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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 $C^2(S^1\setminus\{0, c\})$ 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 $c$.

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