Renormalization of circle diffeomorphisms with a break-type singularity

2015/10/12 by Akhadkulov, Habibulla, Noorani, Mohd Salmi Md, Akhatkulov, Sokhobiddin
#37C15 #37C40 #37E10 #37F25 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1510.03202

Abstract

Let f be an orientation-preserving circle diffeomorphism with irrational rotation number and with a break point ξ₀, that is, its derivative f' has a jump discontinuity at this point. Suppose that f' satisfies a certain Zygmund condition dependent on a parameter γ>0. We prove that the renormalizations of f are approximated by Möbius transformations in C¹-norm if γ∈ (0,1] and they are approximated in C²-norm if γ∈ (1,+∞). It is also shown, that the coefficients of Möbius transformations get asymptotically linearly dependent.

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