
-
Previous Article
A general law of large permanent
- DCDS Home
- This Issue
-
Next Article
Examples of minimal set for IFSs
A locally integrable multi-dimensional billiard system
Steklov Mathematical Institute, 8 Gubkina St. Moscow, 119991, Russia |
We consider a multi-dimensional billiard system in an $(n+1)$-dimensional Euclidean space, the direct product of the "horizontal" hyperplane and the "vertical" line. The hypersurface that determines the system is assumed to be smooth and symmetric in all coordinate hyperplanes. Hence there exists a periodic orbit $γ$ of period 2 moving along the "vertical" coordinate axis. The question we ask is as follows. Is it possible to choose such a system to have the dynamics locally (near $γ$) conjugated to the dynamics of a linear map?
Since the problem is local, the billiard hypersurface can be determined as the graphs of the functions $± f$, where $f$ is even and defined in a neighborhood of the origin on the "horizontal" coordinate hyperplane. We prove that $f$ exists as a formal Taylor series in the non-resonant case and give numerical evidence for convergence of the series.
References:
[1] |
A. Avila, J. De Simoi and V. Kaloshin,
An integrable deformation of an ellipse of small
eccentricity is an ellipse, Annals of Mathematics, 184 (2016), 527-558.
doi: 10.4007/annals.2016.184.2.5. |
[2] |
M. Bialy and A. E. Mironov, Angular Billiard and Algebraic Birkhoff conjecture, Adv. Math.,
313 (2017), 102-126, arXiv: 1601.03196
doi: 10.1016/j.aim.2017.04.001. |
[3] |
G. D. Birkhoff,
Dynamical Systems American Mathematical Society Colloquium Publications, Vol. Ⅸ American Mathematical Society, Providence, R. I. 1966 |
[4] |
S. V. Bolotin, Integrable Birkhoff billiards, Vestnik Moskov. Univ. Ser. I Mat. Mekh. , 2
(1990), 33-36, (in Russian); translated in Mosc. Univ. Mech. Bull., 2 (1990), 10-13. |
[5] |
S. V. Bolotin and D. V. Treschev,
The anti-integrable limit, Russian Math. Surveys, 70 (2015), 975-1030.
doi: 10.4213/rm9692. |
[6] |
B. Beauzamy, E. Bombieri, P. Enflo and H. L. Montgomery,
Products of polynomials in many
variables, Journal of Number Theory, 36 (1990), 219-245.
doi: 10.1016/0022-314X(90)90075-3. |
[7] |
A. Delshams, Yu. Fedorov and R. Ramirez-Ros,
Homoclinic billiard orbits inside symmetrically perturbed ellipsoids, Nonlinearity, 14 (2001), 1141-1195.
doi: 10.1088/0951-7715/14/5/313. |
[8] |
A. Glutsyuk and E. Shustin On polynomially integrable planar outer billiards and curves with symmetry property, preprint arXiv: 1607.07593. |
[9] |
V. V. Kozlov,
Two-link billiard trajectories: Extremal properties and stability, J. Appl. Math. Mech., 64 (2000), 903-907.
doi: 10.1016/S0021-8928(00)00121-0. |
[10] |
V. V. Kozlov,
Problem of stability of two-link trajectories in a multidimensional Birkhoff
billiard, Proc. Steklov Inst. Math., 273 (2011), 196-213.
doi: 10.1134/S0081543811040092. |
[11] |
V. V. Kozlov,
Polynomial conservation laws for the Lorentz gas and the Boltzmann-Gibbs
gas, Russian Math. Surveys, 71 (2016), 253-290.
doi: 10.4213/rm9707. |
[12] |
V. V. Kozlov and D. V. Treshchev,
Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts Translations of Mathematical Monographs, 89 Amer. Math. Soc., Providence, RI, 1991. |
[13] |
S. Tabachnikov,
Geometry and Billiards Student Mathematical Library, 30 Providence, RI -Amer. Math. Soc, 2005.
doi: 10.1090/stml/030. |
[14] |
D. Treschev,
Billiard map and rigid rotation, Phys. D, 255 (2013), 31-34.
doi: 10.1016/j.physd.2013.04.003. |
[15] |
D. V. Treschev,
On a conjugacy problem in billiard dynamics, Proc. Steklov Inst. Math., 289 (2015), 291-299.
doi: 10.1134/S0081543815040173. |
[16] |
H. Whitney, Analytic extensions of functions defined in closed sets, Transactions of the American Mathematical Society, American Mathematical Society, 36 (1934), 63-89.
doi: 10. 1090/S0002-9947-1934-1501735-3. |
show all references
References:
[1] |
A. Avila, J. De Simoi and V. Kaloshin,
An integrable deformation of an ellipse of small
eccentricity is an ellipse, Annals of Mathematics, 184 (2016), 527-558.
doi: 10.4007/annals.2016.184.2.5. |
[2] |
M. Bialy and A. E. Mironov, Angular Billiard and Algebraic Birkhoff conjecture, Adv. Math.,
313 (2017), 102-126, arXiv: 1601.03196
doi: 10.1016/j.aim.2017.04.001. |
[3] |
G. D. Birkhoff,
Dynamical Systems American Mathematical Society Colloquium Publications, Vol. Ⅸ American Mathematical Society, Providence, R. I. 1966 |
[4] |
S. V. Bolotin, Integrable Birkhoff billiards, Vestnik Moskov. Univ. Ser. I Mat. Mekh. , 2
(1990), 33-36, (in Russian); translated in Mosc. Univ. Mech. Bull., 2 (1990), 10-13. |
[5] |
S. V. Bolotin and D. V. Treschev,
The anti-integrable limit, Russian Math. Surveys, 70 (2015), 975-1030.
doi: 10.4213/rm9692. |
[6] |
B. Beauzamy, E. Bombieri, P. Enflo and H. L. Montgomery,
Products of polynomials in many
variables, Journal of Number Theory, 36 (1990), 219-245.
doi: 10.1016/0022-314X(90)90075-3. |
[7] |
A. Delshams, Yu. Fedorov and R. Ramirez-Ros,
Homoclinic billiard orbits inside symmetrically perturbed ellipsoids, Nonlinearity, 14 (2001), 1141-1195.
doi: 10.1088/0951-7715/14/5/313. |
[8] |
A. Glutsyuk and E. Shustin On polynomially integrable planar outer billiards and curves with symmetry property, preprint arXiv: 1607.07593. |
[9] |
V. V. Kozlov,
Two-link billiard trajectories: Extremal properties and stability, J. Appl. Math. Mech., 64 (2000), 903-907.
doi: 10.1016/S0021-8928(00)00121-0. |
[10] |
V. V. Kozlov,
Problem of stability of two-link trajectories in a multidimensional Birkhoff
billiard, Proc. Steklov Inst. Math., 273 (2011), 196-213.
doi: 10.1134/S0081543811040092. |
[11] |
V. V. Kozlov,
Polynomial conservation laws for the Lorentz gas and the Boltzmann-Gibbs
gas, Russian Math. Surveys, 71 (2016), 253-290.
doi: 10.4213/rm9707. |
[12] |
V. V. Kozlov and D. V. Treshchev,
Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts Translations of Mathematical Monographs, 89 Amer. Math. Soc., Providence, RI, 1991. |
[13] |
S. Tabachnikov,
Geometry and Billiards Student Mathematical Library, 30 Providence, RI -Amer. Math. Soc, 2005.
doi: 10.1090/stml/030. |
[14] |
D. Treschev,
Billiard map and rigid rotation, Phys. D, 255 (2013), 31-34.
doi: 10.1016/j.physd.2013.04.003. |
[15] |
D. V. Treschev,
On a conjugacy problem in billiard dynamics, Proc. Steklov Inst. Math., 289 (2015), 291-299.
doi: 10.1134/S0081543815040173. |
[16] |
H. Whitney, Analytic extensions of functions defined in closed sets, Transactions of the American Mathematical Society, American Mathematical Society, 36 (1934), 63-89.
doi: 10. 1090/S0002-9947-1934-1501735-3. |

[1] |
Nicolas Bedaride. Entropy of polyhedral billiard. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 89-102. doi: 10.3934/dcds.2007.19.89 |
[2] |
Pavel Bachurin, Konstantin Khanin, Jens Marklof, Alexander Plakhov. Perfect retroreflectors and billiard dynamics. Journal of Modern Dynamics, 2011, 5 (1) : 33-48. doi: 10.3934/jmd.2011.5.33 |
[3] |
David Cowan. A billiard model for a gas of particles with rotation. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 101-109. doi: 10.3934/dcds.2008.22.101 |
[4] |
Mason A. Porter, Richard L. Liboff. The radially vibrating spherical quantum billiard. Conference Publications, 2001, 2001 (Special) : 310-318. doi: 10.3934/proc.2001.2001.310 |
[5] |
David Cowan. Rigid particle systems and their billiard models. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 111-130. doi: 10.3934/dcds.2008.22.111 |
[6] |
Eugenii Shustin. Dynamics of oscillations in a multi-dimensional delay differential system. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 557-576. doi: 10.3934/dcds.2004.11.557 |
[7] |
Alexey Glutsyuk, Yury Kudryashov. No planar billiard possesses an open set of quadrilateral trajectories. Journal of Modern Dynamics, 2012, 6 (3) : 287-326. doi: 10.3934/jmd.2012.6.287 |
[8] |
Jianlu Zhang. Suspension of the billiard maps in the Lazutkin's coordinate. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2227-2242. doi: 10.3934/dcds.2017096 |
[9] |
Yang Shen, Jiazhong Yang. Hearing the shape of right triangle billiard tables. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5537-5549. doi: 10.3934/dcds.2021087 |
[10] |
Yeping Li. Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 345-360. doi: 10.3934/dcdsb.2011.16.345 |
[11] |
Ming Mei, Yong Wang. Stability of stationary waves for full Euler-Poisson system in multi-dimensional space. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1775-1807. doi: 10.3934/cpaa.2012.11.1775 |
[12] |
Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035 |
[13] |
Dan Li. Global stability in a multi-dimensional predator-prey system with prey-taxis. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1681-1705. doi: 10.3934/dcds.2020337 |
[14] |
Salah Drabla, Salim A. Messaoudi, Fairouz Boulanouar. A general decay result for a multi-dimensional weakly damped thermoelastic system with second sound. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1329-1339. doi: 10.3934/dcdsb.2017064 |
[15] |
M. Bauer, A. Lopes. A billiard in the hyperbolic plane with decay of correlation of type $n^{-2}$. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 107-116. doi: 10.3934/dcds.1997.3.107 |
[16] |
W. Patrick Hooper. Lower bounds on growth rates of periodic billiard trajectories in some irrational polygons. Journal of Modern Dynamics, 2007, 1 (4) : 649-663. doi: 10.3934/jmd.2007.1.649 |
[17] |
Jianlu Zhang. Coexistence of period 2 and 3 caustics for deformative nearly circular billiard maps. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6419-6440. doi: 10.3934/dcds.2019278 |
[18] |
Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993 |
[19] |
Fuyi Xu, Meiling Chi, Lishan Liu, Yonghong Wu. On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2515-2559. doi: 10.3934/dcds.2020140 |
[20] |
Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks and Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]