# Slices of Parameter Space for Meromorphic Maps with Two Asymptotic Values

## TL;DR

This paper fully describes the parameter space of a family of exponential meromorphic functions with two asymptotic values and an attracting fixed point, advancing the understanding of complex dynamical systems with singularities.

## Contribution

It extends the analysis of parameter spaces from rational functions to certain transcendental functions with essential singularities, showing similar structural properties.

## Key findings

- Complete description of parameter space for the specific exponential family
- Demonstration of similar dynamic structures to rational functions despite singularities
- Adaptation of techniques from rational to transcendental dynamics

## Abstract

This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work in [GK] on degree 2 rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further, is that, despite the presence of the essential singularity, our new functions exhibit a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted.