Finite Graphs and Inverse Limits with Set-Valued Functions on Intervals - Nall, Van; Vidal-Escobar, Ivon (Topology Proceedings, 2020)
If the inverse limit of upper semi-continuous set-valued functions from $[0, 1]$ to the closed subsets of $[0, 1]$ is a finite graph $G$, then for some $N$ and all $n ≥ N$, the projection onto the first $n$ coordinates of that inverse limit is a projection onto a finite graph that is homeomorphic to $G$.