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Singular measures of piecewise smooth circle homeomorphisms with two break points

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  • 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.
    Mathematics Subject Classification: Primary: 37E10, 37C40; Secondary: 37A05, 37E45, 37C15.


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