Nonrigidity of piecewise-smooth circle maps

2013/02/27 by Habibulla Akhadkulov, Akhadkulov, Habibulla, Akhtam Dzhalilov +3
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1302.6691

openalex publication_date 2013/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f_i, i=1,2 be piecewise-smooth C¹ circle homeomorphisms with two break points, log Df_i, i=1,2 are absolutely continuous on each continuity intervals of Df_i and Dlog Df_i∈ L^p for some p>1. Suppose, the jump ratios of f₁ and f₂ at their break points do not coincide but have the same total jumps (i.e. the product of jump ratios) and identical irrational rotation number of bounded type. Then the conjugation h between f₁ and f₂ is a singular function, i.e. it is continuous on S¹, but Dh(x)=0 a.e. with respect to Lebesgue measure.

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