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Let $f_i\in C^{2+\alpha}(S^1\setminus \{a_i,b_i\}), \alpha >0, i=1,2$ be circle homeomorphisms with two break points $a_i,b_i$, i.e. discontinuities in the derivative $f_i$, with identical irrational rotation number $rho$ and $\mu_1([a_1,b_1])= \mu_2([a_2,b_2])$, where $\mu_i$ are invariant measures of $f_i$. Suppose the products of the jump ratios of $Df_1$ and $Df_2$ do not coincide, i.e. $\frac{Df_1(a_1-0)}{Df_1(a_1+0)}\times \frac{Df_1(b_1-0)}{Df_1(b_1+0)}\neq \frac{Df_2(a_2-0)}{Df_2(a_2+0)}\times \frac{Df_2(b_2-0)}{Df_2(b_2+0)}$. Then the map $\psi$ conjugating $f_1$ and $f_2$ is a singular function, i.e. it is continuous on $S^1$, but $D\psi = 0$ a.e. with respect to Lebesgue measureComment: 16 pages, 2 figures, to appear in Ergodic Theory and Dynamical System

Let f be a circle class P homeomorphism with two break points 0 and c. If the rotation number of f is of bounded type and f is C 2 (S 1 \ {0, c}) then the unique f -invariant probability measure is absolutely continuous with respect to the Lebesgue measure if and only if 0 and c are on the same orbit and the product of their f -jumps is 1. We indicate how this result extends to class P homeomorphisms of rotation number of bounded type and with a finite number of break points such that f admits at least two break points 0 and c not on the same orbit and that the jump of f at c is not the product of some f -jumps at breaks points not belonging to the orbits of 0 and c.

No abstract

Let T f : S 1 → S 1 be a circle homeomorphism with two break points a b , c b , that means the derivative Df of its lift f : R → R has discontinuities at the points ãb , cb which are the representative points of a b , c b in the interval [0, 1), and irrational rotation number ρ f . Suppose that Df is absolutely continuous on every connected interval of the set [0, 1]\{ã b , cb }, that DlogDf ∈ L 1 ([0, 1]) and the product of the jump ratios of Df at the break points is nontrivial, i.e.Df + (c b ) = 1. We prove that the unique T finvariant probability measure µ f is then singular with respect to Lebesgue measure on S 1 .