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

TL;DR

This paper constructs a specific entire function demonstrating a counterexample to a problem posed by Rubel, showing a family of functions can be normal without factoring through a single parameter, thus addressing a Liouville-type rigidity issue.

Contribution

It provides the first explicit example of a normal family of entire functions that does not factor through a single parameter, solving Rubel's problem and introducing new geometric constructions.

Findings

Constructed an entire function with a normal family not factoring through one parameter.

Used Fatou--Bieberbach domains within a specific thin region in C^2.

Applied the basin of attraction of a polynomial automorphism and Rosay-Rudin's theorem.

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.