Open Problems in Arithmetic Dynamics — Question 35

Definitions and mathematical statement

Write $K=\mathbb{C}_v$ for the field appearing in the supplied source statement, and let $f\in K(z)$ be a rational function of degree $d\geq 2$. The supplied excerpt does not specify the class of fields denoted by $\mathbb{C}_v$, the ambient analytic space in which components are taken, or the meanings of "indifferent component" and "disk." Consequently, the formulation below is conditional on those intended source conventions; these missing definitions must be recovered before the statement is mathematically complete.

Under those conventions, a periodic cycle of components is a finite orbit $(U_0,\ldots,U_{n-1})$ of distinct components under the action induced by $f$, where $n\geq 1$ and $U_i$ maps to $U_{i+1\bmod n}$. Cyclic relabelings describe the same cycle. Let $N_K(f)$ be the number, possibly infinite, of such cycles whose components are indifferent and are not disks, interpreted according to the source conventions.

Question 35 (Benedetto, DeMarco, Ingram, Jones et al., 2018). Can $N_K(f)$ be infinite? Equivalently, does the finiteness assertion \[ \forall f\in K(z),\qquad \deg(f)\geq 2\ \Longrightarrow\ N_K(f)<\infty \] hold for each field $K=\mathbb{C}_v$ allowed by the source?

If finiteness holds, is the number bounded in terms of the degree? For a fixed admissible field $K$, this asks whether \[ \forall d\in\mathbb{Z}_{\geq 2}\ \exists B_K(d)\in\mathbb{Z}_{\geq 0}\ \forall f\in K(z),\qquad \deg(f)=d\ \Longrightarrow\ N_K(f)\leq B_K(d). \] Determine whether the answers depend on $K$. In particular, distinguish a bound allowed to depend on $K$ from a single function $B\colon\mathbb{Z}_{\geq 2}\to\mathbb{Z}_{\geq 0}$ satisfying \[ N_K(f)\leq B(\deg f) \] for every admissible $K$ and every $f\in K(z)$ of degree at least two. The supplied excerpt does not resolve which field-uniform interpretation of the requested bound is intended.

Short English statement

For a rational map of degree at least two over the source's field $\mathbb{C}_v$, ask whether there can be infinitely many periodic cycles of indifferent components that are not disks. If there are always finitely many, ask whether their number admits a bound depending only on the degree, and how the answers depend on the field. The field class and analytic terminology require clarification from the source.

Sources

• Local UnsolvedMath registry: permanent ID $800412$, exact source problem code catalogue entry, title "Open Problems in Arithmetic Dynamics --- Question 35." The supplied registry labels the problem open; this is supplied status metadata, not an independently verified mathematical or literature assessment.
• Benedetto, DeMarco, Ingram, Jones et al., Current Trends and Open Problems in Arithmetic Dynamics (2018), Question 35. Supplied source URL: https://arxiv.org/abs/1806.04980. Attribution and year are reproduced as supplied.
• The source record reports access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; no independent source visit or verification is claimed.