*Iteration of quadrilateral foldings.* Starting with a quadrilateral *q*_{0}=(*A*_{1},*A*_{2},*A*_{3},*A*_{4}) of ℝ^{2}, one constructs a sequence of quadrilaterals *q*_{n}=(*A*_{4n+1},...,*A*_{4n+4}) by iteration of foldings: *q*_{n}=ϕ_{4}°ϕ_{3}°ϕ_{2}°ϕ_{1}(*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 max*a*_{j}+min*a*_{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).