Boshernitzan, Michael2019-05-172019-05-172018-052018-04-18May 2018Bernazzani, Daniel. "Centralizers and Conjugacy Classes in the Group of Interval Exchange Transformations." (2018) Diss., Rice University. <a href="https://hdl.handle.net/1911/105759">https://hdl.handle.net/1911/105759</a>.https://hdl.handle.net/1911/105759We study the group G of interval exchange transformations. Firstly, we will classify, up to conjugation in G, those interval exchange transformations which arise as towers over rotations. Secondly, expanding on previous work of Novak and Li, we will classify, up to conjugation in G, minimal interval exchange transformations which exhibit bounded discontinuity growth. As an application of this classification, we will prove that no infinite order interval exchange transformation could be conjugate in G to one of its proper powers. Thirdly, we will develop broadly applicable, generic conditions which ensure that an interval exchange transformation has no roots in G and that its centralizer is torsion-free. By combining these results with a result of Novak, we will show that a typical interval exchange transformation does not commute with any interval exchange transformations other than its powers. Finally, we will completely describe the possible centralizers of a minimal three-interval exchange transformation.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Interval Exchange TransformationsThesis2019-05-17