We characterize when an inverse limit of a set-valued function is a Cantor set. Given a set-valued function $F:X\to 2^X$, we define the set $D(F)=\bigcap _{n=1}^{\mathrm{\infty }}{F}^n(X)$. It is known that $\underleftarrow{\lim }F=\underleftarrow{\lim }{F}_{D(F)}$, so we only need to consider the $F_{D(F)}$. When $D(F)$ is finite, $\underleftarrow{\lim }F$ is a shift of finite type, so we focus on the case where $D(F)$ is infinite, and we give a characterization for $\underleftarrow{\lim }F$ to be a Cantor set for this context. We go on to examine the entropy of a set-valued function on a countable domain and how that relates to the inverse limit being a Cantor set.

This includes joint work with L. Alvin and S. Greenwood.