Notes on conjugacies and renormalisations of circle diffeomorphisms with breaks

Abstract

Let f be an orientation-preserving circle diffeomorphism with irrational “rotation number” of bounded type and finite number of break points, that is, the derivative f ′ has discontinuities of first kind at these points. Suppose f ′ satisfies a certain Zygmund condition which be dependent on parameter γ > 0 on each continuity intervals. We prove that the Rauzy-Veech renormalisations of f are approximated by Mobius transformations in C1 -norm if γ ∈(0,1] and in C2 -norm if γ ∈(1,∞) . In particular, we show that if f has zero mean nonlinearity, renormalisation of such maps approximated by piecewise affine interval exchange maps. Further, we consider two circle homeomorphisms with the same irrational “rotation number” of bounded type and finite number of break points. We prove that if they are not break equivalent then the conjugating map between these two maps is singular.