Abstract
We consider generalized interval exchange transformations (GIETs) of d ≥ 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 C2 distance to the subspace of IETs, there exists an exponent 0 < α < 1 such that h is C1+α . 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 C1 to C1+α rigidity): for almost every irreducible IET T0 with d = 4 or d = 5 , for any GIET which is topologically conjugate to T0 via a homeomorphism h and has vanishing boundary, the topological conjugacy h is actually a C1+α diffeomorphism, i.e. a diffeomorphism h with derivative Dh which is α -Hölder continuous.