Ergodic dynamical systems over the Cartesian power of the ring of p-adic integers

For any 1-lipschitz ergodic map $F:; $\mathbb{Z}$^{k}{p} \mapsto $\mathbb{Z}$^{k}{p},;k>1\in$\mathbb{N}$,$thereare1-lipschitzergodicmap$G:; $\mathbb{Z}${p} \mapsto $\mathbb{Z}${p}$andtwobijection$H_k$,$T_{k,;P}$that$$G = H_{k} \circ T_{k,;P}\circ F\circ H^{-1}{k} $\text{ and }$ F = H^{-1}{k} \circ T_{k,;P^{-1}}\circ G\circ H_{k}.$$