Locally Omitted Values in Completely Invariant Fatou Components

Definitions and mathematical statement

Let $f:{\mathbb{C}}\to{\mathbb{C}}$ be an entire function. Write $f^n$ for its $n$-fold iterate, and define its Fatou set $F(f)$ to be the set of points having a neighborhood on which $\{f^n:n\geq 0\}$ is a normal family with respect to the spherical metric. A Fatou component is a nonempty connected component of $F(f)$. A Fatou component $D$ is completely invariant if \[ f^{-1}(D)=D. \] Thus both $f(D)\subseteq D$ and $f^{-1}(D)\subseteq D$ hold; surjectivity of $f:D\to D$ is not required.

For $a\in{\mathbb{C}}$ and $\varepsilon>0$, put \[ U(a,\varepsilon)=\{z\in{\mathbb{C}}:|f(z)-a|<\varepsilon\}. \] Following the supplied statement, call $a$ a locally omitted value of $f$ if there exist $\varepsilon>0$ and a nonempty connected component $G$ of $U(a,\varepsilon)$ such that \[ G\cap f^{-1}(\{a\})=\varnothing. \] The existential quantifier on $\varepsilon$ makes explicit the implicit quantifier in the supplied wording. No additional condition involving nested components or smaller radii is imposed here.

Question. For every entire function $f:{\mathbb{C}}\to{\mathbb{C}}$ and every completely invariant Fatou component $D$ of $f$, is it true that \[ \forall a\in{\mathbb{C}},\qquad \left[ \exists\varepsilon>0\ \exists G\text{ a connected component of }U(a,\varepsilon) \text{ with }G\cap f^{-1}(\{a\})=\varnothing \right] \Longrightarrow a\in D? \] The source statement says entire, without explicitly requiring transcendentality or finite order; those hypotheses are therefore not added to the question. The supplied dated curation reports a finite-order result and describes the general assertion as a conjecture. This is a statement-only draft, and that literature assessment has not been independently verified.

Short English statement

If an entire function has a completely invariant Fatou component $D$, must every finite value omitted on some connected component of the inverse image of a sufficiently chosen disk centered at that value belong to $D$? The supplied curation labels the problem partially solved and reports the assertion for finite-order entire functions.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001392$, title Locally Omitted Values in Completely Invariant Fatou Components, source problem code catalogue entry, collection Oberwolfach Reports Open Problems. No individual proposer is supplied. The recorded proposed year is 2009.
• Supplied source citation: Mini-Workshop: The Escaping Set in Transcendental Dynamics (2010), Oberwolfach Reports, pp. 2927--2964, DOI 10.4171/owr/2009/54. This is also the literature-source entry labeled "original OWR report": https://doi.org/10.4171/owr/2009/54. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source entry labeled "Bergweiler Eremenko direct singularities": https://doi.org/10.1215/ijm/1242414130. The supplied assessment attributes to Bergweiler and Eremenko a proof of the locally omitted-value assertion for finite-order entire functions and says their paper states the general assertion as a conjecture.
• Literature assessment date: 2026-08-21. The status "partially solved" and the preceding literature claims are supplied dated curation derived from the OWR collection, not independently verified here. The supplied record also labels the statement corrected_verified and reports checking its mathematical content against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification description is supplied provenance, not an independent verification by this draft.