Skip to main content
    • Aa
    • Aa

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.

Hide All
[1]Baker I. N., Multiply connected domains of normality in iteration theory. Math. Z. 81 (1963), 206214.
[2]Beardon A. F. and Pommerenke Ch. The Poincaré metric of plane domains. J. London Math. Soc. (2) 18 (1978), 475483.
[3]Bergweiler W. Iteration of quasiregular mappings. Comput. Methods Funct. Theory. 10 (2010), 455481.
[4]Bergweiler W., Fletcher A., Langley J. K. and Meyer J. The escaping set of a quasiregular mapping, Proc. Amer. Math. Soc. 137, no. 2 (2009), 641651.
[5]Bergweiler W. and Zheng J. H. On the uniform perfectness of the boundary of multiply connected wandering domains, arxiv preprint
[6]Carleson L. and Gamelin T. Complex Dynamics (Springer-Verlag, 1993).
[7]Eremenko A. Julia sets are uniformly perfect, Preprint (1992).
[8]Fletcher A. and Nicks D. A. Quasiregular dynamics on the n-sphere. Ergodic Theory and Dynam. Syst. 31 (2011), 2331.
[9]Hinkkanen A. Julia sets of rational functions are uniformly perfect. Math. Proc. Camb. Phil. Soc. 113 (1993), 543559.
[10]Hinkkanen A., Martin G. and Mayer V. Local dynamics of uniformly quasiregular mappings. Math. Scand. 95, no. 1 (2004), 80100.
[11]Iwaniec T. and Martin G. Quasiregular semigroups. Ann. Acad. Sci. Fenn. 21, no. 2 (1996), 241254.
[12]Järvi P. and Vuorinen M. Uniformly perfect sets and quasiregular mappings. J. London Math. Soc., II. Ser. 54, No. 3 (1996), 515529.
[13]Mane R. and Da Rocha L. Julia sets are uniformly perfect. Proc. Amer. Math. Soc. 116 (1992), 251257.
[14]Mayer V. Uniformly quasiregular mappings of Lattès type. Conform. Geom. Dyn. 1 (1997), 104111.
[15]Mayer V. Quasiregular analogues of critically finite rational functions with parabolic orbifold. J. Anal. Math. 75 (1998), 105119.
[16]Miniowitz R. Normal families of quasimeromorphic mappings. Proc. Amer Math. Soc. 84 (1982), 3543.
[17]Pommerenke Ch. Uniformly Perfect Sets and the Poincaré Metric. Arch. Math. 32 (1979), 192199.
[18]Rickman S. Quasiregular mappings. (Springer-Verlag 1993).
[19]Siebert H. Fixpunkte und normale Familien quasiregulärer Abbildungen. Dissertation (CAU Kiel, 2004).
[20]Sugawa T. An explicit bound for uniform perfectness of the Julia sets of rational maps. Math. Z. 238, No. 2 (2001), 317333.
[21]Sugawa T. Uniformly perfect sets: analytic and geometric aspects. Sugaku Expositions 16, no. 2 (2003), 225242.
[22]Tukia P. and Väisälä J. Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 5, no. 1 (1980), 97114.
[23]Vuorinen M. Conformal Geometry and Quasiregular Mappings. (Springer-Verlag, 1988).
[24]Zheng J. H. On uniformly perfect boundary of stable domains in iteration of meromorphic functions II. Math. Proc. Camb. Phil. Soc. 132, no. 3 (2002), 531544.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 4 *
Loading metrics...

Abstract views

Total abstract views: 28 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st October 2017. This data will be updated every 24 hours.