A. Hinkkanen, B. Krauskopf, H. Kriete
Growing a Baker domain from attracting islands I: The dynamics of the limit function
Preprint series: Mathematica Gottingensis

MSC:

30D05 Functional equations in the complex domain, iteration and composition of analytic functions

30D20 Entire functions, general theory

Abstract: We consider the dynamics of the entire transcendental function
$f(z) = z-1 + (1-2z)e^z$.
This function features the origin as a superattracting fixed point
together with a Baker domain $\cB$
(on which the family of iterates converges to the constant function $\infty$).
The function $f$ can be approximated by the polynomials
$p_d(z) = z-1 + (1-2z)(1+z/d)^d$ and we are interested in the convergence
of the associated Julia and Fatou sets.
To this end, we pursue a detailed study of the dynamics of $f$.
In particular, we establish that $f$ has no periodic components other than
the Baker domain $\cB$ and the immediate basin of attraction $A^*_f(0)$
of the superattracting fixed point $z_0 = 0$, and no wandering domain.
We also prove a theorem on the structure of the Baker domain $\cB$.
Finally, we study the location of the critical points, the critical values,
and the fixed points.
Keywords: Baker domain, exponential function, Fatou set, iteration, Julia set, transcendental function