On the conjugation of piecewise smooth circle homeomorphisms with a finite number of break points

Abstract

Let fi, i = 1, 2, be orientation preserving circle homeomorphisms with a finite number of break points , at which the first derivatives Dfi have jumps, and whose irrational rotation numbers coincide. Denote by the jump ratio of fi at a break point and by its total jump ratio, given by the product of the jump ratios over all break points .We prove that, for two such circle homeomorphisms fi, i = 1, 2, which on each interval of continuity of Dfi are C2+ε, ε > 0, and whose total jump ratios and do not coincide, the map h conjugating f1 and f2 is a singular function.