# Characterization of minimal sequences associated with self-similar interval exchange maps

## TL;DR

This paper characterizes minimal sequences linked to self-similar interval exchange maps, providing conditions for wandering intervals and exploring the role of eigenvalues in their structure.

## Contribution

It introduces a characterization of minimal sequences under the unique representation property for potentials from non-real eigenvalues, and analyzes conditions for wandering intervals.

## Key findings

- Characterization of minimal sequences for non-real eigenvalues
- Conditions determining the existence of wandering intervals
- Relation between eigenvalues and affine extensions

## Abstract

The construction of affine interval exchange maps with wandering intervals that are semi-conjugate with a given self-similar interval exchange map is strongly related with the existence of the so called minimal sequences associated with local potentials, which are certain elements of the substitution subshift arising from the given interval exchange map. In this article, under the condition called unique representation property, we characterize such minimal sequences for potentials coming from non-real eigenvalues of the substitution matrix. We also give conditions on the slopes of the affine extensions of a self-similar interval exchange map that determine whether it exhibits a wandering interval or not.