Julia sets of uniformly quasiregular mappings are uniformly perfect
arXiv:1012.1378 · doi:10.1017/S0305004111000478
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.
10 pages