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Transactions of the American Mathematical Society

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Area and Hausdorff dimension of Julia sets of entire functions


Author: Curt McMullen
Journal: Trans. Amer. Math. Soc. 300 (1987), 329-342
MSC: Primary 30D05; Secondary 58F08, 58F20
DOI: https://doi.org/10.1090/S0002-9947-1987-0871679-3
MathSciNet review: 871679
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Abstract: We show the Julia set of $ \lambda \sin (z)$ has positive area and the action of $ \lambda \sin (z)$ on its Julia set is not ergodic; the Julia set of $ \lambda \exp (z)$ has Hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero.


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  • [A] L. Ahlfors, Conformal invariants, McGraw-Hill, New York, 1973. MR 0357743 (50:10211)
  • [B1] I. N. Baker, The domains of normality of an entire function, Ann. Acad. Fenn. Ser. AI 1 (1975), 277-283. MR 0402044 (53:5867)
  • [B2] -, Some entire functions with multiply-connected wandering domains, Ergodic Theory Dynamical Systems 5 (1985), 163-169. MR 796748 (86i:30031)
  • [BR] I. N. Baker and P. J. Rippon, Iteration of exponential functions, Ann. Acad. Fenn. Ser. AI 9 (1984), 49-77. MR 752391 (86d:58065)
  • [DK] R. Devaney and M. Krych, Dynamics of $ \operatorname{Exp} (z)$, Ergodic Theory Dynamical Systems 4 (1984), 35-52. MR 758892 (86b:58069)
  • [D] A. Douady, Systèmes dynamiques holomorphes, Asterisque 105 (1983), 39-63; Ergodic Theory Dynamical Systems 4 (1984), 35-52. MR 728980 (85h:58090)
  • [EL] A. È. Eremenko and M. Yu. Lyubich, Iterates of entire functions, Soviet Math. Dokl. 30 (1984), 592-594.
  • [F] M. P. Fatou, Sur l'iteration des fonctions transcendentes entières, Acta Math. 47 (1926), 337-370. MR 1555220
  • [G] F. Gehring, Injectivity of local quasi-isometries, Comm. Math. Helv. 57 (1982), 202-220. MR 684113 (84b:30018)
  • [GGS] E. Ghys, L. Goldberg and D. Sullivan, On the measurable dynamics of $ z \to {e^z}$, Ergodic Theory Dynamical Systems 5 (1985), 329-335. MR 805833 (87e:58117)
  • [GK] L. Goldberg and L. Keen, A finiteness theorem for a dynamical class of entire functions, Ergodic Theory Dynamical Systems 6 (1986), 183-192. MR 857196 (88b:58126)
  • [J] F. John, On quasi-isometries. I, Comm. Pure Appl. Math. 21 (1968), 77-110. MR 0222666 (36:5716)
  • [KS] J. P. Kahane and R. Salem, Ensembles parfaits et séries trigonométriques, Paris, Hermann, 1963, p. 27. MR 0160065 (28:3279)
  • [M] M. Misiurewicz, On iterates of $ {e^z}$, Ergodic Theory Dynamical Systems 1 (1981), 103-106. MR 627790 (82i:58058)
  • [S] D. Sullivan, Conformal dynamical systems, Geometric Dynamics, Lecture Notes in Math., vol. 1007, Springer-Verlag, 1983, pp. 725-752. MR 730296 (85m:58112)

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DOI: https://doi.org/10.1090/S0002-9947-1987-0871679-3
Article copyright: © Copyright 1987 American Mathematical Society

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