Henon Maps: Open Problems — Question 11

Definitions and mathematical statement

The supplied record names a family of maps $f_{c,b}$, but does not specify its formula, phase space, or admissible parameter values. It also does not define the quantities $\lambda^-(p)$ and $\lambda^+(p)$. These omissions remain unresolved below.

In the usual setting of a diffeomorphism, a periodic saddle is a hyperbolic periodic point with both stable and unstable directions. For such a point $p$, let $\mathcal O(p)$ denote its periodic orbit, and let $W^s(\mathcal O(p))$ and $W^u(\mathcal O(p))$ denote the stable and unstable manifolds of that orbit. Under the conventional definition, two periodic saddles $p,q$ are homoclinically related when both \[ W^u(\mathcal O(p))\pitchfork W^s(\mathcal O(q))\ne\varnothing, \qquad W^u(\mathcal O(q))\pitchfork W^s(\mathcal O(p))\ne\varnothing, \] where $\pitchfork$ indicates a transverse intersection. The supplied excerpt does not specify the ambient setting or explicitly confirm this convention.

Question 11 (Berger, Bedford, 2023). For every map $f_{c,b}$ in the intended family and every infinite sequence $(p_n)_{n\ge1}$ of its periodic saddles such that no two distinct terms are homoclinically related, must \[ \lim_{n\to\infty}\min\left\{|\lambda^-(p_n)|,|\lambda^+(p_n)|\right\}=0? \] Equivalently, must every $\varepsilon>0$ admit $N\ge1$ such that \[ \min\left\{|\lambda^-(p_n)|,|\lambda^+(p_n)|\right\}<\varepsilon \qquad\text{for all }n\ge N? \] This is a statement template retaining the source notation. A fully specified mathematical question requires the missing family, domains, parameter restrictions, and definitions of $\lambda^\pm$; in particular, no choice between eigenvalue multipliers and Lyapunov exponents is made here.

Short English statement

The question asks whether every infinite sequence of pairwise homoclinically unrelated periodic saddles of a fixed map $f_{c,b}$ must have the smaller absolute value of its two quantities $\lambda^-$ and $\lambda^+$ tend to zero. The supplied excerpt leaves the map family and these quantities undefined. Its status is recorded as open in the supplied registry; this is not an independently verified assessment.

Sources

• Local UnsolvedMath registry, permanent ID $801059$, exact source problem code catalogue entry, title "Henon Maps: Open Problems --- Question 11." The record attributes the problem to Berger, Bedford, with proposed year 2023, and supplies the status "open."
• Berger, Bedford, Henon Maps: A List of Open Problems (2023), Question 11. Supplied source URL: https://arxiv.org/abs/2312.03907. This reference is supplied by the registry and was not independently consulted for this draft.
• The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. This is supplied dated provenance, not independently verified source access or literature assessment. The record identifies upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.