**MSC:**- 30D05 Functional equations in the complex domain, iteration and composition of analytic functions, See also {34A20, 39-XX, 58F08, 58F23}
- 30D20 Entire functions, general theory

$f(z) = z-1 + (1-2z)e^z$.

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.