This research demonstrates a quasiregular mapping's unique properties in R³, highlighting its quasi-Fatou set and Julia set implications.
We construct a quasiregular mapping in R 3 R^3 that is the first to illustrate several important dynamical properties: the quasi-Fatou set contains wandering components; these quasi-Fatou components are bounded and hollow; and the Julia set has components that are genuine round spheres. The key tool in this construction is a new quasiregular interpolation in round rings in R 3 R^3 between power mappings of differing degrees on the boundary components. We also exhibit the flexibility of constructions based on these interpolations by showing that we may obtain quasiregular mappings which grow as quickly, or as slowly, as desired.
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Burkart et al. (2025) studied this question.
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