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 for the triple lattice PETs on the parameter space (0, 1). We analyze the limit set Λ_φ with respect to the parameter $\varphi =\frac{-1+\sqrt{5}}{2}$. By renormalization, we show that Λ_φ is the nested intersection of a countable sequence of finite unions of isosceles trapezoids. The Hausdorff dimension of Λ_φ satisfies the inequality $1<{dim}_H\left({\Lambda }_{\varphi }\right)=log(\sqrt{2}-1)/log\left(\varphi \right)<2$.

KEYWORDS:

• polytope exchange transformations
• piecewise isometries
• renormalization

2000 AMS SUBJECT CLASSIFICATION:

• 37E99

Acknowledgments

The author would like to thank her advisor Professor Richard Schwartz for his constant support, encouragement, and guidance throughout this project. The author would also like to thank Patrick Hooper and Yuhan Wang for helpful suggestions at different steps of the paper.