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The main aim of this paper is to give topological obstructions to Cantor setsin $\mathbb{R}^3$ being Julia sets of uniformly quasiregular mappings. Our maintool 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 thenshow that this Cantor set can be realized as the Julia set of a uniformlyquasiregular mapping. These are the first such Cantor Julia sets constructedfor $g\geq 3$. We then turn to our dynamical applications and show that everyCantor 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 densesubset of the Julia set; and that there do exist Cantor Julia sets where thelocal genus is non-constant.

Genus $g$ Cantor sets and germane Julia sets