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 $$F_b{F}_{a,z}-F_a{F}_{b,z}\ne 0\text{on }{\mathbb{C}}^3.$$ 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.

10 pages, all comments are welcome!