Discontinuity growth of interval exchange maps

TL;DR

This paper investigates the growth of discontinuities in interval exchange maps, establishing a dichotomy that leads to insights about group actions, centralizers, and automorphisms within the interval exchange group.

Contribution

It introduces a dichotomy in the growth of discontinuities, enabling classification of centralizers and computation of the automorphism group of interval exchange groups.

Findings

Discontinuity growth is either linear or bounded.

Interval exchange groups lack distortion elements.

Automorphism group of the interval exchange group is computed.

Abstract

For an interval exchange map, the number of discontinuities of its iterates either exhibits linear growth or is bounded. This dichotomy is used to prove that the group of interval exchanges does not contain distortion elements, giving examples of groups that do not act faithfully via interval exchanges. As a further application of this dichotomy, a classification of centralizers in the group of interval exchanges is given. This classification of centralizers is used to compute the automorphism group of the interval exchange group.