P-sets and minimal right ideals in N*

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 $(β\mathbb N,+)$ 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 $\mathbb N^*$.

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