On conjugations of PP-homeomorphisms

Let TfTf and TgTg be PP-homeomorphisms of a circle with break point singularities that is, differentiable except in many countable points where the derivative has a jump. Let TfTf and TgTg have the same irrational rotation number. We provide a sufficient condition for the ββ-Hölder continuity of a conjugating map between TfTf and TgTg. Moreover, we provide a sufficient and a necessary condition for the Lipschitz continuity of the conjugating map. And also we provide a sufficient and necessary condition for the C1C1-smoothness of the conjugating map between the break equivalent homeomorphisms TfTf and TgTg.