ИСТИНА

ABSTRACT: During the course, by the automaton we mean a transducer, i.e., a sequential machine which maps symbols of a finite input alphabet to symbols of a finite output alphabet so that any output symbol depends on corresponding input symbol and of current state of the machine, whereas any input symbol changes current state of the machine. An automaton whose input and output alphabets consist of p symbols can naturally be associated to a mapping from p-adic integers to p-adic integers; the mapping is 1-Lipschitz w.r.t. p-adic metric. Moreover, any 1-Lipschitz mapping from p-adic integers to p-adic integers is a mapping associated with some automaton. Therefore one can study behaviour of automata by studying dynamics of corresponding 1-Lipschitz mappings, the automata functions. The mini-course is devoted to dynamical (especially, ergodic) and other properties of automata functions as functions from p-adic integers to p-adic integers and to various applications of corresponding theory in computer science, cryptography, pseudorandom numbers, physics, etc. https://www.math.cinvestav.mx/p-adic2020/courses