Metric properties of the Julia set of some meromorphic functions with an asymptotic value eventually mapped onto a pole

Abstract

We study the dynamics of non-entire transcendental meromorphic functions with a finite asymptotic value mapped after some iterations onto a pole. This situation does not appear in the case of rational or entire functions. We consider the family of non-entire functions \[ f(z)=\frac{a\exp(z^p)+b\exp(-z^p)}{c\exp(z^p)+d\exp(-z^p)} \] with this property, i.e. there exists a finite asymptotic value $\xi$ and a positive natural number $q$ such that $f^q(\xi)\,{=}\,\infty$, where $p\,{\geq}\,1$, $a,b,c,d\,{\in}\,\mathbb{C}$. We show that the Hausdorff dimension of the set of points in the Julia set with bounded trajectory is strictly greater than one. We then impose two conditions on functions from this family. If the first condition holds, then the Lebesgue measure of the Julia set is zero, but the Hausdorff dimension is equal to two. On the other hand, if a function $f$ satisfies the second condition, then the Lebesgue measure of the Julia set is positive. Finally, we describe the measurable dynamics of these functions and prove that there does not exist an invariant measure on the Julia set absolutely continuous with respect to the Lebesgue measure and finite on compact subsets of $\mathbb{C}$.