Research Problems in Function Theory — Problem 2.56

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an entire function $u:\mathbb C^n\to\mathbb C$, set $G_u=u(\mathbb C^n)$. Given a sequence $z_j\in\mathbb C^n$ with $\|z_j\|\to\infty$, say that $u$ has the $L$-property on this sequence if for every compact $K\subset G_u$, only finitely many $u(z_j)$ lie in $K$. An ordered $L$-pair $(u,v)$ means that, on every such sequence, the $L$-property of $u$ implies that of $v$. A nonconstant $f$ is an $L$-atom if every entire $u$ for which $(u,f)$ is an ordered $L$-pair can be written $u=\phi\circ f$ for a holomorphic function $\phi$ on $G_f$. These definitions are the $\mathbb C^n$ version of the range-escape definitions in Problem 5.55.

Statement recorded in the supplied source.

For $n\ge1$, is $f:\mathbb C^n\to\mathbb C$ an $L$-atom if and only if there are entire functions $f_2,\ldots,f_n$ such that \[F=(f,f_2,\ldots,f_n):\mathbb C^n\longrightarrow\mathbb C^n\] is a biholomorphic automorphism? [S2] The source attributes the conjecture to Rubel and states that he can prove the one-variable case. The ordering direction is the one in Problem 5.55: escape under $u$ must imply escape under the proposed atom $f$. [S2]

Short English statement

Are the entire functions that are atomic for the range-escape relation exactly the coordinate functions of holomorphic automorphisms of complex affine space?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302056 (AMR-022-2056). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.56 and Update 2.56; Problem 5.55 (L-property and L-atom). https://arxiv.org/abs/1809.07200.