The Tetrahedral Twists

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 $[\frac{1}{2},1)$.