Annular Recurrent Fatou Components of Hénon Maps

Definitions and mathematical statement

Let $F\colon{\mathbb{C}}^2\to{\mathbb{C}}^2$ be a polynomial Hénon map in the class intended by the cited source. A Fatou component $U$ is a connected component of the open set on which the forward iterates $\{F^n:n\geq0\}$ form a normal family. Recurrence means that there exist $z\in U$, integers $n_j\to\infty$, and $w\in U$ such that $F^{n_j}(z)\to w$.

For parameters $0\leq r<R\leq\infty$, write \[ A(r,R)=\{\zeta\in{\mathbb{C}}:r<|\zeta|<R\}. \] An irrational rotation on this domain is the map $\zeta\mapsto e^{2\pi i\alpha}\zeta$, where $\alpha\in{\mathbb{R}}\setminus{\mathbb{Q}}$. A finite-modulus annulus has $0<r<R<\infty$. The supplied record does not specify whether its use of "annulus" also permits punctured or infinite-modulus conformal types.

Existence question, subject to the source qualifications below. Does there exist a polynomial Hénon map $F$ having a recurrent Fatou component $U$ for which the associated one-dimensional rotation domain in case 2 of the cited classification is conformally equivalent to an annulus?

Writing $S$ for that associated one-dimensional domain, the annular rotation condition means that, for an appropriate return iterate $F^k$, there are a biholomorphism $\phi\colon S\to A(r,R)$ and $\alpha\in{\mathbb{R}}\setminus{\mathbb{Q}}$ such that \[ \phi\circ(F^k|_S)\circ\phi^{-1}(\zeta) =e^{2\pi i\alpha}\zeta \qquad(\zeta\in A(r,R)). \] Here $k\geq1$, and the admissible annulus types and the precise relationship between $S$ and $U$ must be taken from the source classification.

Unresolved formulation qualification. A Fatou component of a map on ${\mathbb{C}}^2$ is open in ${\mathbb{C}}^2$, hence has complex dimension two. It therefore cannot itself be biholomorphic to a one-dimensional annulus. The original wording asks about the one-dimensional rotation domain associated with case 2, whereas the cleaned wording identifies the Fatou component with that domain. The associated-domain reading above preserves the original target, but the supplied record does not reproduce case 2, its hypotheses, or its definition of the associated domain. Thus the displayed conjugacy is an explanation of annular irrational rotation, not a complete replacement for the missing classification conditions.

Short English statement

The question asks whether the one-dimensional rotation domain associated with a recurrent Fatou component of a polynomial Hénon map can be an annulus. The ambient Fatou component has complex dimension two. The precise classification hypotheses, the relationship between the component and its rotation domain, and the allowed annulus types require source review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30001760$, source problem code catalogue entry, title "Annular Recurrent Fatou Components of Hénon Maps." The record assigns the problem to 2011 and supplies no individual proposer.
• Geometric Methods of Complex Analysis (2011), Oberwolfach Reports, DOI 10.4171/owr/2011/19, pp. 1047--1116. Supplied report citation and literature-source link: https://doi.org/10.4171/owr/2011/19. The record attributes the problem to workshop open-problem discussions in this report; it does not supply a problem-specific page or the text of case 2.
• Bedford--Smillie, "classification of invariant Fatou components" (description supplied by the registry): https://math.stonybrook.edu/preprints/ims12-07.pdf. No fuller bibliographic identification or theorem number is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the recurrent Fatou-component classification is known, while occurrence of the annular basin for polynomial Hénon maps remains open. This is supplied dated curation, not an independently verified literature assessment. Both links above are listed among the supplied sources checked by that curation; neither source has been independently visited or verified for this draft.