Cantor's onion: a uniformly quasiregular map whose Julia set is a Cantor set of spheres

arXiv:2611.08773v1 Announce Type: new Abstract: For integers $d,m\ge2$ with $1/d+1/m<1$, we construct a uniformly quasiregular map of $\mathbb{S}^3$ of degree $d^2+m^2$ whose Julia set is a Cantor set of concentric round spheres. To our knowledge, this is the first example of a uniformly quasiregular map in dimension three with infinitely many Fatou components. The map is a three-dimensional analogue of McMullen's maps $z^d+\lambda/z^m$, built from two Zorich-type power maps glued across a single shell by a slit construction. Its Julia set has Hausdorff dimension $2+s$, where $d^{-s}+m^{-s}=1$, and the restriction of the map to its Julia set is a skew product over the shift on two symbols and has topological entropy $\log(d^2+m^2)$. We also prove that uniformly quasiregular maps that expand uniformly near their Julia set, possibly with branching there, have no wandering domains and only finitely many periodic Fatou components, all of them attracting basins. The map constructed here belongs to this class.

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