Regular subdivision in $\mathbf{Z}[\frac{1+\sqrt{5}}{2}]$

Abstract

In the ring $\mathbf{Z}[\frac{1+\sqrt{5}}{2}]$, there is a natural subdivision technique analogous to regular subdivision in rational algebraic rings like $\mathbf{Z}[\frac{1}{2}]$. The properties of this subdivision process are developed using the matrix associated to the Fibonacci substitution tiling. These properties are applied to prove some finiteness properties for a discrete group of piecewise-linear homeomorphisms.