Ergodic Transformations in the Space of p-adic Integers

Let L1 be the set of all mappings f : Zp → Zp of the space of all p-adic integers Zp into itself that satisfy Lipschitz condition with a constant 1. We prove that the mapping f ∈L1 is ergodic with respect to the normalized Haar measure on Zp if and only if f induces a single cycle permutation on each residue ring Z/pZ modulo p, 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 L1. 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.