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Dissipative outer billiards: a case study. (English) Zbl 1351.37158

Summary: We study dissipative polygonal outer billiards, i.e. outer billiards about convex polygons with a contractive reflection law. We prove that dissipative outer billiards about any triangle and the square are asymptotically periodic, i.e. they have finitely many global attracting periodic orbits. A complete description of the bifurcations of the periodic orbits as the contraction rates vary is given. For the square billiard, we also show that the asymptotic periodic behaviour is robust under small perturbations of the vertices and the contraction rates. Finally, we describe some numerical experiments suggesting that dissipative outer billiards about regular polygon are generically asymptotically periodic.

MSC:

37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37E15 Combinatorial dynamics (types of periodic orbits)
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