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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