  We discuss a model in which a quantum particle passes through $\delta$
potentials arranged in an increasingly sparse way. For infinitely many barriers
we derive conditions, expressed in terms ergodic properties of wave function
phases, which ensure that the point and absolutely continuous parts are absent
leaving a purely singularly continuous spectrum. For a finite number of
barriers, the transmission coefficient shows extreme sensitivity to the
particle momentum with fluctuation in many different scales. We discuss a
potential application of this behavior for erasing the information carried by
the wave function.
