Wandering intervals in affine extensions of self-similar interval exchange maps: the cubic Arnoux-Yoccoz map

TL;DR

This paper establishes conditions under which certain self-similar interval exchange maps with complex eigenvalues admit affine extensions with wandering intervals, exemplified by the cubic Arnoux-Yoccoz map.

Contribution

It provides new algebraic criteria for the existence of wandering intervals in affine extensions of self-similar interval exchange maps, including the cubic Arnoux-Yoccoz map.

Findings

Conditions for wandering intervals are based on eigenvalue properties.

The cubic Arnoux-Yoccoz map satisfies these conditions.

Affine maps with wandering intervals are constructed under these criteria.

Abstract

In this article we provide sufficient conditions on a self-similar interval exchange map, whose renormalization matrix has complex eigenvalues of modulus greater than one, for the existence of affine interval exchange maps with wandering intervals and semi-conjugate with it. These conditions are based on the algebraic properties of the complex eigenvalues and the complex fractals built from the natural substitution emerging from self-similarity. We show that the cubic Arnoux-Yoccoz interval exchange map satisfies these conditions.