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

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:Zp→Zpf:Zp→Zp. 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:Zp→Zpf:Zp→Zp 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.