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

Abstract

Let ${\mathcal{L}}_1$ be the set of all mappings $f:{\mathbb{Z}}_p{\mathbb{Z}}_p$ of the space of all $p$-adic integers ${\mathbb{Z}}_p$ 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 ${\mathbb{Z}}_p$ if and only if $f$ induces a single cycle permutation on each residue ring $\mathbb{Z}/p^k\mathbb{Z}$ 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.

To be published in Proceedings of the 2-nd Int'l Conference on p-adic Mathematical Physics (15-25 Sept., 2005, Belgrade)

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