In this paper, we prove that for every compact set of the unit disk of
logarithmic capacity 0, there exists a Schur function both in the disk algebra
and in the Dirichlet space such that the associated composition operator is in
all Schatten classes (of the Dirichlet space), and for which the set of points
whose image touches the unit circle is equal to this compact set. We show that
for every bounded composition operator on the Dirichlet space and for every
point of the unit circle, the logarithmic capacity of the set of point having
this point as image is 0. We show that every compact composition operator on
the Dirichlet space is compact on the gaussian Hardy-Orlicz space; in
particular, it is in every Schatten class on the usual Hilbertian Hardy space.
On the other hand, there exists a Schur function such that the associated
composition operator is compact on the gaussian Hardy-Orlicz space, but which
is not even bounded on the Dirichlet space. We prove that the Schatten classes
on the Dirichlet space can be separated by composition operators. Also, there
exists a Schur function such that the associated composition operator is
compact on the Dirichlet space, but in no Schatten class
