# American Institute of Mathematical Sciences

June  2009, 24(2): 381-403. doi: 10.3934/dcds.2009.24.381

## Singular measures of piecewise smooth circle homeomorphisms with two break points

 1 Faculty of Mathematics, Samarkand State University, Boulevard st. 15, 703004 Samarkand, Uzbekistan 2 Laboratoire Paul Painlevé, Universite Lille I, F-59655 Villeneuve d'Ascq, France 3 Institut für Theoretische Physik, Technische Universität Clausthal, Abteilung Statistische Physik und Nichtlineare Dynamik, Arnold Sommerfeld Straße 6, 38678 Clausthal–Zellerfeld

Received  July 2008 Revised  November 2008 Published  March 2009

Let $T_{f}$ : S1S1 be a circle homeomorphism with two break points ab, cb that means the derivative $Df$ of its lift $f\ :\ \mathbb{R}\rightarrow\mathbb{R}$ has discontinuities at the points ã b, ĉb, which are the representative points of ab, cb in the interval $[0,1)$, and irrational rotation number ρf. Suppose that $Df$ is absolutely continuous on every connected interval of the set [0,1]\{ãb, ĉb}, that DlogDf ∈ L1([0,1]) and the product of the jump ratios of $Df$ at the break points is nontrivial, i.e. $\frac{Df_{-}(\tilde{a}_{b})}{Df_{+}(\tilde{a}_{b})}\frac{Df_{-}(\tilde{c}_{b})}{Df_{+}(\tilde{c}_{b})} \ne1$. We prove, that the unique Tf - invariant probability measure $\mu_{f}$ is then singular with respect to Lebesgue measure on S1.
Citation: Akhtam Dzhalilov, Isabelle Liousse, Dieter Mayer. Singular measures of piecewise smooth circle homeomorphisms with two break points. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 381-403. doi: 10.3934/dcds.2009.24.381
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