We show that if two piecewise C2 Generalized Interval Exchange
Transformations
f and g of generic rotation number with mean-nonlinearity 0 are
homeomorphic, boundary
equivalent and their renormalizations approach affine interval
exchange transformations, then their respective renormalizations
converge to each other and the conjugating map is C1. 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 C1-conjugated as circle
diffeomorphisms.
This is a joint work with Frank Trujillo.

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