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Circle homeomorphisms with two break points. (English) Zbl 1147.37025
Summary: 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.

MSC:
37E10Maps of the circle
37C15Topological and differentiable equivalence, conjugacy, invariants, moduli, classification