The theory of p-adic (and more generally non-Archimedean) dynamical systems arose from the mixing of various (and very different) research flows (see e.g. [20], [14], [31], [32], [33], [34], [35], [37], [38], [39], [36], [49], [73], [74], [75], [110], [142], [122], [141], [149], [151], [150], [192], [193], [223], [227], [249], [250], [247], [248], [264], [265], [266], [312], [382], [383], [384], [453], [454], [452], [455], and the references therein):

• number-theoretic methods in the study of monomial dynamical systems

• theory of ergodic dynamical systems

• p-adic (and more generally non-Archimedean) mathematical physics

• p-dynamical systems in cryptography

• p-adic modeling of cognition and psychology.

The aim of this chapter is to present some recent results about p-adic dynamics. Here discrete dynamical systems based on iterations of functions belonging to a special functional class, namely 1-Lipschitz functions, will be considered. The importance of this class for theory of p-adic dynamical systems was emphasized in a series of pioneering works by V. Anashin [31], [32], [33]. Then some interesting results about such discrete dynamics were obtained in joint works by V. Anashin, A. Khrennikov, and E.Yurova [34], [35], [37], [38], [39], [36], [265], [266], [453], [454], [452], [455].

Let be a ring of p-adic integers. We recall that the space is equipped with a natural probability measure, namely the Haar measure μp normalized as. Balls B−r (a) of non-zero radii constitute the base of the corresponding σ-algebra of measurable subsets, μp (B−r (a)) = p−p. The measure μp is a regular Borel measure, so all continuous transformations are measurable with respect to μp. As usual, a measurable mapping is called measure-preserving if for each measurable subset S ⊂ Zp A measure-preserving mapping is called ergodic implies either

Consider a map. The theory of (discrete) dynamical system studies trajectories (orbits), i.e. sequences of iterations of f,.