Local Discrete Holomorphic Dynamics — Open Problem OP7

Definitions and mathematical statement

Let $n\geq 1$, let $O$ denote the origin of ${\mathbb{C}}^n$, and let $\operatorname{End}({\mathbb{C}}^n,O)$ denote the holomorphic map germs $f:({\mathbb{C}}^n,O)\to({\mathbb{C}}^n,O)$ fixing $O$. A germ is superattracting if $Df(O)=0$. Let $f$ be such a germ with a first nonzero homogeneous term, and write \[ f(z)=H(z)+\sum_{j>k}H_j(z),\qquad k\geq 2, \] where each $H_j$ is a homogeneous polynomial map of degree $j$, and $H\not\equiv 0$ is homogeneous of degree $k$. Assume that $H$ is degenerate in the precise sense \[ \exists v\in{\mathbb{C}}^n\setminus\{O\}\quad H(v)=O. \] The critical locus is $\operatorname{Crit}(f)=\{z:\det Df(z)=0\}$, interpreted locally. Write $\mathcal P_f$ for the postcritical set intended by the source. The usual forward-image expression is \[ \bigcup_{m\geq 1} f^{\circ m}\left(\operatorname{Crit}(f)\right), \] with images taken where the iterates are defined. The supplied record does not specify whether this expression, its closure, or another local interpretation is intended; thus the convention for $\mathcal P_f$ remains unresolved. An analytic cone at $O$ means a complex analytic set germ invariant under complex scalar multiplication, understood as a germ.

Question (Abate, Problem OP7). For every germ $f$ satisfying the preceding hypotheses, if its postcritical set $\mathcal P_f$ is an analytic cone, must there exist a biholomorphic germ $\Phi:({\mathbb{C}}^n,O)\to({\mathbb{C}}^n,O)$ such that \[ \Phi\circ f\circ\Phi^{-1}=H? \] This formulation retains the source's unresolved postcritical-set convention. No finiteness, dominance, or isolated-zero hypothesis on $f$ or $H$ is supplied.

Short English statement

Abate asks whether a superattracting holomorphic germ whose first nonzero homogeneous term vanishes at a nonzero vector must be locally holomorphically conjugate to that term whenever its postcritical set is an analytic cone. The precise local convention for the postcritical set requires clarification.

Sources

• Local UnsolvedMath registry, permanent ID $800789$, source problem code catalogue entry, title "Local Discrete Holomorphic Dynamics --- Open Problem OP7." The supplied record attributes the problem to Abate, with proposed year 2011, and labels it open; that label is supplied registry information, not an independently verified status assessment.
• Abate, Open Problems in Local Discrete Holomorphic Dynamics (2011), Problem OP7. Supplied source URL: https://arxiv.org/abs/1107.4876. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not access or verification performed for this draft.
• Supplied upstream record ID $800789$, revision d371ee917ac9647fbb06cd6fc529deec6f3dda82; registry creation and update dates are both 2026-10-16.