A Symbolic coding of the Morse boundary

Let $X$ be a proper geodesic metric space. We give a new construction of the Morse Boundary that realizes its points as equivalence classes of functions on $X$ which behave similar to the “distance to a point” function. When $G=\langle S \rangle $isafinitelygeneratedgroupand$X=Cay(G,S)$, we use this construction to give a symbolic presentation of the Morse boundary as a space of “derivatives” on $Cay(G,S)\mathrm{.}$ The collection of such derivatives naturally embeds in the shift space ${\mathcal{A}}^G$ for some finite set $\mathcal{A}\mathrm{.}$