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 $$F_bF_{a,z}-F_aF_{b,z}\neq 0 \qquad\text{on }\mathbb{C}^3.$$ The geometric core of the construction is a Fatou--Bieberbach domain contained in the thin region $$\{(u,v)\in\mathbb{C}^2:|u-v^2|<1+|v|\}.$$ 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.