Circle homeomorphisms with two break points

Abstract

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 C2(S1∖{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.