), 470–479. Google Scholar

[10]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in Handbook of Dynamical Systems, Elsevier, 1 (2002), 1015–1089. doi: 10.1016/S1874-575X(02)80015-7.  Google Scholar

[11]

Y. Shen, Hearing the Shape of Some Polygon Billiard Tables, Ph.d. Thesis, Peking University, 2021. Google Scholar

[12]

S. Tabachnikov, Billiards, Panor. Synth., 1 (1995), vi+142pp.  Google Scholar

[13]

G. W. Tokarsky, Galperin's triangle example, Comm. Math. Phys., 335 (2015), 1211-1213.  doi: 10.1007/s00220-015-2336-6.  Google Scholar

[14]

S. Troubetzkoy, Recurrence and periodic billiard orbits in polygons, Regular and Chaotic Dynamics, 9 (2004), 1-12.  doi: 10.1070/RD2004v009n01ABEH000259.  Google Scholar

[15]

S. Troubetzkoy, Periodic billiard orbits in right triangles, Annales de l'institut Fourier, 55 (2005), 29-46.  doi: 10.5802/aif.2088.  Google Scholar

show all references

References:
[1]

J. Bobok and S. Troubetzkoy, Does a billiard orbit determine its (polygonal) table?, Fundamenta Mathematicae, 212 (2011), 129-144.  doi: 10.4064/fm212-2-2.  Google Scholar

[2]

J. Bobok and S. Troubetzkoy, Code and order in polygonal billiards, Topology and its Applications, 159 (2012), 236-247.  doi: 10.1016/j.topol.2011.09.007.  Google Scholar

[3]

A. Calderon, S. Coles, D. Davis, J. Lanier and A. Oliveira, How to hear the shape of a billiard table, preprint, arXiv: 1806.09644, 2018. Google Scholar

[4]

M. Duchin, V. Erlandsson, C. J. Leininger and C. Sadanand, You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces, preprint, arXiv: 1804.05690, 2019. Google Scholar

[5]

G. Galperin, Non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons, Comm. Math. Phys., 91 (1983), 187-211.  doi: 10.1007/BF01211158.  Google Scholar

[6]

G. GalperinT. Krüger and S. Troubetzkoy, Local instability of orbits in polygonal and polyhedral billiards, Comm. Math. Phys., 169 (1995), 463-473.  doi: 10.1007/BF02099308.  Google Scholar

[7]

P. Hooper, Periodic billiard paths in right triangles are unstable, Geom. Dedicata, 125 (2007), 39-46.  doi: 10.1007/s10711-007-9129-9.  Google Scholar

[8]

M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly, 73 (1966), 1-23.  doi: 10.1080/00029890.1966.11970915.  Google Scholar

[9]

A. Katok and B. Hasselblatt, Flows on surface and related dynmical systems, in Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, (1995), 470–479. Google Scholar

[10]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in Handbook of Dynamical Systems, Elsevier, 1 (2002), 1015–1089. doi: 10.1016/S1874-575X(02)80015-7.  Google Scholar

[11]

Y. Shen, Hearing the Shape of Some Polygon Billiard Tables, Ph.d. Thesis, Peking University, 2021. Google Scholar

[12]

S. Tabachnikov, Billiards, Panor. Synth., 1 (1995), vi+142pp.  Google Scholar

[13]

G. W. Tokarsky, Galperin's triangle example, Comm. Math. Phys., 335 (2015), 1211-1213.  doi: 10.1007/s00220-015-2336-6.  Google Scholar

[14]

S. Troubetzkoy, Recurrence and periodic billiard orbits in polygons, Regular and Chaotic Dynamics, 9 (2004), 1-12.  doi: 10.1070/RD2004v009n01ABEH000259.  Google Scholar

[15]

S. Troubetzkoy, Periodic billiard orbits in right triangles, Annales de l'institut Fourier, 55 (2005), 29-46.  doi: 10.5802/aif.2088.  Google Scholar

Figure 2.1.  After a reflection, $ P_{n} $ becomes $ P_{n+1} $ in Case ($ i $) and $ P_{n-1} $ in Case ($ ii $)
Figure 2.2.  From $ B(\gamma^+) = ({r}, {b}, {r}, {g}, {b}, {r}, {b}, {r}, {b}, {g}, {r}, \cdots) $ to get the return sequence $ R(\gamma^+) = (1, 2, 1, 2, \cdots) $ and then to the get the level sequence $ L(\gamma^+) = (0, 1, 0, -1, 0, \cdots) $
Figure 3.1.  The visual diagram of $ \theta_{n} $, $ \theta_{n+1} $ and $ \theta_{n+2} $
Figure 3.2.  Cases ($ i $) and ($ ii $) correspond to $ LM_n = ({r}, {g}, {r}) $ and $ \theta_{n}\in(\alpha, \pi-\alpha) $; Cases ($ iii $) and ($ iv $) correspond to $ LM_n = ({r}, {b}, {r}) $ and $ \theta_{n}\in[0, \alpha)\bigcup\, (\pi-\alpha, \pi] = (-\alpha, \alpha)\, (\bmod\, \pi) $
Figure 3.3.  In Case ($ i $), $ \theta_{n_0} = \pi-\alpha $. In Case ($ ii $), $ \theta_{n_0} = \alpha $
Figure 3.6.  The left-hand part is a right triangle billiard table $ Rt $ and a periodic orbit: $ p_{0} $ $ \longrightarrow $ $ p_{1} $ $ \longrightarrow $ $ p_{2} $ $ \longrightarrow $ $ \cdots $ $ \longrightarrow p_{9} $ $ \longrightarrow $ $ p_{10} $ $ \longrightarrow $ $ \cdots $ in $ Rt $. The right-hand part is the unfolding floors of the corresponding rhombus of $ Rt $ along the periodic orbit
Figure 3.7.  The left-hand part is a right triangle billiard table $ Rt $ and a periodic orbit: $ p_{0} $ $ \longrightarrow $ $ p_{1} $ $ \longrightarrow $ $ p_{2} $ $ \longrightarrow $ $ \cdots $ $ \longrightarrow $ $ p_{6} $ $ \longrightarrow $ $ \cdots $ in $ Rt $. The right-hand part is the unfolding floors of the corresponding rhombus of $ Rt $ along the periodic orbit
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