  We study electronic eigenstates on quasiperiodic lattices using a
tight-binding Hamiltonian in the vertex model. In particular, the
two-dimensional Penrose tiling and the three-dimensional icosahedral
Ammann-Kramer tiling are considered. Our main interest concerns the decay form
and the self-similarity of the electronic wave functions, which we compute
numerically for periodic approximants of the perfect quasiperiodic structure.
In order to investigate the suggested power-law localization of states, we
calculate their participation numbers and structural entropy. We also perform a
multifractal analysis of the eigenstates by standard box-counting methods. Our
results indicate a rather different behaviour of the two- and the
three-dimensional systems. Whereas the eigenstates on the Penrose tiling
typically show power-law localization, this was not observed for the
icosahedral tiling.
