Henon Maps: Open Problems — Question 6

Definitions and mathematical statement

Let $f_{c,b}$ denote the real Hénon family intended by the source, and let $\mathcal H_{{\mathbb{R}}}\subseteq{\mathbb{R}}^2$ denote its admissible parameter pairs. The supplied excerpt does not specify the formula for $f_{c,b}$, the admissible parameter domain, or the definition of mild dissipativity. Consequently, the following is a formal statement of the intended question, pending those definitions.

Define \[ \mathcal M=\left\{(c,b)\in\mathcal H_{{\mathbb{R}}}: f_{c,b}\text{ is mildly dissipative in the source's sense}\right\}. \] Determine $\mathcal M$. In particular, is it true that \[ \forall (c,b)\in\mathcal H_{{\mathbb{R}}},\qquad |b|<1\ \Longrightarrow\ (c,b)\in\mathcal M\,? \] The treatment of $b=0$ must be specified: the displayed inequality includes zero, but the excerpt does not say whether degenerate, noninvertible members belong to the intended family.

The supplied source reports that Wiman's theorem yields mild dissipativity for Hénon maps with $|b|<1/4$, describing this as information for all such maps whose Jacobian lies in $(-1/4,1/4)$, beyond a regime expressed as $|b|\ll1$. This is a supplied background assertion, not a result verified here. The source expects the property to extend beyond that bound.

The excerpt further reports that, under an antecedent assumption not included in the record, restriction to any open topological disc $D\subset{\mathbb{R}}^2$ compactly mapped inside itself is semiconjugate to a nontrivial continuous map of a real tree. It also mentions a closing lemma and a description of zero entropy dynamics. The missing assumption, the meaning of nontriviality, and the precise hypotheses of these background consequences remain unspecified; they are not asserted here as unconditional statements.

Short English statement

Determine which real Hénon maps $f_{c,b}$ are mildly dissipative, and in particular whether every admissible real member with $|b|<1$ has this property. The supplied excerpt reports the bound $|b|<1/4$ as established background. A complete formulation requires the source's family convention, definition of mild dissipativity, and treatment of $b=0$.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $801054$, exact source problem code catalogue entry, titled Henon Maps: Open Problems --- Question 6. The registry attributes the question to Berger and Bedford, with proposed year 2023, and labels it open. This is a supplied status label, not an independently verified assessment.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Question 6. 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 access or verification here.
• The excerpt invokes Wiman's theorem and cites [DL, MR3213832] as related background. No complete bibliographic entry for that citation, nor a precise formulation of the invoked theorem, is supplied.