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Julia sets of uniformly quasiregular mappings are uniformly perfect


It is well known that the Julia set J(f) of a rational map f: is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this paper we prove that an analogous result is true in higher dimensions; namely, that the Julia set J(f) of a uniformly quasiregular mapping f: nn is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.

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[6] L. Carleson and T. Gamelin Complex Dynamics (Springer-Verlag, 1993).

[18] S. Rickman Quasiregular mappings. (Springer-Verlag1993).

[23] M. Vuorinen Conformal Geometry and Quasiregular Mappings. (Springer-Verlag, 1988).

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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