Geodesic
Billiards and nonholonomic distributions
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems. Part VIII, Tome 300 (2003), pp. 56-64 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this note, billiards with full families of periodic orbits are considered. It is shown that construction of a convex billiard with a “rational” caustic (i.e., carrying only periodic orbits) can be reformulated as a problem of finding a closed curve tangent to a $(N-1)$-dimensional distribution on a $(2N-1)$-dimensional manifold. The properties of this distribution are described as well as some important consequences for the billiards with rational caustics. A very particular application of our construction states that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist. 
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Y. Baryshnikov; V. Zharnitsky. Billiards and nonholonomic distributions. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems. Part VIII, Tome 300 (2003), pp. 56-64. http://geodesic.mathdoc.fr/item/ZNSL_2003_300_a5/

[1] V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, Springer, 1983 | MR

[2] G. D. Birkhoff, “On the periodic motions of dynamical systems”, Acta Math., 50 (1927), 359–379 | DOI | MR | Zbl

[3] G. D. Birkhoff, Dynamical systems, With an addendum by Jurgen Moser, AMS Colloquium Publications, IX, AMS, Providence, RI, 1966 | MR

[4] P. A. Griffiths, Exterior differential systems and the calculus of variations, Progress in Mathematics, 25, Birkhäuser, Boston, Mass., 1983 | MR | Zbl

[5] A. Katok, E. Gutkin, “Caustics for inner and outer billiards”, Comm. Math. Phys., 173:1 (1995), 101–133 | DOI | MR | Zbl

[6] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, 91, AMS, Providence, RI, 2002 | MR | Zbl

[7] A. M. Vershik, V. Y. Gershkovich, “Nonholonomic dynamical systems, geometry of distributions and variational problems”, Sovrem. Probl. Mat., Fundam. Napravleniya, 16, Moscow, 1987, 5–85 | MR | Zbl

[8] M. Ghandehari, “An optimal control formulation of the Blashke–Lebesgue theorem”, J. Math. Anal. Appl., 200:2 (1996), 322–331 | DOI | MR | Zbl

[9] V. Zharnitsky, “Invariant tori in the systems of billiard type”, Commun. Math. Phys., 211 (2000), 289–302 | DOI | MR | Zbl

[10] Nobuhiro Innami, “Convex curves whose points are vertices of billiard triangles”, Kodai Math. J., 11 (1988), 17–24 | DOI | MR | Zbl

[11] C. Berge, Unpublished result, communicated to the authors by S. Tabachnikov

[12] M. Rychlik, “Periodic points of the billiard ball map in a convex domain”, J. Diff. Geom., 30:1 (1989), 191–205 | MR | Zbl