  In this note we consider a symmetric random walk defined by a $(f,f^{-1})$
Kalikow type system, where $f$ is the time-one map of the geodesic flow
corresponding to an hyperbolic manifold. We provide necessary and sufficient
conditions for the existence of an stationary measure for the walk that is
equivalent to the volume in the corresponding unit tangent bundle. Some
dynamical consequences for the random walk are deduced in these cases.
