Renormalization of Polygon Exchange Maps arising from Corner Percolation

TL;DR

This paper studies a family of polygon exchange maps derived from Corner Percolation, revealing how their periodicity properties depend on parameters and introducing a renormalization approach applicable to tilings.

Contribution

It introduces a renormalization framework for polygon exchange maps from Corner Percolation, analyzing periodicity and measure properties across parameter space.

Findings

Almost all parameters yield predominantly periodic points.

A dense set of irrational parameters leads to non-periodic behavior.

The measure of non-periodic points can be made arbitrarily close to full measure.

Abstract

We describe a 2 parameter family of polygon exchange transformations parameterized by points in a square. Whenever the two parameters are irrational, the polygon exchange has periodic orbits of arbitrarily large period. We show that for almost all parameters, the polygon exchange map has the property that almost every point is periodic. However, there is a dense set of irrational parameters for which this fails. By choosing parameters carefully, the measure of non-periodic points can be made arbitrarily close to full measure. These results are powered by a notion of renormalization which holds in a more general setting. Namely, we consider a renormalization of tilings arising from the Corner Percolation Model.