  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.
