Density of convex billiards with rational caustics. (English) Zbl 1400.37043
Summary: We show that in the space of all convex billiard boundaries, the set of boundaries with rational caustics is dense. More precisely, the set of billiard boundaries with caustics of rotation number 1/\(q\) is polynomially sense in the smooth case, and exponentially dense in the analytic case.
MSC:
| 37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37J50 | Action-minimizing orbits and measures (MSC2010) |
| 70H08 | Nearly integrable Hamiltonian systems, KAM theory |
References:
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