Henon Maps: Open Problems — Question 4

Definitions and mathematical statement

Let $\mathcal H_{\mathrm{real,cons}}$ denote the class referred to in the supplied source as real, conservative, polynomial Hénon maps. A polynomial map with real coefficients acts on ${\mathbb{R}}^2$ and also defines a complexification on ${\mathbb{C}}^2$. The supplied excerpt does not specify which ambient space is intended for the rotation domain, the precise Hénon class allowed, or whether conservativity requires Jacobian determinant $1$ or permits determinant $-1$.

Write $\mathsf{AnnRot}(f)$ for the property that $f$ has an annular rotation domain in the intended source sense. The excerpt supplies no definition of this property: the ambient space, regularity of the domain and conjugacy, rotation model, and any irrationality requirement remain unspecified. Thus this notation records an unresolved source predicate rather than a complete definition.

The two questions attributed to Berger and Bedford (2023), Question 4, have the following quantifier structure: \[ \exists f\in\mathcal H_{\mathrm{real,cons}}\quad \mathsf{AnnRot}(f), \] and, for the intended topology $\tau$ on this class, \[ \exists U\in\tau\quad U\ne\varnothing \quad\text{and}\quad \forall f\in U,\ \mathsf{AnnRot}(f). \] The source does not specify $\tau$, a fixed degree, or a parameter space, so the second question cannot yet be made topologically precise.

The preceding source text also asks whether an unidentified "third condition" is superfluous and says that a result labeled [OC23] suggests this may be the case. Neither that condition nor the content of [OC23] is supplied. Their relationship to the two existence questions therefore remains unresolved; no hypothesis is removed on the basis of that remark.

Short English statement

Do any real, conservative, polynomial Hénon maps have annular rotation domains, and does a nonempty open family of such maps exist? The supplied excerpt leaves the rotation-domain definition, ambient space, conservativity convention, topology, and preceding "third condition" unresolved.

Sources

• Local UnsolvedMath registry, supplied AMR Open Problem Lists record: permanent ID $801052$, exact source code catalogue entry, title "Henon Maps: Open Problems --- Question 4." The registry attributes the problem to Berger and Bedford, with proposed year 2023, and labels it open; this label is supplied metadata, not an independently verified mathematical assessment.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Question 4. Supplied source URL: https://arxiv.org/abs/2312.03907. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these provenance claims have not been independently verified here.
• The supplied excerpt mentions a result labeled [OC23], but provides no bibliographic entry, author identification, title, URL, or statement for that reference.