  The primary aim of this paper is to give topological obstructions to Cantor
sets in $\mathbb{R}^3$ being Julia sets of uniformly quasiregular mappings. Our
main tool is the genus of a Cantor set. We give a new construction of a genus
$g$ Cantor set, the first for which the local genus is $g$ at every point, and
then show that this Cantor set can be realized as the Julia set of a uniformly
quasiregular mapping. These are the first such Cantor Julia sets constructed
for $g\geq 3$. We then turn to our dynamical applications and show that every
Cantor Julia set of a hyperbolic uniformly quasiregular map has a finite genus
$g$; that a given local genus in a Cantor Julia set must occur on a dense
subset of the Julia set; and that there do exist Cantor Julia sets where the
local genus is non-constant.
