A Solution to a Problem of Rubel on Two-Parameter Normal Families of Entire Functions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915880601911296 |
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| author | He, Yixin Tang, Quanyu Zhang, Teng |
| author_facet | He, Yixin Tang, Quanyu Zhang, Teng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20883 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Solution to a Problem of Rubel on Two-Parameter Normal Families of Entire Functions He, Yixin Tang, Quanyu Zhang, Teng Complex Variables 30D45, 32A19, 32M17 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. |
| title | A Solution to a Problem of Rubel on Two-Parameter Normal Families of Entire Functions |
| topic | Complex Variables 30D45, 32A19, 32M17 |
| url | https://arxiv.org/abs/2603.20883 |