Local structure of the karyon tilings

Karyon tilings $\mathcal{T}$ of the torus $\mathbb{T}^d$ of arbitrary dimension $d$ are considered. The prototype of such tilings is one-dimensional Fibonacci tilings and their two-dimensional analog the Rauzy tiling. Tilings $\mathcal{T}$ are important for applications to multidimensional continued fractions. In this article, we examine the local properties of karyon tilings $\mathcal{T}$.