Renormalization of circle diffeomorphisms with a break-type singularity**This paper is dedicated to Professors Akhtam Abdurakhmanovich Dzhalilov and Konstantin Mikhailovich Khanin on the occasion of their 60th birthdays.

Abstract

Let f be an orientation-preserving circle diffeomorphism with an irrational rotation number and with a break point that is, its derivative has a jump discontinuity at this point. Suppose that satisfies a certain Zygmund condition dependent on a parameter We prove that the renormalizations of f are approximated by Möbius transformations in C1-topology if and in C2-topology if Moreover, it is shown that, in case of the coefficients of Möbius transformations get asymptotically linearly dependent. Further, consider two circle diffeomorphisms with a break point, with the same size of the break and satisfying Zygmund condition with We prove that, under a certain technical condition on rotation numbers, the renormalizations of these diffeomorphisms approach each other in C2-topology.