# American Institute of Mathematical Sciences

October  2017, 37(10): 5271-5284. doi: 10.3934/dcds.2017228

## A locally integrable multi-dimensional billiard system

 Steklov Mathematical Institute, 8 Gubkina St. Moscow, 119991, Russia

Received  January 2017 Revised  May 2017 Published  June 2017

Fund Project: The research is supported by the RNF grant 14-50-00005.

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.

Citation: Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
##### 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.

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##### 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.
The graph of $b^{-1/2}_\infty$ as a function of $\alpha/(2\pi)$. Two "gaps" correspond to the resonances $\frac\alpha{2\pi} = 3/10$ and $\frac\alpha{2\pi} = 1/3$
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