Illinois Journal of Mathematics

Hausdorff dimension of radial and escaping points for transcendental meromorphic functions

Janina Kotus and Mariusz Urbański

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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


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