Henon Maps: Open Problems — Question 30

Definitions and mathematical statement

The supplied question concerns Hénon maps but does not specify the ambient map, its parameter restrictions, or whether saddle points must be fixed points or may be periodic points. The following notation uses the conventional complex setting, conditionally on that interpretation.

Let $f\colon{\mathbb{C}}^2\to{\mathbb{C}}^2$ be a complex Hénon polynomial automorphism in the intended, unspecified class. Its forward filled set is \[ K^+(f)=\left\{z\in{\mathbb{C}}^2:\sup_{n\geq 0}\|f^n(z)\|<\infty\right\}. \] A periodic point $p$ of least period $m_p\geq1$ is a saddle if the eigenvalues of $D(f^{m_p})_p$ can be labeled $\lambda_s,\lambda_u$ with \[ 0<|\lambda_s|<1<|\lambda_u|. \] Its unstable manifold, with its intrinsic complex manifold topology, is \[ W^u(p)=\left\{z\in{\mathbb{C}}^2:f^{-km_p}(z)\longrightarrow p\text{ as }k\longrightarrow\infty\right\}. \] If the intended saddle points are fixed points, restrict this definition to $m_p=1$.

Question 30 (Berger, Bedford, 2023). Under which circumstances can there exist saddle points $p,q$ of the same map $f$, and relatively open subsets \[ U\subseteq W^u(p),\qquad V\subseteq W^u(q), \] such that $U\cap K^+(f)$ is biholomorphic to $V\cap K^+(f)$?

The meaning of "biholomorphic" here requires clarification: the intersections are not asserted to be complex manifolds. One possible interpretation is that a biholomorphism $h\colon U\to V$ satisfies \[ h\left(U\cap K^+(f)\right)=V\cap K^+(f), \] but the supplied record does not state that an extension to all of $U$ and $V$ is required. This interpretation is therefore not imposed as part of the question. The record also does not require $p\neq q$, nonempty open subsets, nonempty intersections, or any exclusion of the identity correspondence. Such conditions must be specified if a nontrivial classification is intended.

Short English statement

Berger and Bedford ask when portions of the forward filled set lying in unstable manifolds of saddle points of a Hénon map can be biholomorphically equivalent. The permitted maps, the meaning of equivalence on these subsets, and any conditions excluding trivial examples remain unspecified in the supplied excerpt.

Sources

• Local UnsolvedMath registry: permanent ID $801078$, source problem code catalogue entry, title "Henon Maps: Open Problems --- Question 30." The supplied record attributes the question to Berger and Bedford (2023) and labels it open; this status is supplied registry information, not an independently verified mathematical assessment.
• Berger, Bedford, Henon Maps: A List of Open Problems (2023), Question 30. Supplied source URL: https://arxiv.org/abs/2312.03907. The record reports access on 2026-10-16 and extraction by arxiv_latex; these provenance details are supplied and have not been independently verified.