We consider generalized interval exchange transformations (GIETs) of \(d\ge 2\) intervals which are linearizable, i.e. differentiably conjugated to standard interval exchange maps (IETs) via a diffeomorphism h of [0, 1] and study the regularity of the conjugacy h. Using a renormalization operator obtained accelerating Rauzy–Veech induction, we show that, under a full measure condition on the IET obtained by linearization, if the orbit of the GIET under renormalization converges exponentially fast in a \({\mathcal {C}}^2\) distance to the subspace of IETs, there exists an exponent \(0<\alpha <1\) such that h is \({\mathcal {C}}^{1+\alpha }\) . Combined with the results proved by the authors in [4], this implies in particular the following improvement of the rigidity result in genus two proved in [4] (from \({\mathcal {C}}^1\) to \({\mathcal {C}}^{1+\alpha }\) rigidity): for almost every irreducible IET \(T_0 \) with \(d=4\) or \(d=5\) , for any GIET which is topologically conjugate to \(T_0\) via a homeomorphism h and has vanishing boundary, the topological conjugacy h is actually a \({\mathcal {C}}^{1+\alpha }\) diffeomorphism, i.e. a diffeomorphism h with derivative Dh which is \(\alpha \) -Hölder continuous.
