Persistence of Wandering Intervals in Self-Similar Affine Interval Exchange Transformations

TL;DR

This paper proves that self-similar affine interval exchange transformations often have wandering intervals, extending the existence of Denjoy counterexamples under broad algebraic conditions.

Contribution

It demonstrates the frequent occurrence of wandering intervals in self-similar affine interval exchanges with certain algebraic properties, generalizing previous results.

Findings

Existence of wandering intervals in broad classes of affine interval exchanges

Semi-conjugation between affine and self-similar interval exchanges

Generalization of Denjoy counterexamples in this context

Abstract

In this article we prove that given a self-similar interval exchange transformation T, whose associated matrix verifies a quite general algebraic condition, there exists an affine interval exchange transformation with wandering intervals that is semi-conjugated to it. That is, in this context the existence of Denjoy counterexamples occurs very often, generalizing the result of M. Cobo in [C].