Piecewise linear periodic maps of the plane with integer coefficients

TL;DR

This paper investigates periodic piecewise linear maps of the plane with integer coefficients, revealing restrictions on possible periods for maps with two pieces and demonstrating the construction of maps with arbitrary periods using binary trees.

Contribution

It characterizes the period restrictions for two-piece maps and introduces a method to construct maps with any period using binary trees and admissible sequences.

Findings

Two-piece maps have limited possible periods.

Maps with more than two pieces can have any period.

Construction method using binary trees and admissible sequences.

Abstract

We study periodic, piecewise linear maps on the plane starting with the Mort Brown's map. We show that if the number of pieces is two, there is only a short list of possible periods (this fact can be seen as the crystallographic restriction for this class of maps). Otherwise, without the restriction on the number of pieces, a map can have any period. We show how to construct such maps using binary trees and so called admissible sequences.