Henon Maps: Open Problems — Question 25

Definitions and mathematical statement

The supplied source asks whether the locus $\mathcal{H}_{\mathbb{C}}$ is connected. It does not define this locus or specify its topology. Consequently, let $(\mathcal{H}_{\mathbb{C}},\tau)$ denote the intended topological space, with both its membership conditions and its topology left unresolved.

Question 25. Is $\mathcal{H}_{\mathbb{C}}$ connected? Explicitly, is it true that for every $U,V\in\tau$, \[ \left(U\cup V=\mathcal{H}_{\mathbb{C}}\ \text{and}\ U\cap V=\varnothing\right) \quad\Longrightarrow\quad \left(U=\varnothing\ \text{or}\ V=\varnothing\right)? \] This expresses the connectivity question but does not specify a concrete parameter locus until the missing definitions are supplied.

The preceding source fragment refers to the fundamental group of $\mathcal{H}_{\mathbb{C}}$, with base point $(c,b)=(-4,0)$, and asserts surjectivity onto the group of shift-commuting automorphisms of $\{0,1\}^{\mathbb{Z}}$, citing references labeled [A,BS1]. Here $\{0,1\}^{\mathbb{Z}}$ is the set of bi-infinite binary sequences, and the shift is $\sigma((x_n)_{n\in\mathbb{Z}})=(x_{n+1})_{n\in\mathbb{Z}}$. The fragment begins mid-sentence and does not identify the asserted map or define the intended automorphism group. Its assertion is retained as supplied background, without treating it as a verified theorem or a complete additional formulation. The role of the parameter value $b=0$ and its inclusion in the intended locus also require clarification.

Short English statement

Question 25 asks whether the locus $\mathcal{H}_{\mathbb{C}}$ is connected, meaning that it cannot be partitioned into two nonempty open subsets. The supplied extraction omits the definition and topology of this locus and truncates the preceding background assertion. The registry labels the problem open; this is a supplied status, not an independently verified assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $801073$, source problem code catalogue entry, in the AMR Open Problem Lists collection. The record attributes the problem to Berger and Bedford, with proposed year 2023, and supplies the status open.
• Berger, Bedford, Henon Maps: A List of Open Problems (2023), Question 25. 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 independent source access or verification.
• The extracted background cites references labeled [A,BS1], but the supplied record contains no corresponding bibliographic entries. Their identities and their support for the truncated surjectivity assertion remain unresolved.