Apparaît dans la collection : Jean Morlet Chair - School -Renormalization and Visualization for packing, Billiard and Surfaces / Chaire Jean-Morlet - Ecole - Renormalisation et visualisation pour des problèmes d'empilement, de billard et de surfaces

The applications of renormalization ideas in Dynamical Systems became increasingly popular after 1979, and, since then, they played an important role in the study of several classes of low-dimensional systems.Very roughly speaking, the philosophy of renormalization is that, after appropriate rescalings, the long time behaviors at short scales of certain systems are dictated by other systems within a fixed class S of systems. In particular, such a renormalization procedure can iterated and, as it turns out, the phrase portraits of those systems whose successive renormalizations tend to stay in a compact portion of S can often be reasonably described (”plough in the dynamical plane to harvest in the parameter space”, A. Douady).In this minicourse, we shall illustrate these ideas by explaining the com-mon strategy of ”recurrence of renormalization to compact sets” behind two different results: 1.the solutions of Masur and Veech in 1982 to Keane's conjecture of unique ergodicity of almost all interval exchange transformations; 2. the solution of Moreira–Yoccoz in 2001 to Palis' conjecture on the prevalence of stable intersections of pairs of dynamical Cantor sets whose Hausdorff dimensions are large.