Henon Maps: Open Problems — Question 41

Definitions and mathematical statement

Let $\operatorname{Aut}_{\mathrm{poly}}({\mathbb{C}}^3)$ denote the group of polynomial maps ${\mathbb{C}}^3\to{\mathbb{C}}^3$ having polynomial inverses. Define $\mathrm{Tame}(3)$ to be its subgroup generated by affine automorphisms $z\mapsto Az+b$, where $A\in\operatorname{GL}_3({\mathbb{C}})$ and $b\in{\mathbb{C}}^3$, and triangular automorphisms. Using an upper triangular coordinate convention, the latter have the form \[ (z_1,z_2,z_3)\longmapsto \left(a_1z_1+P_1(z_2,z_3),\ a_2z_2+P_2(z_3),\ a_3z_3+c\right), \] where $a_1,a_2,a_3\in{\mathbb{C}}\setminus\{0\}$, $c\in{\mathbb{C}}$, $P_1\in{\mathbb{C}}[z_2,z_3]$, and $P_2\in{\mathbb{C}}[z_3]$.

Let $\mathcal C$ be the particular $\mathrm{CAT}(0)$ complex attributed in the supplied source to Lamy and Przytycky [LP], equipped with their isometric action of $\mathrm{Tame}(3)$. Here $\mathrm{CAT}(0)$ means that the geodesic metric satisfies the Euclidean comparison inequality for geodesic triangles. The extracted record does not specify the construction, metric, or action on $\mathcal C$; the question refers to that particular construction, rather than to an arbitrary isometric action on a $\mathrm{CAT}(0)$ space.

Under the conventional interpretation of $\lambda(f)$ as the first dynamical degree, put \[ \deg(g)=\max_{1\leq i\leq3}\deg(g_i), \qquad \lambda(f)=\lim_{n\to\infty}\left(\deg(f^n)\right)^{1/n}, \] where coordinate degrees are total polynomial degrees and $f^n$ is the $n$-fold iterate. The extracted record does not define $\lambda$, so this interpretation requires confirmation against the complete cited source.

Question. Characterize precisely the elements \[ \left\{f\in\mathrm{Tame}(3):\lambda(f)=1\right\} \] in terms of the isometries they induce on $\mathcal C$. Thus the requested characterization should give necessary and sufficient conditions expressed through this action. The supplied question does not propose a specific condition or a particular classification of isometries to use.

For context, the source attributes to Shestakov and Umirbaev [SU] the theorem that $\mathrm{Tame}(3)$ is a proper subgroup of $\operatorname{Aut}_{\mathrm{poly}}({\mathbb{C}}^3)$, contrasting this with equality of the analogous groups in dimension two. These are supplied background assertions; this draft does not independently verify them.

Short English statement

For a tame polynomial automorphism of ${\mathbb{C}}^3$, characterize when its dynamical degree equals one using its action on the particular $\mathrm{CAT}(0)$ complex attributed to Lamy and Przytycky. The meaning of $\lambda$ and the precise complex and action require confirmation from the cited source. The registry labels the question open; this is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $801089$, exact source problem code catalogue entry, title "Henon Maps: Open Problems --- Question 41." The record attributes the problem to Berger and Bedford, proposed in 2023, and supplies the status open; this status is not an independently verified literature assessment.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Question 41. Supplied source URL: https://arxiv.org/abs/2312.03907. The record reports access on 2026-10-16 and extraction by arxiv_latex; these provenance statements are supplied metadata, not a claim of access or verification by this draft.
• Reference label [SU]: Shestakov and Umirbaev, cited in the extracted question for the properness of $\mathrm{Tame}(3)$. No title, date, URL, or full bibliographic entry is supplied for [SU].
• Reference label [LP]: Lamy and Przytycky, as named in the extracted question, cited for the $\mathrm{CAT}(0)$ complex and isometric action. No title, date, URL, full bibliographic entry, or construction is supplied for [LP].