Starting with a quadrilateral q0=(A1,A2,A3,A4) of , one constructs a sequence of quadrilaterals qn=(A4n+1,…,A4n+4) by iteration of foldings: qn=ϕ4∘ϕ3∘ϕ2∘ϕ1(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 exists.
– When none of the qn is isometric to q0, then the drift v is zero if and only if one has , where a1,…,a4 are the sidelengths of q0.
– For Lebesgue almost all q0 the sequence (qn−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 (qn−nv)n⩾1 is unbounded.
Partant d'un quadrilatère q0=(A1,A2,A3,A4) de , on construit une suite qn=(A4n+1,…,A4n+4) de quadrilatères par itération de pliages : qn=ϕ4∘ϕ3∘ϕ2∘ϕ1(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:
@article{CRMATH_2004__338_3_235_0, author = {Benoist, Yves and Hulin, Dominique}, title = {It\'eration de pliages de quadrilat\`eres}, journal = {Comptes Rendus. Math\'ematique}, pages = {235--238}, publisher = {Elsevier}, volume = {338}, number = {3}, year = {2004}, doi = {10.1016/j.crma.2003.12.011}, language = {fr}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2003.12.011/} }
TY - JOUR AU - Benoist, Yves AU - Hulin, Dominique TI - Itération de pliages de quadrilatères JO - Comptes Rendus. Mathématique PY - 2004 SP - 235 EP - 238 VL - 338 IS - 3 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2003.12.011/ DO - 10.1016/j.crma.2003.12.011 LA - fr ID - CRMATH_2004__338_3_235_0 ER -
%0 Journal Article %A Benoist, Yves %A Hulin, Dominique %T Itération de pliages de quadrilatères %J Comptes Rendus. Mathématique %D 2004 %P 235-238 %V 338 %N 3 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2003.12.011/ %R 10.1016/j.crma.2003.12.011 %G fr %F CRMATH_2004__338_3_235_0
Benoist, Yves; Hulin, Dominique. Itération de pliages de quadrilatères. Comptes Rendus. Mathématique, Volume 338 (2004) no. 3, pp. 235-238. doi : 10.1016/j.crma.2003.12.011. http://archive.numdam.org/articles/10.1016/j.crma.2003.12.011/
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