A Family of Minimal and Renormalizable Rectangle Exchange Maps

TL;DR

This paper constructs an infinite family of minimal, renormalizable rectangle exchange maps using cut-and-project sets related to Galois lattices and PV numbers, and develops a renormalization scheme for them.

Contribution

It introduces a new family of minimal, renormalizable DEMs based on cut-and-project sets associated with Galois lattices and PV numbers, with a renormalization framework.

Findings

Constructed an infinite family of DEMs associated with PV numbers.

Developed a renormalization scheme for these DEMs.

Demonstrated how to compose DEMs for multistage renormalization.

Abstract

A domain exchange map (DEM) is a dynamical system defined on a smooth Jordan domain which is a piecewise translation. We explain how to use cut-and-project sets to construct minimal DEMs. Specializing to the case in which the domain is a square and the cut-and-project set is associated to a Galois lattice, we construct an infinite family of DEMs in which each map is associated to a PV number. We develop a renormalization scheme for these DEMs. Certain DEMs in the family can be composed to create multistage, renormalizable DEMs.