Piecewise-smooth circle homeomorphisms with several break points. (English. Russian original) Zbl 1242.37030

Dzhalilov, A. A.; Mayer, D.; Safarov, U. A.

Izv. Math. 76, No. 1, 94-112 (2012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 76, No. 1, 101-120 (2012).

The paper is devoted to the study of circle homeomorphisms with several break points, that is, maps that are smooth everywhere except for several singular points at which the first derivative has a jump. It is shown that the invariant probability measure of an ergodic piecewise-smooth circle homeomorphism with several break points and the product of the jumps at break points being non-trivial is singular with respect to the Lebesque measure. The paper is a continuation of the papers [the first author and I. Liousse, Nonlinearity 19, No. 8, 1951–1968 (2006; Zbl 1147.37025)] and [the first two authors and I. Liousse, Discrete Contin. Dyn. Syst. 24, No. 2, 381–403 (2009; Zbl 1168.37009)].
Some open problems resulting from the discussion are posed.

Reviewer: Georgy Osipenko (St. Peterburg)

MSC:

37E10	Dynamical systems involving maps of the circle
37C40	Smooth ergodic theory, invariant measures for smooth dynamical systems
37C15	Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37E45	Rotation numbers and vectors

Keywords:

circle homeomorhism; Poincaré rotation number; invariant measure

Citations:

Zbl 1147.37025; Zbl 1168.37009