Connected Escaping Sets of Postsingularly Finite Entire Functions

Definitions and mathematical statement

Let $f\colon{\mathbb{C}}\to{\mathbb{C}}$ be a transcendental entire function, meaning an entire function that is not a polynomial. Write $f^0=\operatorname{id}_{{\mathbb{C}}}$ and $f^{n+1}=f\circ f^n$ for $n\geq 0$. Let $S(f)$ be the closure in ${\mathbb{C}}$ of the set of critical values and finite asymptotic values of $f$. A critical value is a value $f(c)$ with $f'(c)=0$; a finite asymptotic value is a number $a\in{\mathbb{C}}$ for which there is a continuous curve $\gamma\colon[0,\infty)\to{\mathbb{C}}$ such that $|\gamma(t)|\to\infty$ and $f(\gamma(t))\to a$ as $t\to\infty$. Define the postsingular set by \[ P(f)=\overline{\bigcup_{n=0}^{\infty}f^n(S(f))}, \] where the closure is taken in ${\mathbb{C}}$. Let $F(f)$ be the set of points having a neighbourhood on which the family $\{f^n:n\geq0\}$ is normal with respect to the spherical metric, and let $J(f)={\mathbb{C}}\setminus F(f)$ be the Julia set. Define the escaping set by \[ I(f)=\{z\in{\mathbb{C}}:|f^n(z)|\longrightarrow\infty\text{ as }n\to\infty\}. \] Assume that \[ \#P(f)<\infty\qquad\text{and}\qquad P(f)\subseteq J(f). \] Question. Under which additional hypotheses on such a function $f$, if any, is $I(f)$ connected in the topology inherited from ${\mathbb{C}}$? Here connectedness means that $I(f)$ cannot be expressed as the union of two disjoint nonempty relatively open subsets.

Unresolved qualifications. The supplied question does not specify a candidate additional hypothesis or whether it seeks sufficient conditions, a necessary-and-sufficient characterization, or a decision about sufficiency of the displayed assumptions alone. It therefore supplies a research question rather than a uniquely specified conjectural implication. The accompanying literature assessment also mentions unresolved immediate-basin and connected-preimage questions, but supplies neither their formulations nor their relationship to this question; no such assertions are added here.

Short English statement

For a transcendental entire function with a finite postsingular set contained in its Julia set, ask what further hypotheses ensure that all points whose iterates tend to infinity form a connected subset of ${\mathbb{C}}$. The intended additional hypotheses and the scope of the requested characterization remain unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001394$, title Connected Escaping Sets of Postsingularly Finite Entire Functions, collection Oberwolfach Reports Open Problems, exact source problem code catalogue entry. No individual proposer is identified; the supplied 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. Source URL and supplied literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2009/54. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and describes it as an open-problem marker whose commentary identifies unresolved immediate-basin and connected-preimage questions. This is supplied dated curation, not an independently verified assessment. The source record explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified. Its accompanying verification note reports that the mathematical content was checked against the official Oberwolfach report and that notation, context, and question form were normalized without changing the source-level claim. This draft preserves that reported qualification without claiming independent source verification.