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On an invariant measure for homeomorphisms of a circle with a point of break. (English. Russian original) Zbl 0921.58035
Funct. Anal. Appl. 32, No. 3, 153-161 (1998); translation from Funkts. Anal. Prilozh. 32, No. 3, 11-21 (1998).
The authors study homeomorphisms of the circle \(S^1=[0,1)\) with a simple point of break, that is, with a jump of the first derivative at a point. As is well known in the case of generic rotation numbers, the invariant measure for a \(C^{2+\nu}\)-diffeomorphism is absolutely continuous with respect to the Lebesgue measure. The authors show that the situation becomes diametrically opposite in the presence of a single point of break-type singularity, namely, they prove that an invariant measure is always singular with respect to the Lebesgue measure.

MSC:
37A99 Ergodic theory
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