Several Complex Variables Problem List — Problem 66

Definitions and mathematical statement

For each $c\in{\mathbb{C}}$, let \[ F_c:{\mathbb{C}}^2\longrightarrow{\mathbb{C}}^2,\qquad F_c(z,w)=(z^2+c-w,z). \] The source specifies that $F_c$ is a symplectic automorphism: it is biholomorphic and satisfies \[ F_c^*({\,\mathrm{d}} z\wedge{\,\mathrm{d}} w)={\,\mathrm{d}} z\wedge{\,\mathrm{d}} w. \] For an integer $k\geq1$, a periodic orbit is represented by points $p_0,\ldots,p_k\in{\mathbb{C}}^2$ satisfying \[ p_{i+1}=F_c(p_i)\quad(0\leq i<k),\qquad p_k=p_0. \] Here $k$ need not be the least period; the source does not impose a least-period requirement.

Question attributed to M. Herman. Do there exist $c\in{\mathbb{C}}$, an integer $k\geq1$, and such a periodic orbit for which $p_0$ belongs to a Siegel domain?

Unresolved qualification. The supplied record does not define "Siegel domain" in this dynamical setting. In particular, it does not specify the required conjugacy or rotation conditions, or whether the domain is associated with $F_c$ itself or with an iterate. The question above retains this source terminology; a unique self-contained mathematical formulation requires clarification of that term.

Short English statement

Can some member of the family $F_c(z,w)=(z^2+c-w,z)$, with $c\in{\mathbb{C}}$, have a periodic point lying in a Siegel domain? The intended meaning of "Siegel domain" remains unspecified in the supplied record.

Sources

• Local UnsolvedMath registry, permanent ID $800267$, source code catalogue entry, title "Several Complex Variables Problem List --- Problem 66." Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 66. Supplied source URL: https://arxiv.org/abs/math/9509202. The source question is signed M. Herman; the registry's proposed_by field names Buzzard, Fornaess, and Krantz.
• The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. This is supplied provenance, not an independent source visit or verification. The registry labels the problem "open"; that label is supplied curation, not an independently verified mathematical or literature assessment.