Complete Invariance of Singular-Value Basins

Definitions and mathematical statement

Let $f\colon{\mathbb{C}}\to{\mathbb{C}}$ be an entire function. Define its set of critical values by \[ \operatorname{CV}(f)=\{f(c):c\in{\mathbb{C}},\ f'(c)=0\}. \] Its set of finite asymptotic values is \[ \operatorname{AV}(f)=\left\{a\in{\mathbb{C}}:\begin{array}{l}\text{there exists a continuous curve }\gamma\colon[0,\infty)\to{\mathbb{C}}\\ |\gamma(t)|\to\infty\text{ and }f(\gamma(t))\to a\text{ as }t\to\infty\end{array}\right\}. \] Here a region means a nonempty connected open subset of ${\mathbb{C}}$. For $D\subseteq{\mathbb{C}}$, its full preimage is \[ f^{-1}(D)=\{z\in{\mathbb{C}}:f(z)\in D\}. \] We use complete invariance to mean forward and backward invariance: \[ f(D)\subseteq D\quad\text{and}\quad f^{-1}(D)\subseteq D, \] or, equivalently, $f^{-1}(D)=D$. This convention does not require surjectivity of $f|_D$.

Question 1, with an unresolved basin qualification. For every entire function $f$ and every immediate basin $D$ of $f$, does \[ \operatorname{CV}(f)\cup\operatorname{AV}(f)\subseteq D \quad\Longrightarrow\quad f^{-1}(D)=D \] hold? The supplied record does not specify the type or period of the orbit associated with the immediate basin. For example, for an attracting fixed point $a$, the basin is $B(a)=\{z\in{\mathbb{C}}:f^{\circ n}(z)\to a\}$, and its immediate basin is the connected component of $B(a)$ containing $a$. This example defines one usual case; it does not resolve whether the source intends only this case, attracting periodic basins more generally, or also parabolic basins. Thus the quantification over immediate basins in Question 1 remains subject to source clarification.

Question 2. Is it true that for every entire function $f\colon{\mathbb{C}}\to{\mathbb{C}}$ and every nonempty connected open set $D\subseteq{\mathbb{C}}$, \[ \operatorname{CV}(f)\cup\operatorname{AV}(f)\subseteq D \quad\Longrightarrow\quad f^{-1}(D)\text{ is connected}? \] The containment hypotheses concern the values themselves; no containment of their closure, boundedness, simple connectivity, properness of $D$, or forward invariance of $D$ is explicitly assumed in Question 2. The record says entire, without an explicit restriction to nonconstant or transcendental entire functions.

Short English statement

The questions ask whether an immediate basin containing every critical and finite asymptotic value must be completely invariant, and whether every region containing those values has connected full preimage. The intended type and period of the immediate basin require clarification. The supplied record labels these questions open; this is a supplied assessment, not an independently verified status.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001393$, title Complete Invariance of Singular-Value Basins, source problem code catalogue entry, collection Oberwolfach Reports Open Problems. No individual proposer is supplied.
• Supplied source citation: Mini-Workshop: The Escaping Set in Transcendental Dynamics (2010), Oberwolfach Reports, pp. 2927--2964, DOI 10.4171/owr/2009/54. Original OWR report: https://doi.org/10.4171/owr/2009/54. The registry separately records proposed year 2009.
• Supplied related reference, labeled "related direct singularities paper": https://doi.org/10.1215/ijm/1242414130. No authors, title, or specific supporting result are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the questions open and reports that the cited related results do not give a general resolution in the original formulation. This is supplied dated curation, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The registry assigns statement status corrected_verified and reports comparison with the official Oberwolfach report, with notation, context, and question form normalized. This verification claim is supplied metadata and has not been independently checked here.