On the renormalizations of circle homeomorphisms with several break\n points

2017/06/12 by Kleyber Cunha, Cunha, Kleyber, Akhtam Dzhalilov +3
Mathematics · Physics and Astronomy · #37B10 #37C05 #37E05 #37E10 #37E20 #Advanced Differential Equations and Dynamical Systems #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1706.03654

openalex publication_date 2017/06/12 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28

Abstract

Let f be an orientation preserving homeomorphisms on the circle with\nseveral break points, that is, its derivative Df has jump discontinuities at\nthese points. We study Rauzy-Veech renormalizations of piecewise smooth circle\nhomeomorphisms, by considering such maps as generalized interval exchange maps\nwith genus one. Suppose that Df is absolutely continuous on the each interval\nof continuity and D\lnDf\∈ mathbbL_p for some p>1. We prove that,\nunder certain combinatorial assumptions on f, renormalizations Rⁿ(f) are\napproximated by piecewise M "obus functions in C^1+L₁-norm, that\nmeans, Rⁿ(f) are approximated in C¹-norm and D²Rⁿ(f) are\napproximated in L₁-norm. In particular, if f has trivial product of size\nof breaks, then the renormalizations are approximated by piecewise affine\ninterval exchange maps.\n

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