A locally integrable multi-dimensional billiard system

Dmitry Treschev

doi:10.3934/dcds.2017228

Article Contents

2017, Volume 37, Issue 10: 5271-5284. Doi: 10.3934/dcds.2017228

This issue Previous Article Examples of minimal set for IFSs Next Article A general law of large permanent

A locally integrable multi-dimensional billiard system

Dmitry Treschev^*,

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

Received: January 2017

Revised: May 2017

Published: October 2017

The research is supported by the RNF grant 14-50-00005

Abstract / Introduction Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 37J10, 37J40; Secondary: 65P10.

Citation:

Full Text(HTML)

Figure 1. 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 $\frac\alpha{2\pi} = 1/3$

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

	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.
	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.
	G. D. Birkhoff, Dynamical Systems American Mathematical Society Colloquium Publications, Vol. Ⅸ American Mathematical Society, Providence, R. I. 1966
	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.
	S. V. Bolotin and D. V. Treschev , The anti-integrable limit, Russian Math. Surveys, 70 (2015) , 975-1030. doi: 10.4213/rm9692.
	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.
	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.
	A. Glutsyuk and E. Shustin On polynomially integrable planar outer billiards and curves with symmetry property, preprint arXiv: 1607.07593.
	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.
	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.
	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.
	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.
	S. Tabachnikov, Geometry and Billiards Student Mathematical Library, 30 Providence, RI -Amer. Math. Soc, 2005. doi: 10.1090/stml/030.
	D. Treschev , Billiard map and rigid rotation, Phys. D, 255 (2013) , 31-34. doi: 10.1016/j.physd.2013.04.003.
	D. V. Treschev , On a conjugacy problem in billiard dynamics, Proc. Steklov Inst. Math., 289 (2015) , 291-299. doi: 10.1134/S0081543815040173.
	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.