Λ-stability of periodic billiard orbits. (2nd August 2018)

• Record Type: Journal Article
• Authors: Gaivão, José Pedro; Troubetzkoy, Serge
• Is Part Of: Nonlinearity. Volume 31:Number 9(2018:Sep.)
• Page Start: 4326
• Page End: 4353
• Publication Date: 2018-08-02
• Subjects: polygonal billiards -- periodic orbits -- stability; 37D50 -- 37C27; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aacc47
• Languages: English
• ISSNs: 0951-7715

Abstract

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 . 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.