Connected Interiors of Conservative Hénon Filled Sets

Definitions and mathematical statement

The record does not explicitly specify the ambient space or the precise class of Hénon maps. Under the standard complex-dynamical interpretation, let $f\colon{\mathbb{C}}^2\to{\mathbb{C}}^2$ be a polynomial automorphism of Hénon type, with constant complex Jacobian determinant $\delta\ne0$. The supplied conservative condition is $|\delta|=1$. Define the forward filled set by \[ K^+(f)=\left\{x\in{\mathbb{C}}^2:\sup_{n\ge0}\|f^n(x)\|<\infty\right\}, \] where $\|\cdot\|$ is the Euclidean norm. Interior and closure are taken in the Euclidean topology of ${\mathbb{C}}^2$.

On an existence reading, the cleaned statement asks whether there exists such a conservative Hénon map satisfying both \[ \operatorname{int}K^+(f)\text{ is connected}, \qquad \overline{\operatorname{int}K^+(f)}=K^+(f). \] No separate nonemptiness assumption on the interior is supplied.

There is an unresolved discrepancy in the target set: the original statement instead asks for \[ \overline{\operatorname{int}K^+(f)}=K(f). \] The record does not define $K(f)$. In standard complex dynamics, one defines \[ K^-(f)=\left\{x\in{\mathbb{C}}^2:\sup_{n\ge0}\|f^{-n}(x)\|<\infty\right\}, \qquad K(f)=K^+(f)\cap K^-(f). \] If this convention was intended, the original and cleaned statements have different targets. Both are retained here pending source verification; the record's supplied correction label does not resolve the discrepancy. The intended Hénon class and the existence interpretation also require confirmation.

Short English statement

The cleaned question asks whether a conservative Hénon map can have connected forward filled-set interior whose closure is the whole forward filled set. The original wording uses an undefined $K$ as the closure target instead of $K^+$, so the intended statement remains unresolved. The complex ambient space and existence quantifier are conditional interpretations.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003922$, source problem code catalogue entry, titled Connected Interiors of Conservative Hénon Filled Sets; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the recorded proposed year is 2018.
• Supplied source citation: Geometric Methods of Complex Analysis (2019), Oberwolfach Reports, pp. 2253--2302, DOI 10.4171/owr/2018/37. Supplied source and literature link, labelled Oberwolfach Report 37/2018: https://doi.org/10.4171/owr/2018/37. The differing year indications are preserved without reconciliation.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes the report's question as unanswered, and reports that no later primary result was located that settles it. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry supplies the statement-status label corrected_verified and says that notation, typography, and sentence structure were repaired while preserving mathematical content. These are supplied editorial assertions, not independent verification of the formulation or mathematical status.