P-adic Deterministic and Random Dynamics

This is the first monograph in the theory of p-adic (and more general non-Archimedean) dynamical systems. The theory of such systems is a new intensively developing discipline on the boundary between the theory of dynamical systems, theoretical physics, number theory, algebraic geometry and non-Archimedean analysis. Investigations on p-adic dynamical systems are motivated by physical applications (p-adic string theory, p-adic quantum mechanics and field theory, spin glasses) as well as natural inclination of mathematicians to generalize any theory as much as possible (e.g., to consider dynamics not only in the fields of real and complex numbers, but also in the fields of p-adic numbers). The main part of the book is devoted to discrete dynamical systems: cyclic behavior (especially when p goes to infinity), ergodicity, fuzzy cycles, dynamics in algebraic extensions, conjugate maps, small denominators. There are also studied p-adic random dynamical system, especially Markovian behavior (depending on p). In 1997 one of the authors proposed to apply p-adic dynamical systems for modeling of cognitive processes. In applications to cognitive science the crucial role is played not by the algebraic structure of fields of p-adic numbers, but by their tree-like hierarchical structures. In this book there is presented a model of probabilistic thinking on p-adic mental space based on ultrametric diffusion. There are also studied p-adic neural network and their applications to cognitive sciences: learning algorithms, memory recalling. Finally, there are considered wavelets on general ultrametric spaces, developed corresponding calculus of pseudo-differential operators and considered cognitive applications.Audience: This book will be of interest to mathematicians working in the theory of dynamical systems, number theory, algebraic geometry, non-Archimedean analysis as well as general functional analysis, theory of pseudo-differential operators; physicists working in string theory, quantum mechanics, field theory, spin glasses; psychologists and other scientists working in cognitive sciences and even mathematically oriented philosophers.