Invariant measures, conjugacies and renormalizations of piecewise-smooth circle maps

The class of orientation preserving circle homeomorphisms is one of important part of one-dimensional dynamical systems. We will consider piecewise-smooth circle homeomorphisms f with finite many break points, that is, maps that are smooth everywhere except for several singular points at which the first derivative Df has a jump. It is well known that the invariant measures of sufficiently smooth circle diffeomorphisms are absolutely continuous w.r.t. Lebesque measure. But in the case maps with break points the results are quite different. We formulate the main results on invariant measures, conjugacies and renormalizations of piecewise-smooth diffeomorphisms with finite many break points.