Dimensions of orbital sets in complex dynamics

Let $E$ be a non-empty compact subset of the Riemann sphere and $T$ be a rational map of degree at least two. We study the associated \emph{orbital set}, that is, the backwards orbit of $E$ under $T$, and study the relationship between the upper box dimension of the orbital set and the upper box dimensions of the Julia set of $T$ and the initial set $E$. Our results extend previous work on inhomogeneous iterated function systems to the setting of complex dynamical systems.