Ergodic Transformations of the Space of $p$-adic Integers

Abstract

Let $\mathcal L_1$ be the set of all mappings $f\colon\Z_p\Z_p$ of the space of all $p$-adic integers $\Z_p$ into itself that satisfy Lipschitz condition with a constant 1. We prove that the mapping $f\in\mathcal L_1$ is ergodic with respect to the normalized Haar measure on $\Z_p$ if and only if $f$ induces a single cycle permutation on each residue ring $\Z/p^k\Z$ modulo $p^k$, for all $k=1,2,3,...$. The multivariate case, as well as measure-preserving mappings, are considered also. Results of the paper in a combination with earlier results of the author give explicit description of ergodic mappings from $\mathcal L_1$. This characterization is complete for $p=2$. As an application we obtain a characterization of polynomials (and certain locally analytic functions) that induce ergodic transformations of $p$-adic spheres. The latter result implies a solution of a problem (posed by A.~Khrennikov) about the ergodicity of a perturbed monomial mapping on a sphere.