Automata as p-adic Dynamical Systems [article]

Livat Tyapaev
2018 arXiv   pre-print
The automaton transformation of infinite words over alphabet F_p={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 Z_p of p-adic integers. The objects of the study are non-Archimedean dynamical systems generated by automata mappings on the space 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
more » ... erators 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.
arXiv:1709.02644v2 fatcat:xzs6j7qpird6hh7e2wnojlrzwi