Konstantin Khanin, Sasa Kocic
Examples of non-rigidity for circle homeomorphisms with breaks
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ABSTRACT. We give examples of analytic circle maps with singularities of break type with the same rotation number and the same size of the break for which no conjugacy is Lipschitz continuous. In the second part of the paper, we discuss a class of rotation numbers for which a conjugacy is $C^1$-smooth, although the numbers can be strongly non-Diophantine (Liouville). For the rotation numbers in this class, we construct examples of analytic circle maps with breaks, for which the conjugacy is not $C^{1+lpha}$ smooth, for any $lpha>0$.