  An important ingredient in the Ferrero--Washington proof of the vanishing of
cyclotomic $\mu$-invariant for Kubota--Leopoldt $p$-adic $L$-functions is an
equidistribution result which they established using the Weyl criterion. The
purpose of our manuscript is to provide an alternative proof by adopting a
dynamical approach. A key ingredient to our methods is studying an ergodic
skew-product map on $\mathbb{Z}_p \times [0,1]$, which is then suitably
identified as a factor of the $2$-sided Bernoulli shift on the sample space
$\{0,1,2,\cdots,p-1\}^{\mathbb{Z}}$.
