Non-Archimedean dynamics of the complex shift

Abstract

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.