Ergodicity for uniformly differentiable modulo $p$ functions on ${\mathbb Z}_p$

TL;DR

This paper establishes an ergodicity criterion for uniformly differentiable functions on p-adic integers, linking ergodic behavior to coefficients in Mahler or van der Put expansions, with new proofs for key coefficient estimates.

Contribution

It provides explicit ergodicity conditions in terms of Mahler or van der Put coefficients and offers a new proof of coefficient estimation results.

Findings

Explicit ergodicity criterion in terms of coefficients

Connection between ergodicity and Mahler/van der Put expansions

Alternative proof of Mahler coefficient estimation

Abstract

We provide an ergodicity criterion for uniformly differentiable modulo $p$ functions on ${\mathbb Z}_p$ in regard to the minimal level of the reduced functions by showing that ergodic conditions are explicitly found in terms of the coefficients of Mahler or van der Put for each prime $p.$ To this end, it is essential to give an alternative, natural proof of Memi$\acute{c}$'s result regarding Mahler's coefficients estimation for uniformly differentiable modulo $p$ functions on ${\mathbb Z}_p.$