On Poncelet’s maps

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Sep 2010Embargo end date: 01 Jan 2008 English Publisher:Elsevier BVJournal:Computers & Mathematics with Applications, volume 60, pages 1,457-1,464 (issn: 0898-1221,

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Authors: Cima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, Víctor;

doi: 10.1016/j.camwa.2010.06.027 , 10.48550/arxiv.0812.2588

arXiv: 0812.2588

handle: 2117/8960

On Poncelet’s maps

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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 this type 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 a natural number such that P^n is the identity, or in other words, the Poncelet's map is conjugated 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 it is not conjugated to a rotation. Finally, we also provide a new proof of Poncelet's result based on dynamical tools.

14 pages; 5 figures on 9 EPS files

Related Organizations

Autonomous University of Barcelona
Spain
Universitat Polit?cnica de Catalunya
Spain
Universitat Politècnica de Catalunya
Spain
Universitat Polite`cnica de Catalunya
Spain

Keywords

Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria, Poncelet's problem, devil's staircase, Geometria projectiva, Poncelet’s theorem, Hyperbolic systems with singularities (billiards, etc.), Dynamical Systems (math.DS), rotation number, Poncelet’s problem, Devil’s staircase, Modelling and Simulation, FOS: Mathematics, Continuous geometries, Mathematics - Dynamical Systems, 51M15, 37C05, 51M04, Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics, :Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics [Àrees temàtiques de la UPC], :51 Geometry::51M Real and complex geometry [Classificació AMS], Sistemes dinàmics diferenciables, Real and complex geometry, Devil's staircase, Circle maps, Computational Mathematics, Computational Theory and Mathematics, Classificació AMS::51 Geometry::51M Real and complex geometry, circle maps, :Matemàtiques i estadística::Geometria [Àrees temàtiques de la UPC], Rotation number

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