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λ-stability of periodic billiard orbits

Abstract : We introduce a new notion of stability for periodic orbits in polygonal billiards. We say that a periodic orbit of a polygonal billiard is λ-stable if there is a periodic orbit for the corresponding pinball billiard which converges to it as λ → 1. This notion of stability is unrelated to the notion introduced by Galperin, Stepin and Vorobets. We give sufficient and necessary conditions for a periodic orbit to be λ-stable and prove that the set of d-gons having at most finite number of λ-stable periodic orbits is dense is the space of d-gons. Moreover, we also determine completely the λ-stable periodic orbits in integrable polygons.
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José Pedro Gaivão, Serge Troubetzkoy. λ-stability of periodic billiard orbits. Nonlinearity, IOP Publishing, 2018, 31 (9), pp.4326-4353. ⟨10.1088/1361-6544/aacc47⟩. ⟨hal-01400339⟩

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