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

Abstract

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.