Mathematische Annalen

, Volume 324, Issue 3, pp 619–656

Conformal, geometric and invariant measures for transcendental expanding functions

  • J. Kotus
  • M. Urbanski

DOI: 10.1007/s00208-002-0356-y

Cite this article as:
Kotus, J. & Urbanski, M. Math. Ann. (2002) 324: 619. doi:10.1007/s00208-002-0356-y

Abstract.

We describe the fractal structure of expanding meromorphic maps of the form \(H\circ\exp\circ Q\), where H and Q are rational functions whose most transparent examples are among the functions of the form \(\frac{A\exp(z^p)+B\exp(-z^p)}{C\exp(z^p)+D\exp(-z^p)}\) with \(AD-BC\ne 0\). In particular we show that depending upon whether the Hausdorff dimension of the Julia set is greater or less than 1, the finite non-zero geometric measure is provided by the Hausdorff or packing measure. In order to describe this fractal structure we introduce and explore in detail Walters expanding conformal maps and jump-like conformal maps.

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • J. Kotus
    • 1
  • M. Urbanski
    • 2
  1. 1.Faculty of Mathematics and Information Sciences, Warsaw University of Technology, Warsaw 00-661, Poland (e-mail: janinak@impan.gov.pl) PL
  2. 2. Department of Mathematics, University of North Texas, P.O. Box 311430, Denton TX 76203-1430, USA (e-mail: urbanski@unt.edu, Web: http://www.math.unt.edu/~urbanski) US