The Triple Lattice PETs

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 ${\Lambda }_{\varphi }$ with respect to the parameter $\varphi =\frac{-1+\sqrt{5}}{2}$. By renormalization, we show that ${\Lambda }_{\varphi }$ is the limit of embedded polygons in ${\mathbb{R}}^2$ and its Hausdorff dimension satisfies the inequality $1<{\dim }_H({\Lambda }_{\varphi })=\log (\sqrt{2}-1)/\log (\varphi )<2$ so that ${\Lambda }_{\varphi }$ has Lebesgue measure zero.

References in corpus (1)

• Truchet Tilings and Renormalization