The Triple Lattice PETs

TL;DR

This paper introduces a new family of polytope exchange transformations called triple lattice PETs, analyzes their renormalization properties, and studies the fractal limit set's dimension and measure.

Contribution

It presents the first detailed analysis of the triple lattice PETs, including their renormalization scheme and the fractal properties of their limit sets.

Findings

Existence of a renormalization scheme for triple lattice PETs.

The limit set $\Lambda_\phi$ has Hausdorff dimension between 1 and 2.

The limit set $\Lambda_\phi$ has Lebesgue measure zero.

Abstract

Polytope exchange transformations (PETs) are higher dimensional generalizations of interval exchange transformations (IETs) which have been well-studied for more than 40 years. A general method of constructing PETs based on multigraphs was described by R. Schwartz in 2013. In this paper, we describe a one-parameter family of multigraph PETs called the triple lattice PETs. We show that there exists a renormalization scheme of the triple lattice PETs in the interval $(0,1)$. We analyze the the limit set $\Lambda_\phi$ with respect to the parameter $\phi=\frac{-1+\sqrt 5}{2}$. By renormalization, we show that $\Lambda_\phi$ is the limit of embedded polygons in $\mathbb R^2$ and its Hausdorff dimension satisfies the inequality $1< \dim_H(\Lambda_\phi) = \log(\sqrt 2-1)/\log(\phi)<2$ so that $\Lambda_\phi$ has Lebesgue measure zero.