Conformal, geometric and invariant measures for transcendental expanding functions

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.