  Monomial mappings, $x\mapsto x^n$, are topologically transitive and ergodic
with respect to Haar measure on the unit circle in the complex plane. In this
paper we obtain an anologous result for monomial dynamical systems over
$p-$adic numbers. The process is, however, not straightforward. The result will
depend on the natural number $n$. Moreover, in the $p-$adic case we never have
ergodicity on the unit circle, but on the circles around the point 1.
