  Multifractal properties of the distribution of topological invariants for a
model of trajectories randomly entangled with a nonsymmetric lattice of
obstacles are investigated. Using the equivalence of the model to random walks
on a locally nonsymmetric tree, statistical properties of topological
invariants, such as drift and return probabilities, have been studied by means
of a renormalization group (RG) technique. The comparison of the analytical
RG--results with numerical simulations as well as with the rigorous results of
P.Gerl and W.Woess demonstrates clearly the validity of our approach. It is
shown explicitly by direct counting for the discrete version of the model and
by conformal methods for the continuous version that multifractality occurs
when local uniformity of the phase space (which has an exponentially large
number of states) has been broken.
