Conjugation between circle maps with several break points

Abstract

Let$f$and$g$be two class$P$-homeomorphisms of the circle$S^{1}$with break point singularities, which are differentiable maps except at some singular points where the derivative has a jump. Assume that$f$and$g$have irrational rotation numbers and the derivatives$\text{Df}$and$\text{Dg}$are absolutely continuous on every continuity interval of$\text{Df}$and$\text{Dg}$, respectively. We prove that if the product of the$f$-jumps along all break points of$f$is distinct from that of$g$then the homeomorphism$h$conjugating$f$and$g$is a singular function, i.e. it is continuous on$S^{1}$, but$\text{Dh}(x)=0$ almost everywhere with respect to the Lebesgue measure. This result generalizes previous results for one and two break points obtained by Dzhalilov, Akin and Temir, and Akhadkulov, Dzhalilov and Mayer. As a consequence, we get in particular Dzhalilov–Mayer–Safarov’s theorem: if the product of the$f$-jumps along all break points of$f$is distinct from$1$, then the invariant measure$\unicode[STIX]{x1D707}_{f}$is singular with respect to the Lebesgue measure.