  Deterministic walks over a random set of points in one and two dimensions
(d=1,2) are considered. Points (``cities'') are randomly scattered in R^d
following a uniform distribution. A walker (a ``tourist''), at each time step,
goes to the nearest neighbor city that has not been visited in the past \tau
steps. Each initial city leads to a trajectory composed of a transient part and
a final p-cycle attractor. The distribution of transient times, p-cycles and
number of cities per attractor are studied. It is shown numerically that
transient times (for d=1,2) follow a Poisson law with a $\tau$ dependent decay
but the density of p-cycles follows a power law D(p) \propto p^{-\alpha(\tau)}
for d=2. For large \tau, the expoent tends to \alpha ~ 5/2. Some analytical
results are given for the d=1 case. Since the power law is robust and does not
depend on free parameters, this system presents ``generic scale invariance''.
Applications to animal exploratory behavior and other local minimization
problems are suggested.
