An entire transcendental family with a persistent Siegel disc

TL;DR

This paper investigates a family of entire transcendental maps with a persistent Siegel disc, analyzing their parameter space, stability components, and topological properties of the Siegel disc.

Contribution

It introduces a one-parameter family of maps with specific critical and asymptotic properties, exploring their parameter plane and Siegel disc topology.

Findings

Identification of stable components in the parameter plane

Description of the family interpolating between exponential and quadratic maps

Topological properties of Siegel discs in relation to parameters

Abstract

We study the class of entire transcendental maps of finite order with one critical point and one asymptotic value, which has exactly one finite pre-image, and having a persistent Siegel disc. After normalisation this is a one parameter family $f_a$ with $a\in\mathbb{C}^*$ which includes the semi-standard map $\lambda ze^z$ at $a=1$, approaches the exponential map when $a\to0$ and a quadratic polynomial when $a\to\infty$. We investigate the stable components of the parameter plane (capture components and semi-hyperbolic components) and also some topological properties of the Siegel disc in terms of the parameter.