Regularity of Conjugacies of Linearizable Generalized Interval Exchange Transformations

We consider generalized interval exchange transformations (GIETs) of $$ intervals which are linearizable, i.e. differentiably conjugated to standard interval exchange maps (IETs) via a diffeomorphism h of [0, 1] and study the regularity of the conjugacy h. Using a renormalization operator obtained accelerating Rauzy-Veech induction, we show that, under a full measure condition on the IET obtained by linearization, if the orbit of the GIET under renormalization converges exponentially fast in a $$ distance to the subspace of IETs, there exists an exponent $$ such that h is $$ . 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 $$ to $$ rigidity): for almost every irreducible IET $$ with $$ or $$ , for any GIET which is topologically conjugate to $$ via a homeomorphism h and has vanishing boundary, the topological conjugacy h is actually a $$ diffeomorphism, i.e. a diffeomorphism h with derivative Dh which is $$ -Hölder continuous.