Henon Maps: Open Problems — Question 18

Definitions and mathematical statement

Let $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$. For $a\in\mathbb D\setminus\{0\}$, define the dissipative Hénon map \[ H_a:{\mathbb{C}}^2\longrightarrow{\mathbb{C}}^2,\qquad H_a(x,y)=(ay,x+y^2). \] Here dissipativity means that the modulus of the complex Jacobian determinant is less than one; indeed, $\det DH_a=-a$.

A rotation number $\theta\in{\mathbb{R}}/{\mathbb{Z}}$ is of bounded type if its representative $\alpha\in(0,1)$ is irrational and its continued fraction expansion $\alpha=[0;b_1,b_2,\ldots]$ satisfies $\sup_{n\geq1}b_n<\infty$. Write $\operatorname{HR}_{\theta}(F)$ for the source assertion that $F$ has a Herman ring with rotation number $\theta$. This notation retains an unresolved source qualification: the supplied record does not define a Herman ring for the maps under consideration, including whether the ring must be invariant under the map itself or may be periodic with rotation defined using a return map.

Fix the map denoted by $H_2$ in the source and a bounded type rotation number $\theta$. Assume $\operatorname{HR}_{\theta}(H_2)$. The question is whether \[ \forall\varepsilon>0\quad\exists a\in\mathbb D\setminus\{0\}\quad \left(|a-1|<\varepsilon\ \text{and}\ \operatorname{HR}_{\theta}(H_a)\right). \] The map $H_2$ is not defined in the supplied record. In particular, it is not identified here with a member or an iterate of the displayed family. Consequently, the hypothesis and the Herman ring predicate require clarification from the original source before this becomes a fully specified mathematical statement.

Short English statement

Assuming the source's map $H_2$ has a Herman ring with bounded type rotation number $\theta$, can the same rotation number occur on Herman rings of $H_a(x,y)=(ay,x+y^2)$ for nonzero parameters inside the unit disk arbitrarily close to $1$? The meaning of $H_2$ and the precise Herman ring convention remain unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $801066$, source problem code catalogue entry, title "Henon Maps: Open Problems --- Question 18." The supplied record attributes the problem to Berger and Bedford, with proposed year 2023, and labels its status as open; this status is supplied registry information, not an independently verified literature assessment.
• Berger and Bedford, Hénon Maps: A List of Open Problems (2023), Question 18. Supplied source URL: https://arxiv.org/abs/2312.03907. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not evidence of independent source consultation for this draft.