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

Yurova \cite{Yu} and Anashin et al. \cite{AKY1, AKY2} characterize the ergodicity of a 1-Lipschitz function on ${\mathbb{Z}}_2$ 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 ${\mathbb{Z}}_p$. In addition, we provide alternative proofs of two criteria (because of \cite{AKY1, AKY2} and \cite{Yu}) for an ergodic 1-Lipschitz function on ${\mathbb{Z}}_2,$ represented by both the Mahler basis and the van der Put basis.