Using Bernoulli maps to accelerate mixing of a random walk on the torus

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 / ε 2 ) O(1/\varepsilon ^2) , where ε \varepsilon 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  φ \varphi the mixing time becomes O ( | ln ⁡ ε | ) O(\lvert \ln \varepsilon \rvert ) . We also study the dissipation time of this process, and obtain  O ( | ln ⁡ ε | ) O(\lvert \ln \varepsilon \rvert ) upper and lower bounds with explicit constants.