| Summary: | 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.
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