Henon Maps: Open Problems — Question 7

Definitions and mathematical statement

Let $f_{c,b}$ denote the Hénon maps referred to in the supplied record, and let $\mathcal E_0$ denote its source-specific class. The record does not specify the formula for $f_{c,b}$, the parameter domain, the phase space, or the definition of $\mathcal E_0$. It also does not define mild dissipation or infinite renormalizability. These terms must retain their intended source meanings; the following is therefore a conditional formulation pending those definitions.

Once the ambient family $\mathscr H$, its topology $\tau$, and the relevant entropy and Morse--Smale conventions have been specified, write \[ \mathscr H_{+}=\{g\in\mathscr H:h_{\mathrm{top}}(g)>0\}, \qquad \mathscr H_{\mathrm{MS}}=\{g\in\mathscr H:g\text{ is Morse--Smale}\}, \] where $h_{\mathrm{top}}$ denotes topological entropy if this is the entropy intended by the source. The source does not specify that convention or the invariant space on which entropy is evaluated.

The two questions are whether, for every infinitely renormalizable, mildly dissipative map $f_{c,b}\in\mathcal E_0$, respectively, \[ f_{c,b}\in\overline{\mathscr H_{+}}^{\,\tau} \qquad\text{and}\qquad f_{c,b}\in\overline{\mathscr H_{\mathrm{MS}}}^{\,\tau}. \] Here closure means that every $\tau$-neighborhood of the given map contains a map of the indicated type. This expresses an approximation question; the ambient approximation family and topology are unresolved. In particular, the record does not say that the approximating maps must belong to $\mathcal E_0$, have the same $b$, or remain mildly dissipative.

The supplied background places the question in a regime described as $\lvert b\rvert$ being "larger but smaller than $1/4$", which it associates with mild dissipation. No lower threshold or comparison regime is provided, and the inequality alone is not supplied as a definition of mild dissipation. The record attributes to [CPT] a description of the dynamics in $\mathcal E_0$ in this regime, including the assertion that every $f_{c,b}\in\partial\mathcal E_0$ is infinitely renormalizable, with "renormalization periods eventually equal to $2$". It says this resolves a conjecture of Tresser under mild dissipation and that the conjecture remains open without that assumption. These are supplied background and status claims, not independently verified conclusions. The meaning of the stated periods, the topology defining $\partial\mathcal E_0$, and the precise converse suggested by the background require clarification.

Short English statement

For each infinitely renormalizable, mildly dissipative Hénon map in the source class $\mathcal E_0$, ask separately whether it can be approximated by maps with positive entropy and by Morse--Smale maps. The supplied record labels the problem open. Essential definitions and approximation conventions are missing, so this is a conditional statement requiring source clarification and human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $801055$, exact source problem code catalogue entry, title "Henon Maps: Open Problems --- Question 7". The record attributes the problem to Berger and Bedford, with proposed year 2023, and supplies the status label "open".
• Berger, Bedford, Henon Maps: A List of Open Problems (2023), Question 7. Supplied source URL: https://arxiv.org/abs/2312.03907. The provenance records extraction by arxiv_latex and access on 2026-10-16. This access date is supplied provenance; the source has not been independently visited or verified for this draft.
• The supplied statement refers to [CPT] and to a conjecture of Tresser, but provides no bibliographic identification for [CPT] or separate reference for the conjecture. Its claims about their results and open status are retained as source assertions.