Regularity of Conjugacies of Linearizable Generalized Interval Exchange Transformations

Abstract

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.