The simple random walk on $Z$ can be simulated by picking randomly
uniformly in the unit interval, iterating $T(x)=2x \mod 1$, and moving the
"walker" one step to the right every time the orbit lands at $[0,1/2)$ and
one step to the left every time the orbit lands at $[1/2,1)$. What happens
if we replace the chaotic $T(x)=2x \mod 1$ by the zero entropy $R(x)=x+\alpha
\mod 1$ for $\alpha$ irrational? The resulting process, called the
"deterministic random walk", is again a recurrent zero drift walk – but
one that visits zero much more often. This is a joint work with Avila,
Dolgopyat, and Duriev.