Ergodic dynamical systems over the cartesian power of the ring of $2$-adic integers

It is proved that, for any $1$-lipschitz ergodic map $F\colon\mathbb Z^k_2\mapsto\mathbb Z^k_2$, where $k>1$ and $k\in\mathbb N,$ there are $1$-lipschitz ergodic map $G\colon\mathbb Z_2\mapsto\mathbb Z_2$ and two bijections $H_k$, $T_{k,P}$ such that $G=H_k\circ T_{k,P}\circ F\circ H^{-1}_k$ and $F=H^{-1}_k\circ T_{k,P^{-1}}\circ G\circ H_k$.