  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.
