Automata as pp-adic Dynamical Systems

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

Non-Archimedean dynamics of the complex shift

An (asynchronous) automaton transformation of one-sided infinite words over p-letter alphabet Fp = Z/pZ, where p is a prime, is a continuous transformation (w.r.t. the p-adic metric) of the ring of p-adic integers Zp. Moreover, an automaton mapping generates a non-Archimedean dynamical system on Zp. Measure-preservation and ergodicity (w.r.t. the Haar measure) of such dynamical systems play an important role in cryptography (e.g., in stream cyphers). The aim of this paper is to present a novel way of realizing a complex shift in p-adics. In particular, we introduce conditions on the Mahler expansion of a transformation on the p-adics which are sufficient for it to be complex shift. Moreover, we have a sufficient condition of ergodicity of such mappings in terms of Mahler expansion