Area and Hausdorff dimension of the set of accessible points of the Julia sets of λe^z and λ sin(z)

• Autorzy: Bogusława Karpińska
• Języki publikacji: EN

Abstrakty

EN

The Julia set $J_λ$ of the exponential function $ E_λ:z → λ e^z$ for λ ∈ (0,1/e)$ is known to be a union of curves ("hairs") whose endpoints $C_λ$ are the only accessible points from the basin of attraction. We show that for λ as above the Hausdorff dimension of $C_λ$ is equal to 2 and we give estimates for the Hausdorff dimension of the subset of $C_λ$ related to a finite number of symbols. We also consider the set of endpoints for the sine family $F_λ:z → (1/(2i))λ (e^{iz}-e^{-iz})$ for λ ∈ (0,1) and prove that it has positive Lebesgue measure.

Kategorie tematyczne

• 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
• 28A80: Fractals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1999
• Tom: 159
• Numer: 3
• Strony: 269-287

Daty

wydano

1999

otrzymano

1998-05-17

poprawiono

1999-01-05

Twórcy

autor

Bogusława Karpińska

• Institute of Mathematics, Technical University of Warsaw, Pl. Politechniki 1, 00-661 Warszawa, Poland

Bibliografia

• [1] I. N. Baker, Fixpoints and iterates of entire functions, Math. Z. 71 (1959), 146-153.
• [2] R. Bowen, Hausdorff dimension of quasi-circles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11-26.
• [3] R. L. Devaney and L. Goldberg, Uniformization of attracting basins for exponential maps, Duke Math. J. 2 (1987), 253-266.
• [4] R. Devaney and M. Krych, Dynamics of exp(z), Ergodic Theory Dynam. Systems 4 (1984), 35-52.
• [5] R. L. Devaney and F. Tangerman, Dynamics of entire functions near the essential singularity, ibid. 6 (1986), 489-503.
• [6] P. L. Duren, Univalent Functions, Springer, New York, 1983.
• [7] A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), 989-1020.
• [8] N. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. 51 (1985), 369-384.
• [9] J. Mayer, An explosion point for the set of endpoints of the Julia set of λexp(z), Ergodic Theory Dynam. Systems 10 (1990), 177-183.
• [10] C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), 329-342.
• [11] F. Przytycki and M. Urbański, Conformal repellers and ergodic theory, in preparation.
• [12] D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), 99-107.