Systèmes dynamiques
Comptes Rendus. Mathématique, Volume 338 (2004) no. 3, pp. 235-238.

Starting with a quadrilateral q0=(A1,A2,A3,A4) of ${ℝ}^{2}$, one constructs a sequence of quadrilaterals qn=(A4n+1,…,A4n+4) by iteration of foldings: qn=ϕ4ϕ3ϕ2ϕ1(qn−1) where the folding ϕj replaces the vertex number j by its symmetric with respect to the opposite diagonal.

We study the dynamical behavior of this sequence. In particular, we prove that:

– The drift $\mathrm{v}:={\mathrm{lim}}_{\mathrm{n}\to \infty }\frac{1}{n}{q}_{n}$ exists.

– When none of the qn is isometric to q0, then the drift v is zero if and only if one has $\mathrm{max}{a}_{j}+\mathrm{min}{a}_{j}⩽\frac{1}{2}\sum {\mathrm{a}}_{j}$, where a1,…,a4 are the sidelengths of q0.

– For Lebesgue almost all q0 the sequence (qnnv)n⩾1 is dense on a bounded analytic curve with a center, or an axis of symmetry. However, for Baire generic q0, the sequence (qnnv)n⩾1 is unbounded.

Partant d'un quadrilatère q0=(A1,A2,A3,A4) de ${ℝ}^{2}$, on construit une suite qn=(A4n+1,…,A4n+4) de quadrilatères par itération de pliages : qn=ϕ4ϕ3ϕ2ϕ1(qn−1), où le pliage ϕj remplace le sommet numéro j par son symétrique par rapport à la diagonale opposée. Dans cette Note, nous étudions le comportement dynamique de la suite qn.

Accepted:
Published online:
DOI: 10.1016/j.crma.2003.12.011

Yves Benoist 1; Dominique Hulin 2

1 École normale supérieure-CNRS, 45, rue d'Ulm, 75230 Paris, France
2 Université Paris-Sud, bâtiment 425, Orsay 91405, France
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Yves Benoist; Dominique Hulin. Itération de pliages de quadrilatères. Comptes Rendus. Mathématique, Volume 338 (2004) no. 3, pp. 235-238. doi : 10.1016/j.crma.2003.12.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.12.011/`

[1] Y. Benoist, D. Hulin, Itération de pliages de quadrilatères, Preprint, 2003

[2] H. Bos; C. Kers; F. Oort; D. Raven Poncelet's closure theorem, Exposition. Math., Volume 5 (1987), pp. 289-364

[3] K. Charter; T. Rogers The dynamics of quadrilateral folding, Experiment. Math., Volume 2 (1993), pp. 209-222

[4] J. Esch; T. Rogers Dynamics on elliptic curves arising from polygonal folding, Discrete Comput. Geom., Volume 25 (2001), pp. 477-502

[5] P. Griffiths; J. Harris On Cayley's explicit solution to Poncelet porism, Enseign. Math., Volume 24 (1978), pp. 31-40

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