Rauzy Induction, raised by Rauzy and Veech, has been served as an important technique to study interval exchange transformations. In this thesis, based on the classical theory ([RAU][VEE1]), we utilized Rauzy Induction, introduced new ideas, constructed new techniques, and achieved new generic results on interval exchange transformations.
In Chapter 3, we reached this final theorem (Corollary 3.3.3) for any irreducible m-permutation, the measure theoretically generic interval exchange transformation T satisfies: the topologically generic transformation in the commutant of T is rank one. This is a corollary of Theorem 3.0.1, which establishes that for any irreducible m-permutation, the measure theoretically generic interval exchange transformation T has all of its nonzero powers rank one. The connection between these two results is made by applying a theorem of J. King [KIN1]: a rank one transformation generates a dense subgroup of its commutant. When all powers are rank one and rigid, the commutant contains a dense Gdelta-set of rank one transformations.
In Chapter 4, we proved that measure theoretically typically all the symmetric (with permutation (3 2 1)) three interval exchange transformations are whirly. Thus by [GLA, TSI, WEI], near Borel action of the commutant of the corresponding in terval exchange transformation admits no nontrivial spatial factors. We relied on a relation of the return times associated with the Rauzy-Veech induction, and also the idea to study the multiples of the time when the iteration of T is close to identity map. Based on the whirly property of some intervals, we complete the work by a density point argument. Whether the corresponding statement of general cases (m > 3) is true is still an open problem.
In Chapter 5, further discussion on interval exchange transformations were described. A class of pseudo-Anosov inverval exchange transformations were verified to be weak mixing. They are concrete weak mixing examples. Then we also introduced the computational results with Matlab about the Rauzy classes.
In Chapter 6, we studied del Junco-Rudolph's example, proved that all powers of it are rank one. We also reached a proposition about the cylinder sets, which is a positive step toward determining whether the map is whirly or not.