Henon Maps: Open Problems — Question 51

Definitions and mathematical statement

The supplied record refers to a ring $\mathcal{O}_{K,S}$, an ideal sequence $(\mathfrak{a}_n)_{n\geq 1}$, a parameter $d$, and an ideal norm $\operatorname{Norm}$, but does not define these objects or their underlying dynamical construction. In particular, the hypotheses on $K$, $S$, and $d$, and the construction of $\mathfrak{a}_n$, remain unresolved. The following therefore preserves the source questions as a conditional formulation, rather than supplying a fully specified problem.

Determine the circumstances under which \[ \log\operatorname{Norm}(\mathfrak{a}_n)=o(d^n) \qquad (n\to\infty). \] Whenever the logarithm and quotient are defined, this asks for conditions ensuring \[ \lim_{n\to\infty}\frac{\log\operatorname{Norm}(\mathfrak{a}_n)}{d^n}=0. \] Also determine the circumstances under which there exists an ideal $\mathfrak{b}\subset\mathcal{O}_{K,S}$ such that \[ \#\{n\in{\mathbb{Z}}_{\geq 1}:\mathfrak{a}_n\mid\mathfrak{b}\}=\infty. \] The source additionally states that \[ \mathfrak{a}_1\mid\mathfrak{a}_n\qquad\text{for every }n\geq 1, \] and identifies $\mathfrak{b}=\mathfrak{a}_1$ as a reasonable candidate. This candidate is a suggestion in the source, not an asserted conclusion. Under the usual convention for integral ideals, $I\mid J$ means $J=IC$ for some integral ideal $C$; the intended ideal class and norm convention must be confirmed from the missing context.

Short English statement

Characterize when the source's ideal sequence has logarithmic norm growing more slowly than $d^n$, and when infinitely many terms divide one fixed ideal of $\mathcal{O}_{K,S}$. The source suggests the first term as that fixed ideal. The sequence, norm, parameter, and ambient hypotheses are missing, so the intended Hénon-map problem cannot yet be stated completely.

Sources

• Local UnsolvedMath registry, permanent ID $801099$, exact source code catalogue entry, title "Henon Maps: Open Problems --- Question 51." The supplied registry attributes the problem to Berger and Bedford, with proposed year 2023, and labels it open; this status is supplied metadata, not an independently verified assessment.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Question 51. Supplied source URL: https://arxiv.org/abs/2312.03907. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source access or verification.