Henon Maps: Open Problems — Question 16

Definitions and mathematical statement

The supplied record does not define the map $H_2$, give its formula, or specify its parameters. The following question therefore requires identification of that map, with the displayed iterates defined on the relevant points.

For $h>0$, let \[ V_h=\{z\in{\mathbb{C}}/{\mathbb{Z}}:\operatorname{Im}z<h\}, \] where ${\mathbb{C}}/{\mathbb{Z}}$ is the quotient by integer translations. For an irrational rotation class $\theta\in({\mathbb{R}}\setminus{\mathbb{Q}})/{\mathbb{Z}}$, translation $z\mapsto z+\theta$ is well-defined on $V_h$. Following the supplied definition, say that $H_2$ has a Herman ring with rotation number $\theta$ if there exist $h>0$, a holomorphic map $\phi:V_h\to{\mathbb{C}}^2$, and an integer $n\geq2$ such that \[ H_2^{\circ n}(\phi(z))=\phi(z+\theta) \qquad\text{for every }z\in V_h, \] where $H_2^{\circ n}$ denotes the $n$-fold iterate.

Question. Does the map denoted by $H_2$ admit such data for at least one $\theta\in({\mathbb{R}}\setminus{\mathbb{Q}})/{\mathbb{Z}}$?

The domain above retains exactly the supplied one-sided inequality; no lower bound on $\operatorname{Im}z$ is imposed. The supplied definition requires only that $\phi$ be holomorphic, without requiring it to be nonconstant, injective, or an embedding. These omissions leave the intended geometric meaning of "Herman ring" unresolved; no additional conditions have been imposed here.

Short English statement

For the unspecified map $H_2$, ask whether some iterate of order $n\geq2$ intertwines a holomorphic map from the stated annulus with an irrational translation. The map $H_2$ and the intended nondegeneracy requirements on the holomorphic map need clarification.

Sources

• Local UnsolvedMath registry: permanent ID $801064$, source problem code catalogue entry, title "Henon Maps: Open Problems --- Question 16." The supplied record attributes the problem to Berger, Bedford, with proposed year 2023, and labels its status as open; this status is supplied metadata, not an independently verified assessment.
• Berger, Bedford, Henon Maps: A List of Open Problems (2023), Question 16. Supplied source URL: https://arxiv.org/abs/2312.03907. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent source access or verification.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.