ON THE DYNAMICS OF ENDOMORPHISMS OF THE DIRECT PRODUCT OF TWO FREE GROUPS

Abstract 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 $\Phi \in \mathrm {End}(F_n\times F_m)$ , it is decidable whether there is some $k\in \mathbb {N}$ such that $(x,y)\Phi ^k=(z,w)$ (or $(x,y)\Phi ^k\sim (z,w)$ ). 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.