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Entire functions with Julia sets of zero measure

  • Gwyneth M. Stallard (a1)

We extend results of McMullen about the dynamics of entire functions for which the orbits of the critical values stay away from the Julia set. In particular we show that such functions are expanding on their Julia sets which have self-similarity properties. Under suitable further conditions the Julia sets have plane measure zero.

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[5] P. Fatou . Sur l'itération des fonctions transcendantes entières. Acta Math. 47 (1926), 337370.

[7] C. McMullen . Area and Hausdorff dimension of Julia sets of entire functions. Trans. Amer. Math. Soc. 300 (1987), 329342.

[8] D. Sullivan . Conformal dynamical systems. In Geometric Dynamics, Lecture Notes in Math. vol. 1007 (Springer-Verlag, 1983), pp. 725752.

[9] D. Sullivan . Quasiconformal homeomorphisms and dynamics 1. Ann. of Math. (2) 122 (1985), 401418.

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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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