A Numeration System for Fibonacci-like Wang Shifts

Motivated by the study of Fibonacci-like Wang shifts, we define a numeration system for $\mathbb{Z}$ and $\mathbb{Z}^2$ based on the binary alphabet $\{0,1\}$. We introduce a set of 16 Wang tiles that admits a valid tiling of the plane described by a deterministic finite automaton taking as input the representation of a position $(m,n)\in\mathbb{Z}^2$ and outputting a Wang tile.