  Starting from the geometrical interpretation of the R\'enyi entropy, we
introduce further extensive generalizations and study their properties. In
particular, we found the probability distribution function obtained by the
MaxEnt principle with generalized entropies. We prove that for a large class of
dynamical systems subject to random perturbations, including particle transport
in random media, these entropies play the role of Liapunov functionals. Some
physical examples, which can be treated by the generalized R\'enyi entropies
are also illustrated.
