Georgia Institute of Technology College of Sciences

An important question in circle dynamics is regarding the absolute continuity of an invariant measure. We will consider orientation preserving circle homeomorphisms with break points, that is, maps that are smooth everywhere except for several singular points at which the first derivative has a jump. It is well known that the invariant measures of sufficiently smooth circle dieomorphisms are absolutely continuous w.r.t. Lebesgue measure. But in the case of homeomorphisms with break points the results are quite dierent. We will discuss conjugacies between two circle homeomorphisms with break points. Consider the class of circle homeomorphisms with one break point b and satisfying the Katznelson-Ornsteins smoothness condition i.e. Df is absolutely continuous on [b; b + 1] and D2f 2 Lp(S1; dl); p > 1: We will formulate some results concerning the renormaliza- tion behavior of such circle maps.