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Itération de pliages de quadrilatères

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Abstract

Iteration of quadrilateral foldings. Starting with a quadrilateral q 0=(A 1,A 2,A 3,A 4) of ℝ2, one constructs a sequence of quadrilaterals q n =(A 4n+1,...,A 4n+4) by iteration of foldings: q n =ϕ4°ϕ3°ϕ2°ϕ1(q n-1) where the folding ϕ j replaces the vertex number j by its symmetric with respect to the opposite diagonal (see Fig. 1).

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

– The drift \({v:= \lim_{n\rightarrow\infty}}\frac{1}{n} q_n\) exists.

– When none of the q n is isometric to q 0, then the drift v is zero if and only if one has maxa j +mina j ≤1/2∑a j where a 1,...,a 4 are the sidelengths of q 0.

– For Lebesgue almost all q 0 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 q 0, the sequence (q n -nv) n≥1 is unbounded (see Figs. 2 to 7).

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Authors and Affiliations

  1. Ecole Normale Supérieure-CNRS, 45 rue d’Ulm, 75230, Paris, France

    Yves Benoist

  2. Université Paris-Sud, Bâtiment 425, Orsay, 91405, France

    Dominique Hulin

Authors
  1. Yves Benoist
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  2. Dominique Hulin
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Correspondence to Yves Benoist or Dominique Hulin.

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Benoist, Y., Hulin, D. Itération de pliages de quadrilatères. Invent. math. 157, 147–194 (2004). https://doi.org/10.1007/s00222-003-0353-0

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  • DOI: https://doi.org/10.1007/s00222-003-0353-0

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