Itération de pliages de quadrilatères

Starting with a quadrilateral q₀=(A₁,A₂,A₃,A₄) of $\text{R}{}^2$, one constructs a sequence of quadrilaterals q_n=(A_4n+1,…,A_4n+4) by iteration of foldings: q_n=?₄°?₃°?₂°?₁(q_n?1) where the folding ?_j 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 $\text{v:=}\text{lim}{}_{\mathrm{n\to \infty }}\text{1}\text{n}\text{q}{}_{\mathrm{n}}$ exists.– When none of the q_n is isometric to q₀, then the drift v is zero if and only if one has $\text{max}\text{a}{}_{\mathrm{j}}\text{+}\text{min}\text{a}{}_{\mathrm{j}}\text{?}\text{1}\text{2}\text{∑a}{}_{\mathrm{j}}$, where a₁,…,a₄ are the sidelengths of q₀.– For Lebesgue almost all q₀ the sequence (q_n?nv)_n?1 is dense on a bounded analytic curve with a center, or an axis of symmetry. However, for Baire generic q₀, the sequence (q_n?nv)_n?1 is unbounded. To cite this article: Y. Benoist, D. Hulin, C. R. Acad. Sci. Paris, Ser. I 338 (2004).