Criteria of measure-preserving for p -adic dynamical systems in terms of the van der Put basis

Abstract

This paper is devoted to (discrete) p -adic dynamical systems, an important domain of algebraic and arithmetic dynamics. We consider the following open problem from theory of p -adic dynamical systems. Given continuous function f : Z p → Z p . Let us represent it via special convergent series, namely van der Put series. How can one specify whether this function is measure-preserving or not for an arbitrary p ? In this paper, for any prime p , we present a complete description of all compatible measure-preserving functions in the additive form representation. In addition we prove the criterion in terms of coefficients with respect to the van der Put basis determining whether a compatible function f : Z p → Z p preserves the Haar measure. ► A novel approach for study of discrete (in general nonsmooth) p -adic dynamical systems based on usage of van der Put series is elaborated. ► Criteria of measure-preserving are presented. ► The additive form representation of measure-preserving dynamical systems is found.