Renormalizations of circle diffeomorphisms with a break-type singularity

Habibulla Akhadkulov, Mohd Salmi Md Noorani, Sokhobiddin Akhatkulov

Published 2015-10-12Version 1

Let $f$ be an orientation-preserving circle diffeomorphism with irrational rotation number and with a break point $\xi_{0},$ that is, its derivative $f'$ has a discontinuity of the first kind at this point. Suppose that $f'$ satisfies a certain Zygmund condition dependent on a parameter $\gamma>0.$ We prove that the renormalizations of $f$ are approximated by M\"{o}bius transformations in $C^{1}$-norm if $\gamma\in (0,1]$ and they are approximated in $C^{2}$-norm if $\gamma\in (1,+\infty).$ Also it is shown, that the coefficients of M\"{o}bius transformations get asymptotically linearly dependent.