Slices of Parameter Space for Meromorphic Maps with Two Asymptotic Values

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.

arXiv comment47 pages, 12 figures

References in corpus (4)

• Iteration of meromorphic functions
• Stable components in the parameter plane of transcendental functions of finite type
• Monotonicity of entropy and positively oriented transversality for families of interval maps
• Accessible Boundary Points in the Shift Locus of a Familiy of Meromorphic Functions with Two Finite Asymptotic Values

Cited by in corpus (2)

• Stable components in the parameter plane of transcendental functions of finite type
• Accessible Boundary Points in the Shift Locus of a Familiy of Meromorphic Functions with Two Finite Asymptotic Values