Figure 2.1.
After a reflection, $ P_{n} $ becomes $ P_{n+1} $ in Case ($ i $ ) and $ P_{n-1} $ in Case ($ ii $ )
-
Previous Article
Concentration phenomena for magnetic Kirchhoff equations with critical growth
- DCDS Home
- This Issue
-
Next Article
Dynamics of particles on a curve with pairwise hyper-singular repulsion
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 |
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,
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. |
| [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
| [1] |
Xiaohui Liu, Jinhua Wang, Dianhua Wu. Two new classes of binary sequence pairs with three-level cross-correlation. Advances in Mathematics of Communications, 2015, 9 (1) : 117-128. doi: 10.3934/amc.2015.9.117 |
| [2] |
Zhe Zhang, Jiuping Xu. Bi-level multiple mode resource-constrained project scheduling problems under hybrid uncertainty. Journal of Industrial & Management Optimization, 2016, 12 (2) : 565-593. doi: 10.3934/jimo.2016.12.565 |
| [3] |
Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015 |
| [4] |
Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018 |
| [5] |
Walter Briec, Bernardin Solonandrasana. Some remarks on a successive projection sequence. Journal of Industrial & Management Optimization, 2006, 2 (4) : 451-466. doi: 10.3934/jimo.2006.2.451 |
| [6] |
Mario Roldan. Hyperbolic sets and entropy at the homological level. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3417-3433. doi: 10.3934/dcds.2016.36.3417 |
| [7] |
Alexander Bobylev, Åsa Windfäll. Boltzmann equation and hydrodynamics at the Burnett level. Kinetic & Related Models, 2012, 5 (2) : 237-260. doi: 10.3934/krm.2012.5.237 |
| [8] |
Kai-Uwe Schmidt, Jonathan Jedwab, Matthew G. Parker. Two binary sequence families with large merit factor. Advances in Mathematics of Communications, 2009, 3 (2) : 135-156. doi: 10.3934/amc.2009.3.135 |
| [9] |
Matthew Macauley, Henning S. Mortveit. Update sequence stability in graph dynamical systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1533-1541. doi: 10.3934/dcdss.2011.4.1533 |
| [10] |
Wenjun Xia, Jinzhi Lei. Formulation of the protein synthesis rate with sequence information. Mathematical Biosciences & Engineering, 2018, 15 (2) : 507-522. doi: 10.3934/mbe.2018023 |
| [11] |
Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 3017-3025. doi: 10.3934/dcdss.2020465 |
| [12] |
Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim. New design of quaternary LCZ and ZCZ sequence set from binary LCZ and ZCZ sequence set. Advances in Mathematics of Communications, 2009, 3 (2) : 115-124. doi: 10.3934/amc.2009.3.115 |
| [13] |
Wenbing Chen, Jinquan Luo, Yuansheng Tang, Quanquan Liu. Some new results on cross correlation of $p$-ary $m$-sequence and its decimated sequence. Advances in Mathematics of Communications, 2015, 9 (3) : 375-390. doi: 10.3934/amc.2015.9.375 |
| [14] |
Fanghua Lin, Dan Liu. On the Betti numbers of level sets of solutions to elliptic equations. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4517-4529. doi: 10.3934/dcds.2016.36.4517 |
| [15] |
Francisco Sánchez-Sánchez, Miguel Vargas-Valencia. Games with nested constraints given by a level structure. Journal of Dynamics & Games, 2018, 5 (2) : 95-107. doi: 10.3934/jdg.2018007 |
| [16] |
Jun Wu, Shouyang Wang, Wuyi Yue. Supply contract model with service level constraint. Journal of Industrial & Management Optimization, 2005, 1 (3) : 275-287. doi: 10.3934/jimo.2005.1.275 |
| [17] |
David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335 |
| [18] |
Marina Dolfin, Mirosław Lachowicz. Modeling DNA thermal denaturation at the mesoscopic level. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2469-2482. doi: 10.3934/dcdsb.2014.19.2469 |
| [19] |
Alexander Zeh, Antonia Wachter. Fast multi-sequence shift-register synthesis with the Euclidean algorithm. Advances in Mathematics of Communications, 2011, 5 (4) : 667-680. doi: 10.3934/amc.2011.5.667 |
| [20] |
Zilong Wang, Guang Gong. Correlation of binary sequence families derived from the multiplicative characters of finite fields. Advances in Mathematics of Communications, 2013, 7 (4) : 475-484. doi: 10.3934/amc.2013.7.475 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]