Regularity of conjugacies of linearizable generalized interval exchange transformations

Abstract

We consider generalized interval exchange transformations (GIETs) of d intervals ($d\geq 2$) 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 renormalisation 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 renormalisation converges exponentially fast in a $C^2$ distance to the subspace of IETs, there exists an exponent $0 < \alpha < 1$ such that h is $C^{1+{\alpha}}$. 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 previous work by the same authors (from $C^1$ to $C^{1+{\alpha}}$ 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 $C^{1+{\alpha}}$ diffeomorphism, i.e. a diffeomorphism h with derivative Dh which is $\alpha$-H\"older continuous.