Henon Maps: Open Problems — Question 27

Definitions and mathematical statement

Let $f_{c,b}$ denote a real Hénon map in the parameterization intended by the supplied source, and write \[ H(c,b):=h_{\mathrm{top}}\!\left(f_{c,b}|_{{\mathbb{R}}^2}\right). \] Here $h_{\mathrm{top}}$ denotes topological entropy. The supplied record does not specify the formula for $f_{c,b}$, the admissible parameter set, or the convention for entropy on the noncompact space ${\mathbb{R}}^2$; these qualifications remain unresolved.

Requested construction (Question 27). Construct an algorithm which, for every real Hénon map $f_{c,b}$ in the intended family, computes a rigorous, nontrivial upper bound for $H(c,b)$. Thus, on an admissible input representing such a map, its output $U$ must satisfy \[ h_{\mathrm{top}}\!\left(f_{c,b}|_{{\mathbb{R}}^2}\right)\leq U \] with a mathematically rigorous guarantee. The source does not specify the input representation, output representation, termination requirements, or a quantitative meaning of "nontrivial." In particular, it does not explicitly require bounds of arbitrary accuracy or specify for which parameters a bound strictly below $\log 2$ must be obtained.

Supplied background and qualifications. The source cites [NBGM] and [NP] for attempts to obtain lower bounds. It attributes to [NBGM] the rigorous bound \[ h_{\mathrm{top}}\!\left(f_{c,b}|_{{\mathbb{R}}^2}\right)>0.46469 \] for the classical Hénon parameter, whose numerical value and normalization are not supplied, and states that this bound is believed to be close to optimal. It attributes to [BLS] the equivalence \[ h_{\mathrm{top}}\!\left(f_{c,b}|_{{\mathbb{R}}^2}\right)<\log 2 \quad\Longleftrightarrow\quad J(f_{c,b})\not\subseteq{\mathbb{R}}^2, \] where $J(f_{c,b})$ is the Julia set of the associated complex dynamical system. The precise Julia-set convention and hypotheses of this cited equivalence are not given in the record. The source states that no algorithm providing rigorous nontrivial upper bounds is available. These are supplied source assertions and qualifications, not independently verified results or a current literature assessment.

Short English statement

Construct an algorithm that gives a rigorous, nontrivial upper bound for the topological entropy of a real Hénon map. The supplied record labels this problem open. The map normalization, admissible parameters, entropy convention, computational input model, and required strength of the bound need clarification before a fully specified algorithmic statement is possible.

Sources

• UnsolvedMath local registry, permanent ID $801075$, source code catalogue entry, title "Henon Maps: Open Problems --- Question 27." The supplied record attributes the problem to Berger and Bedford, with proposed year 2023, and labels it open. These are registry metadata, not independent verification of attribution or status.
• Berger and Bedford, Hénon Maps: A List of Open Problems (2023), Question 27. Supplied source URL: https://arxiv.org/abs/2312.03907. The record reports access on 2026-10-16 and extraction by arxiv_latex; this is supplied provenance and does not indicate independent source access for this draft.
• References [NBGM] and [NP] are cited in the supplied statement for lower-bound work; [NBGM] is credited with the bound $H(c,b)>0.46469$ at the classical parameter. Reference [BLS] is credited with the entropy/complex-Julia-set equivalence reproduced above. Full bibliographic entries and separate links for these labels are not supplied, so their identities and exact hypotheses remain unresolved.