Orbit structure of interval exchange transformations with flip

TL;DR

This paper establishes bounds on the invariant components of interval exchange transformations with flips, demonstrating typical stability properties and classifying possible configurations for these components.

Contribution

It provides a sharp bound on the sum of periodic and minimal components and characterizes all possible configurations for typical transformations.

Findings

The number of periodic plus twice the minimal components is at most n.

Almost all transformations have stable periodic and robust minimal components.

Complete classification of possible component configurations for typical transformations.

Abstract

A sharp bound on the number of invariant components of an interval exchange transformation is provided. More precisely, it is proved that the number of periodic components n_per and the number of minimal components n_min of an interval exchange transformation of n intervals satisfy n_per+2 n_min\le n. Besides, it is shown that almost all interval exchange transformations are typical, that is, have all the periodic components stable and all the minimal components robust (i.e. persistent under almost all small perturbations). Finally, we find all the possible values for the integer vector (n_per, n_min) for all typical interval exchange transformation of n intervals.