An entire transcendental family with a persistent Siegel disc

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 $a\in {\mathbb{C}}^{\ast }$ which includes the semi-standard map $\lambda z{e}^z$ at $a=1$, approaches the exponential map when $a\to 0$ 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.

36 pages, 17 figures. Revised typos and references. To appear in the Journal of Difference Equations and Applications, special volume in honor of Robert Devaney