Illinois Journal of Mathematics

Hausdorff dimension of radial and escaping points for transcendental meromorphic functions

Janina Kotus and Mariusz Urbański

Full-text: Open access

Enhanced PDF (178 KB)

Abstract

We consider a class of transcendental meromorphic functions $f:\mathbb{C}\mapsto\overline{\mathbb{C}}$ with infinitely many poles. Under some regularity assumption on the location of poles and the behavior of the function near the poles, we provide explicite lower bounds for the hyperbolic dimension (Hausdorff dimension of radial points) of the Julia set and upper bounds for the Hausdorff dimension of the set of escaping points in the Julia set. In particular, the Hausdorff dimension of the latter set is less than the Hausdorff dimension of the former set. Consequently, the Hausdorff dimension of the set of escaping points is less than 2, and the area of this set is equal to zero. The functions under consideration may have infinitely many singular values, and we do not even assume them to belong to the class $\mathcal{B}$. We only require the distance between the set of poles and the set of finite singular values to be positive.

Article information

Source
Illinois J. Math., Volume 52, Number 3 (2008), 1035-1044.

Dates
First available in Project Euclid: 1 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1254403730

Digital Object Identifier
doi:10.1215/ijm/1254403730

Mathematical Reviews number (MathSciNet)
MR2546023

Zentralblatt MATH identifier
1175.37049

Subjects
Primary: 37F35: Conformal densities and Hausdorff dimension
Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04] 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]

Citation

Kotus, Janina; Urbański, Mariusz. Hausdorff dimension of radial and escaping points for transcendental meromorphic functions. Illinois J. Math. 52 (2008), no. 3, 1035--1044. doi:10.1215/ijm/1254403730. https://projecteuclid.org/euclid.ijm/1254403730


Export citation

References

  • I. N. Baker, The domains of normality of an entire function, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), 277–283.
    Mathematical Reviews (MathSciNet): MR0402044
    Zentralblatt MATH: 0329.30019
    Digital Object Identifier: doi:10.5186/aasfm.1975.0101
  • I. N. Baker, J. Kotus and Y. Lü, Iterates of meromorphic functions I, Ergodic. Theory Dynam. Systems 11 (1991), 241–248.
    Mathematical Reviews (MathSciNet): MR1116639
  • W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (1993), 151–188.
    Mathematical Reviews (MathSciNet): MR1216719
    Zentralblatt MATH: 0791.30018
    Digital Object Identifier: doi:10.1090/S0273-0979-1993-00432-4
  • J. Kotus, On the Hausdorff dimension of Julia sets of meromorphic functions II, Bull. Soc. Math. France 123 (1995), 33–46.
    Mathematical Reviews (MathSciNet): MR1330786
    Zentralblatt MATH: 0826.30017
    Digital Object Identifier: doi:10.24033/bsmf.2249
  • J. Kotus and M. Urbański, Hausdorff dimension and Hausdorff measures of Julia sets of elliptic functions, Bull. London Math. Soc. 35 (2003), 269–275.
    Mathematical Reviews (MathSciNet): MR1952406
    Digital Object Identifier: doi:10.1112/S0024609302001686
  • V. Mayer, The size of the Julia set of meromorphic functions, preprint, 2005, to appear in Mathematische Nachrichten.
    Mathematical Reviews (MathSciNet): MR2547715
    Digital Object Identifier: doi:10.1002/mana.200710794
  • R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. 73 (1996), 105–154.
    Mathematical Reviews (MathSciNet): MR1387085
    Digital Object Identifier: doi:10.1112/plms/s3-73.1.105
  • P. J. Rippon and G. Stallard, Escaping points of meromorphic functions with a finite number of poles, J. Anal. Math. 96 (2005), 225–245.
    Mathematical Reviews (MathSciNet): MR2177186
    Zentralblatt MATH: 1087.30022
    Digital Object Identifier: doi:10.1007/BF02787829
  • P. J. Rippon and G. Stallard, Dimensions Julia set of meromorphic functions with a finite number of poles, Ergodic Theory Dynam. Systems 26 (2006), 525–538.
    Mathematical Reviews (MathSciNet): MR2218773
    Digital Object Identifier: doi:10.1017/S0143385705000544
  • B. Skorulski, The existence of conformal measures for some transcendental meromorphic functions, Contemp. Math. 396 (2006), 169–201.
    Mathematical Reviews (MathSciNet): MR2210775
    Zentralblatt MATH: 1207.37031
  • G. Stallard, The Hausdorff dimension of Julia set of entire functions II, Math. Proc. Cambridge Phil. Soc. 119 (1996), 513–536.
    Mathematical Reviews (MathSciNet): MR1357062
    Digital Object Identifier: doi:10.1017/S0305004100074387
  • G. Stallard, The Hausdorff dimension of Julia set of meromorphic functions, J. London. Math. Soc. 49 (1994), 281–295.
    Mathematical Reviews (MathSciNet): MR1260113
    Zentralblatt MATH: 0794.30023
    Digital Object Identifier: doi:10.1112/jlms/49.2.281

Duke University Press

Illinois Journal of Mathematics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more