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