Ergodic Transformations in the Space of p-Adic Integers

• Main Author: Anashin, Vladimir
• Format: Conference Proceeding
• Language: English
• ISSN: 0094-243X
• DOI: 10.1063/1.2193107

Let 1 be the set of all mappings f : p - > p of the space of all p-adic integers p into itself that satisfy Lipschitz condition with a constant 1. We prove that the mapping f 1 is ergodic with respect to the normalized Haar measure on p if and only if f induces a single cycle permutation on each residue ring /pk modulo pk, 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 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 due to A. Khrennikov about the ergodicity of a perturbed monomial mapping on a sphere.