Decidability of the Brinkmann problems for endomorphisms of the free group (Q6614078)

Let \(F_{n}\) be the free group of finite rank \(n \geq 2\) and let \(\mathbb{N}\) denote the set of natural numbers including \(0\). In the paper under review, the authors prove the following: \N\NTheorem. Given two elements \(u,v \in F_{n}\) and an endomorphism \(\varphi \in \mathrm{End}(F_{n})\), it is algorithmically decidable whether\N\begin{itemize} \N\item[(i)] there exists some \(k \in \mathbb{N}\) such that \((u)\varphi^{k}=v\); \N\item[(ii)] there exists some \(k \in \mathbb{N}\) such that \((u)\varphi^{k}\) is conjugate to \(v\).\N\end{itemize}\N\NThis result generalizes the analogous claims on automorphisms and monomorphims of the free group proved by \textit{P. Brinkmann} [J. Algebra 324, No. 5, 1083--1097 (2010; Zbl 1209.20023)] and \textit{A. D. Logan} [``The conjugacy problem for ascending HNN-extensions of free groups'', Preprint , \url{arXiv:2209.04357}], respectively.