Comportamento genérico de difeomorfismos do círculo

Abstract

We will study the generic behavior of circle diffeomorphisms, in the combinatorial, topological and measure-theoretical sense, following the work of Michael Herman. To each order preserving homeomorphism of the circle we can associate a positive real number, called rotation number, which is invariant under conjugacy. We will show that there is a set of irrational numbers with full Lebesgue measure on R such that, if f is a circle diffeomorphism of class Cr, with r greater or equal 3 and with rotation number in that set, then f is at least Cr−2-conjugated to an irrational translation. Moreover, we will show that if ft is a C1-path defined on a interval [a,b] over the set of the circle diffeomorphisms orientation preserving, with r ≥ 3, then the set of parameters where ft is Cr−2-conjugated to a irrational translation has positive Lebesgue measure, since the rotation numbers of fa and fb are distinct.