Henon Maps: Open Problems — Question 42

Definitions and mathematical statement

Let $\mathcal H$ denote the class of maps called generalized Hénon maps in the supplied source record. For $h\in\mathcal H$, retain the source notation $\bar S(h)$. The record does not define either the precise class $\mathcal H$ or the associated object $\bar S(h)$; these definitions must be recovered from the cited source before this formulation can be made self-contained. In particular, the bar in $\bar S(h)$ is retained without assigning it an interpretation.

A biholomorphism is a bijective holomorphic map whose inverse is holomorphic, with respect to the complex structures on the objects concerned. The record does not specify the nature or complex structures of the objects $\bar S(h)$.

For every fixed $f\in\mathcal H$, describe the class \[ \mathcal B(f)=\left\{g\in\mathcal H:\text{there exists a biholomorphism }\Phi:\bar S(f)\longrightarrow\bar S(g)\right\}. \] This is the classification question posed in the record, conditional on the missing source definitions. No additional compatibility requirement on $\Phi$ is stated in the supplied formulation.

Short English statement

For any generalized Hénon map $f$, characterize all generalized Hénon maps $g$ for which the associated objects $\bar S(f)$ and $\bar S(g)$ are biholomorphic. The definitions of generalized Hénon map and $\bar S(h)$ are missing from the supplied record, so the intended classification is not yet self-contained.

Sources

• Local UnsolvedMath registry, permanent ID $801090$, exact source problem code catalogue entry, title "Henon Maps: Open Problems --- Question 42." 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 metadata, not an independently verified mathematical or literature assessment.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Question 42. Supplied source URL: https://arxiv.org/abs/2312.03907. The record reports access on 2026-10-16 and extraction method arxiv_latex; these are supplied provenance statements, not a claim of independent source consultation.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.