In this work we study the renormalization operator acting on piecewise smooth homeomorphisms on the circle, that turns out to be essentially the study of Rauzy-Veech renormalizations of generalized interval exchanges maps with genus one. In particular we show that renormalizations of such maps with zero mean nonlinearity and satisfying certain smoothness and combinatorial assumptions converges to the set of piecewise affine interval exchange maps.

In this work, we find sufficient conditions for two piecewise C 2+ν homeomorphism f and g of the circle to be C 1 conjugate. Besides the restrictions on the combinatorics of the maps (we assume that maps have bounded "rotation number" ), and necessary conditions on the one-side derivatives of points where f and g are not differentiable, we also assume zero mean nonlinearity for f and g.The proof is based on the study of Rauzy-Veech renormalization of genus one generalized interval exchange maps with certain restrictions on its combinatorics.

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