  We study the mixing time of a random walk on the torus, alternated with a
Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the
mixing time of the random walk alone is $O(1/\epsilon^2)$, where $\epsilon$ is
the step size. Our main results show that for a class of Bernoulli maps, when
the random walk is alternated with the Bernoulli map $\varphi$ the mixing time
becomes $O(|\ln \epsilon|)$. We also study the \emph{dissipation time} of this
process, and obtain $O(|\ln \epsilon|)$ upper and lower bounds with explicit
constants.
