Toward the ergodicity of $p$-adic 1-Lipschitz functions represented by the van der Put series

TL;DR

This paper extends the characterization of ergodic 1-Lipschitz functions on p-adic integers using van der Put series, providing new sufficient conditions and alternative proofs for existing criteria.

Contribution

It offers new sufficient conditions for ergodicity of p-adic 1-Lipschitz functions and presents alternative proofs of known criteria using different basis representations.

Findings

Provided new sufficient conditions for ergodicity on df3pe1c integers.

Presented alternative proofs for ergodicity criteria on df3pe1c integers.

Extended previous characterizations to more general df3pe1c settings.

Abstract

Yurova \cite{Yu} and Anashin et al. \cite{AKY1, AKY2} characterize the ergodicity of a 1-Lipschitz function on $\Z_2$ in terms of the van der Put expansion. Motivated by their recent work, we provide the sufficient conditions for the ergodicity of such a function defined on a more general setting $\Z_p$. In addition, we provide alternative proofs of two criteria (because of \cite{AKY1, AKY2} and \cite{Yu}) for an ergodic 1-Lipschitz function on $\Z_2,$ represented by both the Mahler basis and the van der Put basis.