Renormalization of circle diffeomorphisms with a break-type singularity

TL;DR

This paper studies the renormalization behavior of circle diffeomorphisms with a break point, showing approximation by Möbius transformations in different norms depending on a Zygmund condition parameter.

Contribution

It establishes the approximation of renormalizations by Möbius transformations in various norms based on the Zygmund condition parameter, extending understanding of singular circle diffeomorphisms.

Findings

Renormalizations are approximated by Möbius transformations in C^1 norm for γ in (0,1].

Renormalizations are approximated by Möbius transformations in C^2 norm for γ in (1,+∞).

Coefficients of Möbius transformations become asymptotically linearly dependent.

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.