On Local Integrability in Billiard Dynamics

ABSTRACT

We consider a billiard system in a convex domain symmetric in coordinate axes . Hence, there exists a periodic orbit γ of period 2 moving along the y-axis. We ask the following question. Is it possible to choose the domain such that the dynamics of the corresponding billiard map are locally (near γ) conjugated to the dynamics of the rigid rotation by the angle α? In [Treschev 13], numeric evidence for positive answer is given in the case when α/π is a Diophantine number. In this paper, we propose further numerical analysis of the problem. In particular , we show that the relative measure of the domain in the billiard phase space on which the dynamics is conjugated to the rigid rotation can reach 50%.

Keywords:

• billiard system
• integrable billiard system
• superintegrability

Notes

1 One should keep in mind that there are many nonequivalent definitions of integrability.

2 and for multidimensional case in [Treschev nd]

3 Here for brevity we skip some brackets: in particular, the expression τ₋χ τ₊f○χ means (τ₋χ) · (τ₊f○χ), etc.

4 Symmetry of the problem implies that the functions f corresponding to the angles α and π − α both exist and coincide, or both do not exist. Hence, the interval (0.3, 0.5) can be extended to (0.3, 0.7).

5 It has the form χ(B_r).

Additional information

Funding

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