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

TL;DR

This paper explores the structure of ergodic 1-Lipschitz maps over the Cartesian power of p-adic integers, showing they can be conjugated to simpler forms involving bijections and specific transformations.

Contribution

It demonstrates that any ergodic 1-Lipschitz map on \\mathbb{Z}_p^k can be conjugated to a map on \\mathbb{Z}_p using explicit bijections and transformations, extending understanding of p-adic dynamical systems.

Findings

Existence of conjugation between complex and simpler ergodic maps

Explicit construction of bijections and transformations

Structural characterization of ergodic maps over p-adic spaces

Abstract

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 $$G = H_{k} \circ T_{k,\;P}\circ F\circ H^{-1}_{k} \text{ and } F = H^{-1}_{k} \circ T_{k,\;P^{-1}}\circ G\circ H_{k}.$$