  The goal of this article is to show a rigidity property of conjugacies of
generalized interval exchange transformations (GIETs). More precisely, we show
that if two piecewise $C^3$ GIETs $f$ and $g$ of generic rotation number with
mean-non-linearity 0 are homeomorphic, boundary-equivalent and their
renormalizations approach in an appropriate way the set of affine interval
exchange transformations, then their respective renormalizations converge to
each other and the conjugating map is $C^1$. Moreover, if $f$ and $g$ are GIETs
with rotation type combinatorial data, generic rotation number and they are
break-equivalent as piecewise circle diffeomorphisms, they are actually
$C^1$-conjugated as circle diffeomorphisms. These results generalize the work
of K. Cunha and D. Smania \cite{cunha_rigidity_2014} in the case of piecewise
$C^3$ circle maps, where the authors prove an analogous result for GIETs with
rotation type combinatorial data and bounded rotation number.
