On Poncelet's maps
Cimà, Anna (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Mañosa Fernández, Víctor 1971- (Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada III)

Date:	2010
Abstract:	Given two ellipses, one surrounding the other one, Poncelet introduced a map P from the exterior one to itself by using the tangent lines to the interior ellipse. This procedure can be extended to any two smooth, nested and convex ovals and we call these types of maps, Poncelet's maps. We recall what he proved around 1814 in the dynamical systems language: In the two ellipses' case and when the rotation number of P is rational there exists an n € ℕ such that Pn = Id, or in other words, Poncelet's map is conjugate to a rational rotation. In this paper we study general Poncelet's maps and give several examples of algebraic ovals where the corresponding Poncelet's map has a rational rotation number and is not conjugate to a rotation. Finally, we also provide a new proof of Poncelet's result based on dynamical and computational tools.
Grants:	Ministerio de Economía y Competitividad MTM2008-03437 Ministerio de Economía y Competitividad DPI2005-08-668-C03-1
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Poncelet's problem ; Circle maps ; Rotation number ; Devil's staircase
Published in:	Computers and Mathematics with Applications, Vol. 60, Issue 5 (September 2010) , p. 1457-1464, ISSN 0898-1221

DOI: 10.1016/j.camwa.2010.06.027

Postprint 14 p, 663.8 KB

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