Abstract

The applications ofrenormalization 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 common strategy ofrecurrence of renormalization to compact sets” behind two different results: 1. the solutions of Masur and Veech in 1982 to Keane's coniecture ofunique ergod. 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.