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 $f^{\prime }$ has a jump discontinuity at this point. Suppose that $f^{\prime }$ satisfies a certain Zygmund condition dependent on a parameter $\gamma >0.$ We prove that the renormalizations of $f$ are approximated by Möbius 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 Möbius transformations get asymptotically linearly dependent.

arXiv comment26 pages