## Nihonkai Mathematical Journal

### Differentiability of invariant circles for strongly integrable convex billiards

Nobuhiro Innami

#### Abstract

Let $C$ be a closed convex curve of class $C^2$ in the plane. We consider the domain bounded by $C$ a billiard table. Assume that the convex billiard of $C$ is integrable and satisfies a certain property. The property is that the limiting leaves are either closed curves or discrete points in the phase space. Then the set of points with irrational slopes make invariant circles of class $C^1$. If the sets of points with rational slopes do not make invariant circles, then they contains two invariant circles such that they are of class $C^1$ except at finitely many points in $C$.

#### Article information

Source
Nihonkai Math. J., Volume 24, Number 1 (2013), 1-17.

Dates
First available in Project Euclid: 5 September 2013

https://projecteuclid.org/euclid.nihmj/1378408114

Mathematical Reviews number (MathSciNet)
MR3114122

Zentralblatt MATH identifier
1281.53044

Subjects
Secondary: 37E40: Twist maps

#### Citation

Innami, Nobuhiro. Differentiability of invariant circles for strongly integrable convex billiards. Nihonkai Math. J. 24 (2013), no. 1, 1--17. https://projecteuclid.org/euclid.nihmj/1378408114

#### References

• V. Bangert. Mather sets for twist maps and geodesics on tori, In U. Kirchgraber, H. O. Walther (eds.) Dynamics Reported, Vol 1, pp.1–56, John Wiley & Sons and B. G. Teubner, 1988.
• V. Bangert. Geodesic rays, Busemann functions and monotone twist maps, Calc. Var., Vol 2, 49–63 (1994).
• M. Bialy. Convex billiards and a theorem of E. Hopf. Math. Z., 214 (1993) 147–154.
• G. D. Birkhoff. Surface translations and their dynamical applications, Acta Math., Vol 43, 1–119 (1922).
• H. Busemann. The geometry of geodesics, Academic Press, New York, 1955.
• L. Green. A theorem of E. Hopf, Michigan Math. J., Vol 5, 31–34 (1958).
• E. Gutkin and A. Katok. Caustics for inner and outer billiards, Commun. Math. Phys., Vol 173, 101–133 (1995).
• E. Hopf. Closed surfaces without conjugate points, Nat. Acad. Sci. U.S.A., 34 (1948) 47–51.
• N. Innami. Convex curves whose points are vertices of billiard triangles, Kodai Math. J., 11 (1988) 17–24.
• N. Innami. Integral formulas for polyhedral and spherical billiards, J. Math. Soc. Japan, 50, 2 (1998) 339–357.
• N. Innami. Geometry of geodesics for convex billiards and circular billiards. Nihonkai Math. J. 13, 1 (2002) 73–120.
• W. Klingenberg. A course in differential geometry, Berlin, Springer-Verlag, 1978.
• V. F. Lazutkin. The existence of caustics for a billiard problem in a convex domain, Izv. Akad. Nauk SSSR Ser. Mat., 37, 1 (1973) 186–216.
• A. Katok and B. Hasselblatt. Introduction to the modern theory of dynamical systems, Cambridge University Press, New York, 1995.
• M. P. Wojtkowski. Two applications of Jacobi fields to the billiard ball problems, J. Diff. Geom., 40 (1994) 155–164.