Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yixin He, Quanyu Tang, Teng Zhang

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.

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

Abstract

We construct an entire function such that the family of entire functions of is normal on , while 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 The geometric core of the construction is a Fatou--Bieberbach domain contained in the thin region We obtain this domain from the basin of attraction of an explicit polynomial automorphism of , together with the theorem of Rosay and Rudin on attracting basins.
Paper Structure (5 sections, 4 theorems, 67 equations)

This paper contains 5 sections, 4 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Paper organization
  3. A Fatou--Bieberbach domain in a parabolic tube
  4. Normality of the associated affine family
  5. Proof of the main theorem

Key Result

Theorem 1.2

There exists an entire function $F\in \mathcal{O}(\mathbb{C}^3)$ such that

Theorems & Definitions (10)

  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2: Rosay--Rudin
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Remark 4.1