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

TL;DR

This paper provides a complete criterion for identifying measure-preserving functions in $p$-adic dynamical systems using van der Put series coefficients, advancing the understanding of measure-preservation in algebraic dynamics.

Contribution

It introduces a new criterion based on van der Put basis coefficients to determine measure-preserving properties of $p$-adic functions.

Findings

Complete description of measure-preserving functions in additive form

Criterion for measure-preservation in terms of van der Put coefficients

Applicable to all primes $p$

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.