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Dynamics of particles on a curve with pairwise hyper-singular repulsion
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |






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