Julia sets of uniformly quasiregular mappings are uniformly perfect
Alastair Fletcher, Daniel A. Nicks
TL;DR
This paper proves that Julia sets of uniformly quasiregular mappings in higher dimensions are uniformly perfect, extending a known property from rational maps and implying they have positive Hausdorff dimension.
Contribution
It establishes that Julia sets of uniformly quasiregular mappings in R^n are uniformly perfect, generalizing a key property from complex dynamics to higher dimensions.
Findings
Julia sets are uniformly perfect in higher dimensions
Julia sets have positive Hausdorff dimension
Extension of complex dynamics properties to quasiregular mappings
Abstract
It is well-known that the Julia set J(f) of a rational map is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this article we prove that an analogous result is true in higher dimensions; namely, that the Julia set J(f) of a uniformly quasiregular mapping f in R^n is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.
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