Figure 2.1.
After a reflection, $ P_{n} $ becomes $ P_{n+1} $ in Case ($ i $ ) and $ P_{n-1} $ in Case ($ ii $ )
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doi: 10.3934/dcds.2021087
Hearing the shape of right triangle billiard tables
School of Mathematical Sciences, Peking University, Beijing 100871, China |
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.
Mathematics Subject Classification:
Primary: 37C83; Secondary: 70F35.
Citation:
Yang Shen, Jiazhong Yang.
Hearing the shape of right triangle billiard tables. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021087
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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. |
| [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. |
| [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. |
| [6] |
G. Galperin, T. Krüger and S. Troubetzkoy,
Local instability of orbits in polygonal and polyhedral billiards, Comm. Math. Phys., 169 (1995), 463-473.
doi: 10.1007/BF02099308. |
| [7] |
P. Hooper,
Periodic billiard paths in right triangles are unstable, Geom. Dedicata, 125 (2007), 39-46.
doi: 10.1007/s10711-007-9129-9. |
| [8] |
M. Kac,
Can one hear the shape of a drum?, Amer. Math. Monthly, 73 (1966), 1-23.
doi: 10.1080/00029890.1966.11970915. |
| [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. |
| [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. |
| [13] |
G. W. Tokarsky,
Galperin's triangle example, Comm. Math. Phys., 335 (2015), 1211-1213.
doi: 10.1007/s00220-015-2336-6. |
| [14] |
S. Troubetzkoy,
Recurrence and periodic billiard orbits in polygons, Regular and Chaotic Dynamics, 9 (2004), 1-12.
doi: 10.1070/RD2004v009n01ABEH000259. |
| [15] |
S. Troubetzkoy,
Periodic billiard orbits in right triangles, Annales de l'institut Fourier, 55 (2005), 29-46.
doi: 10.5802/aif.2088. |
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. |
| [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. |
| [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. |
| [6] |
G. Galperin, T. Krüger and S. Troubetzkoy,
Local instability of orbits in polygonal and polyhedral billiards, Comm. Math. Phys., 169 (1995), 463-473.
doi: 10.1007/BF02099308. |
| [7] |
P. Hooper,
Periodic billiard paths in right triangles are unstable, Geom. Dedicata, 125 (2007), 39-46.
doi: 10.1007/s10711-007-9129-9. |
| [8] |
M. Kac,
Can one hear the shape of a drum?, Amer. Math. Monthly, 73 (1966), 1-23.
doi: 10.1080/00029890.1966.11970915. |
| [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. |
| [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. |
| [13] |
G. W. Tokarsky,
Galperin's triangle example, Comm. Math. Phys., 335 (2015), 1211-1213.
doi: 10.1007/s00220-015-2336-6. |
| [14] |
S. Troubetzkoy,
Recurrence and periodic billiard orbits in polygons, Regular and Chaotic Dynamics, 9 (2004), 1-12.
doi: 10.1070/RD2004v009n01ABEH000259. |
| [15] |
S. Troubetzkoy,
Periodic billiard orbits in right triangles, Annales de l'institut Fourier, 55 (2005), 29-46.
doi: 10.5802/aif.2088. |
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 3.2.
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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