A Solution to a Problem of Rubel on Two-Parameter Normal Families of Entire Functions

We construct an entire function $F(z,a,b)\in \mathcal{O}({\mathbb{C}}^3)$ such that the family $$\{F(\cdot ,a,b):a,b\in \mathbb{C}\}$$ of entire functions of $z$ is normal on $\mathbb{C}$, while $F$ does not factor through a single entire parameter. This solves a problem of L.~A.~Rubel concerning Liouville-type rigidity. In fact, our example satisfies the stronger condition $$F_b{F}_{a,z}-F_a{F}_{b,z}\mathrm{\ne }0\text{on }{\mathbb{C}}^3\mathrm{.}$$ The geometric core of the construction is a Fatou–Bieberbach domain contained in the thin region $$\{(u,v)\in {\mathbb{C}}^2:\mathrm{\mid }u-v^2\mathrm{\mid }<1+\mathrm{\mid }v\mathrm{\mid }\}\mathrm{.}$$ We obtain this domain from the basin of attraction of an explicit polynomial automorphism of ${\mathbb{C}}^2$, together with the theorem of Rosay and Rudin on attracting basins.