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

Dzhalilov, Akhtam; Liousse, Isabelle

Nonlinearity 19, No. 8, 1951-1968 (2006).

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:

37E10	Dynamical systems involving maps of the circle
37C15	Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems