Ergodic dynamical systems over the Cartesian power of the ring of p-adic integers
This provides a foundational result in pure mathematics for researchers studying p-adic dynamics and ergodic theory, but it is incremental as it extends known reduction techniques to higher dimensions.
The paper tackles the problem of characterizing ergodic dynamical systems over the Cartesian power of p-adic integers by showing that any 1-Lipschitz ergodic map in higher dimensions can be reduced to a 1-Lipschitz ergodic map in one dimension through specific bijections, establishing a structural equivalence.
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 $$G = H_{k} \circ T_{k,\;P}\circ F\circ H^{-1}_{k} \text{ and } F = H^{-1}_{k} \circ T_{k,\;P^{-1}}\circ G\circ H_{k}.$$