  We study magnetic properties in the half-filled Hubbard model on the
Ammann-Beenker tiling. First, we focus on the domain structure with locally
eightfold rotational symmetry to examine the strictly localized confined states
for the tightbinding model. We count the number of vertices and confined states
in the larger domains generated by the deflation operations systematically.
Then, the fraction of the confined states, which plays an important role for
magnetic properties in the weak coupling limit, is obtained as $p=1/2\tau^2$,
where $\tau(=1+\sqrt{2})$ is the silver ratio. It is also found that the wave
functions for confined states are densely distributed in the system and thereby
the introduction of the Coulomb interactions immediately induces the finite
staggered magnetizations. Increasing the Coulomb interactions, the spatial
distribution of the magnetizations continuously changes to those of the
Heisenberg model. We discuss crossover behavior in the perpendicular space
representation and reveal the superlattice structure in the spatial
distribution of the staggered magnetizations.
