The Non-Archimedean Theory of Discrete Systems

In the paper, we study behaviour of discrete dynamical systems (automata) w.r.t. transitivity; that is, speaking loosely, we consider how diverse may be behaviour of the system w.r.t. variety of word transformations performed by the system: we call a system completely transitive if, given arbitrary pair a, b of finite words that have equal lengths, the system $${\mathfrak{A}}$$ , while evolution during (discrete) time, at a certain moment transforms a into b. To every system $${\mathfrak{A}}$$ , we put into a correspondence a family $${\mathcal{F}_{\mathfrak{A}}}$$ of continuous mappings of a suitable non-Archimedean metric space and show that the system is completely transitive if and only if the family $${\mathcal{F}_{\mathfrak{A}}}$$ is ergodic w.r.t. the Haar measure; then we find easy-to-verify conditions the system must satisfy to be completely transitive. The theory can be applied to analyse behaviour of straight-line computer programs (in particular, pseudo-random number generators that are used in cryptography and simulations) since basic CPU instructions (both numerical and logical) can be considered as continuous mappings of a (non-Archimedean) metric space $${\mathbb{Z}_{2}}$$ of 2-adic integers.