Geometric representation of interval exchange maps over algebraic number fields

TL;DR

This paper investigates the geometric and arithmetic properties of interval exchange maps over algebraic number fields, focusing on renormalizability, the drift vector, and the finite decomposition property.

Contribution

It introduces a characterization of renormalizability for these maps and explores the relationship with the drift vector, providing new examples and properties.

Findings

Identified conditions for renormalizability in algebraic number field interval exchanges.

Discovered examples with zero and non-zero drift vectors.

Provided evidence related to the finite decomposition property.

Abstract

We consider the restriction of interval exchange transformations to algebraic number fields, which leads to maps on lattices. We characterize renormalizability arithmetically, and study its relationships with a geometrical quantity that we call the drift vector. We exhibit some examples of renormalizable interval exchange maps with zero and non-zero drift vector, and carry out some investigations of their properties. In particular, we look for evidence of the finite decomposition property: each lattice is the union of finitely many orbits.