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Yurova \cite{Yu} and Anashin et al. \cite{AKY1, AKY2} characterize theergodicity of a 1-Lipschitz function on $\Z_2$ in terms of the van der Putexpansion. Motivated by their recent work, we provide the sufficient conditionsfor 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.

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