On the dynamics of endomorphisms of the direct product of two free groups

We prove that Brinkmann’s problems are decidable for endomorphisms of $F_n\times F_m$: given $(x,y),(z,w)\in F_n\times F_m$ and $\mathrm{\Phi }\in \text{End}(F_n\times F_m)$, it is decidable whether there is some $k\in \mathbb{N}$ such that $(x,y){\mathrm{\Phi }}^k=(z,w)$ (or $(x,y){\mathrm{\Phi }}^k\sim (z,w)$). We also prove decidability of a two-sided version of Brinkmann’s conjugacy problem for injective endomorphisms which, from the work of Logan, yields a solution to the conjugacy problem in ascending HNN-extensions of $F_n\times F_m$. Finally, we study the dynamics of automorphisms of $F_n\times F_m$ at the infinity, proving that that their dynamics at the infinity is asymptotically periodic, as occurs in the free and free-abelian times free cases.