Characterization of Ergodicity of p-Adic Dynamical Systems by Using the van der Put Basis

Characterization of ergodicity of p-adic dynamical systems by using the van der Put Basis is presented. Results for p-adic maps that provide the opportunity to characterize their important properties, including ergodicity and the preservation of the Haar measure, in terms of coefficients with respect to the van der Put basis, are obtained. Put basis is related to the zero-dimensional topology of these fields, consists of characteristic functions of p-adic balls, and the basic point in the construction of this basis is the continuity of the characteristic function of a p-adic ball. The space-p is equipped by the natural probability measure known as the Haar measure that is normalized. It is proved that for every locally compatible function that is uniformly continuous and dense the function is uniquely defined by its values at the points.