In this paper, we present ergodicity criteria for 1-Lipschitz functions on ${\mathbf{Z}}_p$, in terms of the van der Put coefficients as well as the inherent data associated with the function. These criteria are applied to provide sufficient conditions for ergodicity of the 1-Lipschitz p-adic functions with special features, such as everywhere/uniform differentiability with respect to the Mahler expansion. In particular, the ergodicity criteria are obtained for certain 1-Lipschitz functions on ${\mathbf{Z}}_2$ and ${\mathbf{Z}}_3$, which are known as $\mathcal{B}$-functions, in terms of the Mahler and van der Put expansions. These functions are locally analytic functions of order 1 (and therefore contain polynomials). For arbitrary primes $p\ge 5$, an ergodicity criterion of $\mathcal{B}$-functions on ${\mathbf{Z}}_p$ is introduced, which leads to an efficient and practical method of constructing ergodic polynomials on ${\mathbf{Z}}_p$ that realize a given unicyclic permutation modulo p. Thus, a complete description of ergodic polynomials modulo $p^{\mu }$, which are reduced from all ergodic $\mathcal{B}$-functions on ${\mathbf{Z}}_p$, is provided where $\mu =\mu (p)=3$ for $p\in \{2,3\}$ and $\mu =2$ for $p\ge 5$.

在本文中，我们提出了 1-Lipschitz 函数的遍历性标准 ${\mathbf{Z}}_p$，就范德普系数以及与函数相关的固有数据而言。这些标准用于为具有特殊特征的 1-Lipschitz p - adic 函数的遍历性提供充分条件，例如关于马勒展开的处处/均匀可微性。特别是，获得了某些 1-Lipschitz 函数的遍历性准则${\mathbf{Z}}_2$ 和 ${\mathbf{Z}}_3$, 被称为 $\mathcal{乙}$- 函数，就马勒和范德普特展开而言。这些函数是 1 阶局部解析函数（因此包含多项式）。对于任意素数$p\ge 5$, 一个遍历性准则 $\mathcal{乙}$- 功能 ${\mathbf{Z}}_p$ 引入了一种高效实用的构造遍历多项式的方法 ${\mathbf{Z}}_p$实现给定的单环置换模p。因此，对遍历多项式取模的完整描述$p^{\mu }$，从所有遍历 $\mathcal{乙}$- 功能 ${\mathbf{Z}}_p$, 在那里提供 $\mu =\mu (p)=3$ 为了 $p\in \{2,3\}$ 和 $\mu =2$ 为了 $p\ge 5$.