Itération de pliages de quadrilatères (II)

Abstract

Starting with a quadrilateral $q_0=(A_1,A_2,A_3,A_4)$ of $\m R^2$, one constructs a sequence of quadrilaterals $q_n=(A_{4n+1},\ldots ,A_{4n+4})$ by iteration of foldings~: $q_n= \ph_4\circ\ph_3\circ\ph_2\circ\ph_1(q_{n-1})$ where the folding $\ph_j$ replaces the vertex number $j$ by its symmetric with respect to the opposite diagonal. We have studied [1] the dynamical behavior of this sequence. In particular, we have seen that the drift ${\D v(q_0):= \lim_{n\ra\infty}}\frac1n q_n$ exists and, for Lebesgue almost all $q_0$, the sequence $( q_n -nv(q_0))_{n\geq 1}$ is dense on a bounded analytic curve. Here, we prove that, for Baire generic $q_0$, the closure of the same sequence $( q_n -nv(q_0))_{n\geq 1}$ contains all the translates of $q_0$.