Rigidity for piecewise smooth circle homeomorphisms and certain GIETs

In this article, we prove a rigidity property for a class of generalized interval exchange transformations (GIETs), which contains the class of piecewise smooth circle homeomorphisms. More precisely, we show that if two piecewise C3 GIETs f and g with zero mean non-linearity are topologically conjugated, boundary-equivalent, have the same typical combinatorial rotation number and their renormalizations approach in an appropriate way the set of affine interval exchange transformations, then, their respective renormalizations converge to each other exponentially and the conjugating map is of class C1. In addition, if f and g are GIETs with rotation-type combinatorial data (i.e., they naturally define a piecewise smooth circle homeomorphism), have the same typical combinatorial rotation number, and they are break-equivalent as piecewise smooth circle homeomorphisms, then they are C1-conjugated as circle maps. This work provides the first rigidity results for GIETs not smoothly conjugated to IETs, and generalizes a previous result of K. Cunha and D. Smania [5], concerning an exceptional class of circle maps, in the setting of piecewise C3 circle homeomorphisms.