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 $\Lambda_\phi$ 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.