Monday, April 16, 2012 - 11:05
1 hour (actually 50 minutes)
An important question in circle dynamics is regarding the absolute continuity of aninvariant measure. We will consider orientation preserving circle homeomorphisms withbreak points, that is, maps that are smooth everywhere except for several singular pointsat which the rst derivative has a jump. It is well known that the invariant measuresof sufficiently smooth circle dieomorphisms are absolutely continuous w.r.t. Lebesguemeasure. But in the case of homeomorphisms with break points the results are quitedierent. We will discuss conjugacies between two circle homeomorphisms with breakpoints.Consider the class of circle homeomorphisms with one break point b and satisfying theKatznelson-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.