Automata as $p$-adic Dynamical Systems

Abstract

The automaton transformation of infinite words over alphabet $\mathbb F_p=\{0,1,\ldots,p-1\}$, where $p$ is a prime number, coincide with the continuous transformation (with respect to the $p$-adic metric) of a ring $\mathbb Z_p$ of $p$-adic integers. The objects of the study are non-Archimedean dynamical systems generated by automata mappings on the space $\mathbb Z_p$. Measure-preservation (with the respect to the Haar measure) and ergodicity of such dynamical systems plays an important role in cryptography (e.g. for pseudorandom 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.