  Consider deterministic random walks F: I x Z -> I x Z, defined by
F(x,n)=(f(x), K(x)+n), where f is an expanding Markov map on the interval I and
K: I->Z. We study the universality (stability) of ergodic (for instance,
recurrence and transience), geometric and multifractal properties in the class
of perturbations of the type G(x,n)=(f_n(x), L(x,n)+n) which are topologically
conjugate with F and f_n are expanding maps exponentially close to f when |n|
goes to infinity. We give applications of these results in the study of the
regularity of conjugacies between (generalized) infinitely renormalizable maps
of the interval and the existence of wild attractors for one-dimensional maps.
