Cantor's onion: a uniformly quasiregular map whose Julia set is a Cantor set of spheres
Abstract
For integers d,m≥2 with 1/d+1/m<1, we construct a uniformly quasiregular map of 𝕊3 of degree d2+m2 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 zd+λ/zm, 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(d2+m2). 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.
Related
- Examples of Uniformly Quasiregular Mappings with Fatou Set Having Infinitely Many Components
- Schwarz-Pick and Denjoy-Wolff for uniformly quasiregular maps
- The Great Picard theorem for quasiregular values
- Equipotentials and folds for quasiregular Mandelbrot sets
- Averaged derivatives and reverse inequalities in quasiregular Bergman spaces
BibTeX
@misc{fletcher2026cantors,
title = {{Cantor's onion: a uniformly quasiregular map whose Julia set is a Cantor set of spheres}},
author = {Alastair N. Fletcher and Gaven J. Martin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08773},
eprint = {2611.08773},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08773},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cantor's onion: a uniformly quasiregular map whose Julia set is a Cantor set of spheres
AU - Alastair N. Fletcher
AU - Gaven J. Martin
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08773
UR - https://arxiv.org/abs/2611.08773
AB - For integers d,m≥2 with 1/d+1/m<1, we construct a uniformly quasiregular map of 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+λ/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.
ER -