Henon Maps: Open Problems — Question 13

Definitions and mathematical statement

Let $\mathcal{WY}$ denote the parameter locus named in the supplied source, and let $H_{a,b}$ denote its Hénon map for $(a,b)\in\mathcal{WY}$. Write $\mathfrak K_{a,b}$ for the collection of kneading sequences denoted by $\mathfrak K$ for this map in the source. The record does not define $\mathcal{WY}$, specify the formula or normalization of $H_{a,b}$, or define these kneading sequences and their symbolic coding. These remain unresolved prerequisites; in particular, no real or complex parameter domain is stipulated here.

Question 13 (Berger--Bedford, 2023). Under the source's intended definitions, does every pair of distinct kneading sequences belonging to a map with parameters in $\mathcal{WY}$ determine its ordered parameter pair uniquely within $\mathcal{WY}$? If the sequences for different parameters are compared in a common symbolic coding, this asks whether \[ \begin{gathered} \text{for all }(a,b),(a',b')\in\mathcal{WY} \text{ and all }s,t\in\mathfrak K_{a,b}\cap\mathfrak K_{a',b'},\\ s\ne t\quad\Longrightarrow\quad(a,b)=(a',b'). \end{gathered} \] Existence of a realizing parameter pair is assumed by taking the two sequences from one such map; the question does not assert realizability of arbitrary symbolic pairs. The source further asks whether the two sequences thereby govern every other kneading sequence in $\mathfrak K_{a,b}$. This can be read as determination of the entire collection through uniqueness of the parameters, but the record does not specify whether "govern" requires any further reconstruction rule.

Short English statement

Berger and Bedford ask whether any two distinct kneading sequences of a Hénon map with parameters in $\mathcal{WY}$ uniquely identify those parameters and thereby determine all its other kneading sequences. The parameter locus, map normalization, symbolic coding, and meaning of "govern" require clarification from the source.

Sources

• Local UnsolvedMath registry: permanent ID $801061$, source problem code catalogue entry, title Henon Maps: Open Problems --- Question 13. The supplied registry attributes the problem to Berger and Bedford, with proposed year 2023, and labels it open; that label is supplied metadata, not an independently verified mathematical or literature assessment.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Question 13. Supplied source URL: https://arxiv.org/abs/2312.03907. The source record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not access or verification performed for this draft.