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A standard interval exchange map is a one-to-one map of the interval which islocally a translation except at finitely many singularities. We define for suchmaps, in terms of the Rauzy-Veech continuous fraction algorithm, a diophantinearithmetical condition called restricted Roth type which is almost surelysatisfied in parameter space. Let $T_0$ be a standard interval exchange map ofrestricted Roth type, and let $r$ be an integer $\geq 2$. We prove that,amongst $C^{r+3}$ deformations of $T_0$ which are $C^{r+3}$ tangent to $T_0$ atthe singularities, those which are conjugated to $T_0$ by a $C^r$diffeomorphism close to the identity form a $C^1$ submanifold of codimension$(g-1)(2r+1) +s$. Here, $g$ is the genus and $s$ is the number of marked pointsof the translation surface obtained by suspension of $T_0$. Both $g$ and $s$can be computed from the combinatorics of $T_0$.

Linearization of generalized interval exchange maps