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. (C) 2012 Elsevier Inc. All rights reserved.

In this paper we study lower ramification numbers of power series tangent to the identity that are defined over fields of positive characteristics p. Let g be such a series, then g has a fixed point at the origin and the corresponding lower ramification numbers of g are then, up to a constant, the degree of the first non-linear term of p-power iterates of g. The result is a complete characterization of power series g having ramification numbers of the form 2 ( 1 + p + âŠ + p n ) . Furthermore, in proving said characterization we explicitly compute the first significant terms of g at its pth iterate.

In this paper we consider the problem of finding the roots of p-adic functions. In the case, where the function is defined by a polynomial with integer p-adic coefficients, using Hensel's lifting lemma helps us find the roots of the p-adic function.

We generalize Hensel's lifting lemma for a wider class of p -adic functions, namely, the functions which satisfy the Lipschitz condition with constant , in particular, the functions of this class may be non-differentiable. The paper also presents an iterative procedure for finding approximate (in p -adic metric) values of the root of p^{α}-Lipschitz functions, thus generalizing the p-adic analogue of Newton's method for such a class of functions.