Renormalization of circle diffeomorphisms with a break-type singularity

Abstract

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 jump discontinuity 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 $\gamma\in (1,+\infty).$ It is also shown, that the coefficients of M\"{o}bius transformations get asymptotically linearly dependent.