  Recall that a $P$-set is a closed set $X$ such that the intersection of
countably many neighborhoods of $X$ is again a neighborhood of $X$. We show
that if $\mathfrak{t} = \mathfrak{c}$ then there is a minimal right ideal of
$(\beta \mathbb N,+)$ that is also a $P$-set. We also show that the existence
of such $P$-sets implies the existence of $P$-points; in particular, it is
consistent with ZFC that no minimal right ideal is a $P$-set. As an application
of these results, we prove that it is both consistent with and independent of
ZFC that the shift map is (up to isomorphism) the unique chain transitive
autohomeomorphism of $\mathbb N^*$.
