Singular measures of piecewise smooth circle homeomorphisms with two break points. (English) Zbl 1168.37009

Dzhalilov, Akhtam; Liousse, Isabelle; Mayer, Dieter

Discrete Contin. Dyn. Syst. 24, No. 2, 381-403 (2009).

Let \(T f:S^1\rightarrow S^1\) be a circle homeomorphism with two break points (i.e. two points where the derivative of the lift \(D f\) is discontinuous). Assume that:
i) \(T f\) has irrational rotation number,
ii) That \(Df\) is absolutely continuous (apart from the break points),
iii) That \(D(\log(D f))\) is integrable on \([0,1]\),
iv) The product of the jump ratios at the two break points is nontrivial. Then the unique \(T f\) invariant probability measure is singular with respect to Lebesgue measure on \(S^1\).

Reviewer: E. Ahmed (Mansoura)

MSC:

37E10	Dynamical systems involving maps of the circle
37A05	Dynamical aspects of measure-preserving transformations

Keywords:

circle homeomorphism; break points; rotation number; invariant measures