Group Actions via Interval Exchange Transformations

Christopher Novak

doi:https://doi.org/10.21985/N2014C

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Group Actions via Interval Exchange Transformations

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This dissertation addresses the structure of the group of interval exchange transformations. The two primary topics considered are: a) the classification of interval exchange actions for certain groups; and b) properties of the interval exchange group which are reflected in the dynamics of interval exchange maps. In Chapter 3 a classification is given for the continuous interval exchange actions of the group of real numbers. The interval exchange group is endowed with a natural topological group structure, with respect to which any continuous one-parameter action must factor through a toral action generated by disjointly supported rotation groups. In Chapter 4 the asymptotics are classified for the number of discontinuities exhibited by the iterates of an interval exchange. It is seen that the number of discontinuities is either bounded or exhibits linear growth; no intermediate growth rates are possible. It is further shown that any map with bounded discontinuity growth is essentially an element of a toral rotation action. In Chapter 5, the dichotomy in discontinuity growth is used to prove that no finitely generated subgroup of the interval exchange group contains a distortion element. Consequently, no group having a distortion element can act faithfully via interval exchanges. Chapter 6 contains a complete classification of centralizers in the interval exchange group. The structure of an element's centralizer is controlled by three types of dynamic behavior: the existence of periodic points, minimal sets with bounded discontinuity growth, and minimal sets with linear discontinuity growth. The classification of centralizers is used in Chapter 7 to compute the automorphism group of the interval exchange group. Since automorphisms preserve the group structure of centralizers, they also preserve the associated dynamics. Consequently, an automorphism must be induced under conjugation by a map on the circle. It is then seen that the group of outer automorphisms is generated by an order-two orientation reversing map on the circle.

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