Square Turning Maps and their Compactifications

TL;DR

This paper introduces infinite rectangle exchange transformations based on square turning, explores their higher-dimensional compactifications as polytope exchanges, and connects these to lattice structures and hyperbolic groups.

Contribution

It presents a new class of infinite transformations and their compactifications, revealing their structure as polytope exchanges linked to Euclidean lattices and hyperbolic groups.

Findings

Higher-dimensional polytope exchange systems are constructed.

Connections to Euclidean lattices and hyperbolic groups are established.

A family of systems parametrized by dimension is identified.

Abstract

In this paper we introduce some infinite rectangle exchange transformations which are based on the simultaneous turning of the squares within a sequence of square grids. We will show that such noncompact systems have higher dimensional dynamical compactifications. In good cases, these compactifications are polytope exchange transformations based on pairs of Euclidean lattices. In each dimension $8m+4$ there is a $4m+2$ dimensional family of them. Here $m=0,1,2,...$ The case $m=0$, which we studied in depth in an earlier paper, has close connections to the $E_4$ Weyl group and the $(2,4,\infty)$ hyperbolic triangle group.