Herman-ring retracts for Hénon maps

Definitions and mathematical statement

Let $F:\mathbb C^2\to\mathbb C^2$ be a polynomial automorphism, with constant nonzero Jacobian determinant $\delta$. Define \[ K^+(F)=\{z:\sup_{n\ge0}\|F^n(z)\|<\infty\}, \] and let $U$ be a component of its interior which is periodic under $F$. After replacing $F$ by an iterate, assume $F(U)=U$. Suppose that for some sequence $n_j\to\infty$ the iterates $F^{n_j}|_U$ converge locally uniformly to a holomorphic map $g:U\to\mathbb C^2$ of complex rank one.

Can this limiting map be a holomorphic retraction onto an embedded rotation annulus $V\subset U$? Retraction means $g(U)=V$ and $g|_V=\operatorname{id}_V$, equivalently $g^2=g$. The proposed possibilities are that $V$ is biholomorphic either to \[ \{z:r<|z|<1\}\quad(0<r<1), \qquad\text{or to }\{z:0<|z|<1\}, \] and that the restriction of $F$ to $V$ is conjugate to an irrational rotation in the annulus coordinate.

The first possibility is the Herman-ring alternative, and the second is a punctured Siegel disk. Ordinary disk retracts give known rank-one rotation behavior, but do not settle these annular alternatives. The issue is realization by a polynomial automorphism on its actual stable component, not merely an abstract annular holomorphic map. The rank is complex differential rank; convergence and the retraction identity are required on all of $U$, not just along one selected orbit.

Short English statement

Can a polynomial automorphism have a rank-one stable-component limit retracting onto a Herman ring or a punctured Siegel disk?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5300081 (AMR-052-0081), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Stony Brook - Open Problems in Dynamical Systems, 1990 §4 rank-one. Original source indexed by AMR-052-0081. https://www.math.stonybrook.edu/open-problems-dynamical-systems.
• S3 Bielefeld, Conformal Dynamics Problem List (1990), Section4, rank-one case. https://web.pdx.edu/~veerman/bielefeld.pdf.