  A $*$-reduction between two equivalence relations is a Baire measurable
reduction which preserves generic notions, i.e., preimages of meager sets are
meager. We show that a $*$-reduction between orbit equivalence relations
induces generically an embedding between the associated Becker graphs. We
introduce a notion of dimension for Polish $G$-spaces which is generically
preserved under $*$-reductions. For every natural number $n$ we define a free
action of $S_{\infty}$ whose dimension is $n$ on every invariant Baire
measurable non-meager set. We also show that the $S_{\infty}$-space which
induces the equivalence relation $=^{+}$ of countable sets of reals is
$\infty$-dimensional on every invariant Baire measurable non-meager set. We
conclude that the orbit equivalence relations associated to all these actions
are pairwise incomparable with respect to $*$-reductions.
