Consider a compact manifold $M$ (e.g. a torus) equipped with
a smooth measure $\mu$ (e.g. Lebesgue measure in the case
of torus) as a probability space $(M,\mathcal M,\mu)$. Consider
an ergodic map $T:M \to M$ along with a smooth function
$p:M \to (0,1)$. Define a random walk along orbits of $T$ as follows:
a point $x$ jumps to $T x$ with probability $p(x)$ and
to $T^{-1} x$ with probability $1-p(x)$.
Is there a limiting distribution of such a random walk for a generic
initial point? Is it absolutely continuous with respect to $\mu$?
We shall present an answer for several essentially different
maps $T$.