Dynamical decomposition of generalized interval exchange transformations

We develop a renormalization scheme which extends the classical Rauzy-Veech induction used to study interval exchange tranformations (IETs) and allows to study generalized interval exchange transformations (GIETs) $T: [0,1) \to [0,1)$ with possibly more than one quasiminimal component (i.e. not infinite-complete, or, equivalently, not semi-conjugated to a minimal IET). The renormalization is defined for more general maps that we call interval exchange transformations with gaps (g-GIETs), namely partially defined GIETs which appear naturally as the first return map of $C^r$-flows on two-dimensional manifolds to any transversal segment. We exploit this renormalization scheme to find a decomposition of $[0,1)$ into finite unions of intervals which either contain no recurrent orbits, or contain only recurrent orbits which are closed, or contain a unique quasiminimal. This provides an alternative approach to the decomposition results for foliations and flows on surfaces by Levitt, Gutierrez and Gardiner from the $1980s$.