Henon Maps: Open Problems — Question 29

Definitions and mathematical statement

For a polynomial automorphism $f\colon{\mathbb{C}}^2\to{\mathbb{C}}^2$, define its forward filled Julia set and forward Julia set by \[ K^+(f)=\left\{z\in{\mathbb{C}}^2:\sup_{n\in{\mathbb{Z}}_{\geq 0}}\|f^n(z)\|<\infty\right\},\qquad J^+(f)=\partial_{{\mathbb{C}}^2}K^+(f), \] where $f^0$ is the identity and the norm is the Euclidean norm. A nonsingular one-dimensional holomorphic foliation $\mathcal F$ on an open set $U\subseteq{\mathbb{C}}^2$ is a foliation locally represented in holomorphic coordinates by the sets on which one coordinate is constant; its leaves are connected complex curves. A subset of $U$ is saturated by $\mathcal F$ if it is a union of whole leaves.

The supplied Question 29 asks whether there exist a generalized complex Hénon map $f$, an open set $U\subseteq{\mathbb{C}}^2$, and a holomorphic foliation $\mathcal F$ of $U$ such that \[ U\cap J^+(f)\neq\varnothing, \qquad \forall L\in\operatorname{Leaves}(\mathcal F),\quad L\cap J^+(f)\neq\varnothing\ \Longrightarrow\ L\subseteq J^+(f). \] Thus $J^+(f)\cap U$ must be saturated by $\mathcal F$. No invariance of $U$ or of the foliation under $f$ is requested in the supplied statement.

Unresolved source conventions. The supplied record does not define the class of generalized complex Hénon maps, including whether it includes finite compositions of elementary Hénon maps. The quantifier over $f$ retains the source's terminology and requires confirmation of that class. The definition of foliation above uses nonsingular foliations by complex curves; the record does not explicitly specify the dimension or whether singular foliations are permitted. The displayed formulation is subject to confirmation of those conventions.

Short English statement

Does some generalized complex Hénon map have a forward Julia set that, on an open set meeting it, is a union of entire leaves of a holomorphic foliation? The intended map class and foliation conventions require confirmation from the source.

Sources

• Local UnsolvedMath registry, permanent ID $801077$, source code catalogue entry, title "Henon Maps: Open Problems --- Question 29." The supplied record attributes the question to Berger and Bedford, with proposed year 2023.
• Berger and Bedford, Hénon Maps: A List of Open Problems (2023), Question 29. 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 were not independently verified here.
• The local record labels the problem open. This is a supplied registry classification, not an independently verified assessment of the literature or mathematical status.