Henon Maps: Open Problems — Question 21

Definitions and mathematical statement

The supplied record concerns Hénon maps but does not specify the underlying field, the class of maps, or the definitions of quasi-hyperbolicity and unstable slices. Let $\mathcal H_{\mathrm{src}}$ denote the intended class of maps, let $\mathrm{QH}_{\mathrm{src}}(f)$ denote quasi-hyperbolicity in the source's sense, and let $\mathcal U_{\mathrm{src}}(f)$ denote the intended collection of unstable slices. Write $\mathrm{John}_{[\mathrm{BS7}]}(S)$ for the John-type condition intended by the source's reference [BS7]. These symbols are placeholders for missing source definitions, not independently specified mathematical conditions.

The source question can therefore only be recorded schematically as follows: \[ \text{Is it true that }\quad \forall f\in\mathcal H_{\mathrm{src}},\qquad \mathrm{QH}_{\mathrm{src}}(f) \Longrightarrow \left(\forall S\in\mathcal U_{\mathrm{src}}(f),\ \mathrm{John}_{[\mathrm{BS7}]}(S)\right)? \] The supplied evidence does not identify the geometric sets to which the condition applies, the relevant metric, or the permitted dependence of any John-type constants. In particular, no uniformity over slices or maps is asserted here. A fully specified mathematical formulation requires the omitted definitions and the identification of [BS7].

Short English statement

Question 21 asks whether quasi-hyperbolicity of a Hénon map forces its unstable slices to satisfy the John-type condition referenced as [BS7]. The supplied record leaves the map class, slices, condition, and constant dependence unspecified, so this remains a source-dependent formulation requiring review.

Sources

• Local UnsolvedMath registry, permanent ID $801069$, source problem code catalogue entry, title: Henon Maps: Open Problems --- Question 21. The supplied registry attributes the problem to Berger and Bedford, with proposed year 2023, and labels its status as open; this label is not an independently verified mathematical assessment.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Question 21. 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 a claim of access or verification for this draft.
• The statement refers to [BS7], but the supplied record provides no bibliographic entry, URL, or text for that reference. Its intended John-type condition cannot be recovered from the supplied evidence alone.