The Tetrahedral Twists

TL;DR

This paper introduces a new family of piecewise isometries called tetrahedral twists on a regular tetrahedron, studies their dynamics via renormalization, and provides partial results on the existence of a renormalization scheme.

Contribution

It defines tetrahedral twists, compares them to existing PETs, and proves renormalizability on specific subintervals of the parameter space.

Findings

Renormalization scheme exists on certain subintervals.

Tetrahedral twists are similar to Patrick Hooper's PETs.

Partial proof of renormalizability across the entire parameter interval.

Abstract

We introduce a family of piecewise isometries $f_s$ parametrized by $s \in [0,1)$ on the surface of a regular tetrahedron, which we call the tetrahedral twists. This family of maps is similar to the PETs constructed by Patrick Hooper. We study the dynamics of the tetrahedral twists through the notion of renormalization. By the assistance of computer, we conjecture that the renormalization scheme exists on the entire interval $[0,1)$. In this paper, we show that this system is renormalizable in the subintervals $[\frac{53}{128},\frac{29}{70}]$ and $[\frac{1}{2},1)$.