Henon Maps: Open Problems — Question 46

Definitions and mathematical statement

A number field is a finite extension $K/{\mathbb{Q}}$. A standard convention for a generalized Hénon map over $K$ is a polynomial automorphism \[ f:{\mathbb{A}}^2_K\longrightarrow{\mathbb{A}}^2_K,\qquad f(x,y)=(y,p(y)-ax), \] where $p\in K[t]$, $a\in K\setminus\{0\}$, and $d=\deg p\ge 2$. The supplied record does not define its intended class of generalized Hénon maps; in particular, whether compositions are included remains unresolved. Here algebraic degree means the maximum total degree of the coordinate polynomials.

For a map $f$ defined over $K$, a point $P\in{\mathbb{A}}^2(K)$ has exact period $n\ge1$ if \[ f^n(P)=P,\qquad f^j(P)\ne P\quad(1\le j<n). \] Its cycle consists of the $n$ distinct points $P,f(P),\ldots,f^{n-1}(P)$. Write \[ \operatorname{Per}_K(f)=\{P\in{\mathbb{A}}^2(K): f^n(P)=P\text{ for some }n\ge1\}. \]

Source question, with unresolved quantifiers made explicit. Over a number field $K$, can one construct infinite families of generalized Hénon maps of algebraic degree $d$, each possessing a $K$-rational cycle of length at least $d+3$? Under the interpretation that $K$ and $d$ are fixed and a family means pairwise distinct maps, the requested condition is an infinite collection $(f_i)_{i\in I}$ with \[ \deg f_i=d,\qquad P_i\in{\mathbb{A}}^2(K),\qquad \operatorname{per}_{f_i}(P_i)=n_i\ge d+3 \quad(i\in I). \] The record does not specify whether this is requested for every number field and every degree, for particular choices, or with either parameter allowed to vary. It also does not specify how members of a family are to be distinguished.

The further question asks whether one can construct degree-$d$ maps with $N_d$ periodic points such that \[ N_d-d\longrightarrow+\infty, \] or, more strongly, \[ \frac{N_d}{d}\longrightarrow+\infty \qquad(d\longrightarrow\infty). \] If these points are intended to be $K$-rational, a lower-bound formulation is to seek maps $f_d$ and positive integers $N_d$ satisfying \[ \deg f_d=d,\qquad \#\operatorname{Per}_K(f_d)\ge N_d \] and the respective growth condition. The source does not explicitly repeat the rationality requirement here, specify whether $N_d$ is an exact total or a lower bound, state whether $K$ remains fixed, or specify whether maps must exist in every sufficiently large degree or only along an unbounded sequence of degrees. These qualifications remain unresolved.

Short English statement

Question 46 asks for infinite families of degree-$d$ generalized Hénon maps with $K$-rational cycles of length at least $d+3$, and for periodic-point counts whose excess over $d$, or ratio to $d$, tends to infinity. The map class, family equivalence, field and degree quantifiers, and the precise meaning of the later periodic-point count require clarification. The supplied registry labels the problem open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $801094$, source code catalogue entry, title "Henon Maps: Open Problems --- Question 46." The registry attributes the problem to Berger and Bedford, with proposed year 2023, and labels it open.
• Berger and Bedford, Hénon Maps: A List of Open Problems (2023), Question 46. 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 statements, not an independent visit or verification.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry records creation and update on 2026-10-16.