Wandering Cauliflowers

TL;DR

This paper studies wandering domains of a specific entire function, showing their shapes converge to a known parabolic basin and classifying their behavior within a broader framework.

Contribution

It provides a detailed analysis of wandering domains for a particular function and classifies them within a ninefold framework, expanding understanding of their geometric properties.

Findings

Wandering domains' shapes converge to the parabolic basin of z^2 + 1/4.

The wandering domains are contracting with diameters tending to zero.

Classified the wandering domains within a ninefold framework.

Abstract

In this paper we examine an orbit of simply connected wandering domains for the function ${f(z) = z\cos z+2\pi}$. They are noteworthy in that they are non-congruent but arise from a simple closed form function. Moreover, the shape of the wandering domains, suitably scaled, converges in the Hausdorff metric to the filled-in parabolic basin of the quadratic ${z^2+c}$ with $c=\tfrac{1}{4}$, commonly named the ``cauliflower''. We complete our analysis by classifying the wandering domains within the ninefold framework in \cite{benini+2021}, finding they are contracting and the diameters of the wandering domains tend to zero. To conclude we propose an expansion of the analysis to a wider family of functions and discuss some potential results.