  We present numerical calculations of electronic structure and transport in
Penrose approximants. The electronic structure of perfect approximants shows a
spiky density of states and a tendency to localization that is more pronounced
in the middle of the band. Near the band edges the behavior is more similar to
that of free electrons. These calculations of band structure and in particular
the band scaling suggest an anomalous quantum diffusion when compared to normal
ballistic crystals. This is confirmed by a numerical calculation of quantum
diffusion which shows a crossover from normal ballistic propagation at long
times to anomalous, possibly insulator-like, behavior at short times. The time
scale t*(E) for this crossover is computed for several approximants and is
detailed. The consequences for electronic conductivity are discussed in the
context of the relaxation time approximation. The metallic like or non metallic
like behavior of the conductivity is dictated by the comparison between the
scattering time due to defects and the time scale t*(E).
