Automata as $p$-adic Dynamical Systems

The automaton transformation of infinite words over alphabet Fp = {0, 1, . . . , p− 1}, where p is a prime number, coincide with the continuous transformation (with respect to the p-adic metric) of a ring Zp of p-adic integers. The objects of the study are non-Archimedean dynamical systems generated by automata mappings on the space Zp. Measure-preservation (with the respect to the Haar measure) and ergodicity of such dynamical systems plays an important role in cryptography (e.g. for pseudo-random generators and stream cyphers design). The possibility to use p-adic methods and geometrical images of automata allows to characterize of a transitive (or, ergodic) automata. We investigate a measure-preserving and ergodic mappings associated with synchronous and asynchronous automata. We have got criterion of measure-preservation for an n-unit delay mappings associated with asynchronous automata. Moreover, we have got a sufficient condition of ergodicity of such mappings.