Open Problems in Arithmetic Dynamics — Question 1

Definitions and mathematical statement

Let $S$ be an arbitrary set, let $T\subseteq S$ be fixed, and let $f:S\to S$. Write ${\mathbb{N}}_0=\{0,1,2,\ldots\}$, define $f^0=\operatorname{id}_S$ and $f^{n+1}=f\circ f^n$, and, for each $\alpha\in S$, define the hitting-time set \[ H_{f,T}(\alpha)=\{n\in{\mathbb{N}}_0:f^n(\alpha)\in T\}. \] The supplied question asks for a classification of maps $f$ for which every $H_{f,T}(\alpha)$ has a suitably nice structure, giving finite unions of one-sided arithmetic progressions as an example. The phrase "nice structure" is not defined, so this broader request does not specify a unique mathematical property.

Under the positive-step convention, the illustrative classification problem is to characterize all maps $f:S\to S$ satisfying \[ \forall\alpha\in S\quad \exists r_\alpha\in{\mathbb{N}}_0\quad \exists (a_{\alpha,j},d_{\alpha,j})_{j=1}^{r_\alpha} \in\left({\mathbb{N}}_0\times\{1,2,3,\ldots\}\right)^{r_\alpha} \quad H_{f,T}(\alpha) =\bigcup_{j=1}^{r_\alpha} \{a_{\alpha,j}+k d_{\alpha,j}:k\in{\mathbb{N}}_0\}. \] The union is empty when $r_\alpha=0$; its parameters may depend on $\alpha$. The source does not specify whether singleton progressions are permitted. If they are permitted, the displayed condition instead allows $d_{\alpha,j}=0$, with $\{a_{\alpha,j}+k\cdot0:k\in{\mathbb{N}}_0\}=\{a_{\alpha,j}\}$. These conventions give different properties and remain unresolved here. No additional structure on $S$, restriction on $T$, or regularity hypothesis on $f$ is supplied.

Short English statement

For a fixed subset $T$ of a set $S$, classify self-maps whose every forward orbit visits $T$ at times having a specified simple structure. The source suggests finite unions of one-sided arithmetic progressions, but leaves both the broader structural target and the treatment of singleton progressions unspecified.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $800378$, exact source problem code catalogue entry, title "Open Problems in Arithmetic Dynamics --- Question 1." The registry labels the problem open; this is a supplied status, not an independently verified assessment.
• Benedetto, DeMarco, Ingram, Jones et al., Current Trends and Open Problems in Arithmetic Dynamics (2018), Question 1. Supplied source URL: https://arxiv.org/abs/1806.04980. The registry attributes the question to these authors and records proposed year 2018.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance statements; the source has not been independently accessed or verified for this draft.