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Zbl 1147.37025
Dzhalilov, Akhtam; Liousse, Isabelle
Circle homeomorphisms with two break points. (English)
[J] Nonlinearity 19, No. 8, 1951-1968 (2006). ISSN 0951-7715; ISSN 1361-6544/e

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 $c$.

MSC 2000:: *37E10 Maps of the circle
37C15 Topological and differentiable equivalence, etc.

Cited in: Zbl 1242.37030 Zbl 1147.37024

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