Ergodic function on Z_p

초록

In this paper, we present ergodicity criteria for $1$-Lipschitz functions on $\text{Zp}$, 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 ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_3$, 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 $\text{Zp}$ is introduced, which leads to an efficient and practical method of constructing ergodic polynomials on ${\mathbb{Z}}_p$ 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 $\text{Zp},$ is provided where $\mu =\mu (p)$ = 3 for $p\in \{2,3\}$ and $\mu =2$ for $p\ge 5.$