# Stable components in the parameter plane of transcendental functions of finite type

## TL;DR

This paper investigates the structure of parameter spaces called shell components in families of transcendental entire and meromorphic functions, revealing their topological properties and the role of virtual centers, with implications for dynamical plane behavior.

## Contribution

It introduces the concept of shell components in parameter planes of transcendental maps, proving their simple connectivity, local connectedness, and the existence of virtual centers, extending previous studies to broader classes.

## Key findings

- Shell components are simply connected and have locally connected boundaries.
- No superattracting cycles exist in shell components; instead, virtual centers are identified.
- Basins containing only one asymptotic value are simply connected in the dynamical plane.

## Abstract

We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we call {\em shell components}, for which the free asymptotic value tends to an attracting cycle of non-constant multiplier. The exponential and the tangent families are examples that have been studied in detail, and the hyperbolic components in those parameter planes are shell components. Our results apply to slices of both entire and meromorphic maps. We prove that shell components are simply connected, have a locally connected boundary and have no center, i.e., no parameter value for which the cycle is superattracting. Instead, there is a unique parameter in the boundary, the {\em virtual center}, which plays the same role. For entire slices, the virtual center is always at infinity, while for meromorphic ones it maybe finite or infinite. In the dynamical plane we prove, among other results, that the basins of attraction which contain only one asymptotic value and no critical points are simply connected. Our dynamical plane results apply without the restriction of finite type.