P-sets and minimal right ideals in N*

TL;DR

This paper investigates the existence of $P$-sets within minimal right ideals of the Stone-Čech compactification of natural numbers, linking set-theoretic assumptions to topological and dynamical properties.

Contribution

It establishes conditions under which minimal right ideals are $P$-sets and explores their implications for the existence of $P$-points and the uniqueness of certain homeomorphisms.

Findings

Existence of $P$-sets in minimal right ideals under $rak{t} = rak{c}$

Implication that $P$-sets imply $P$-points

Consistency results regarding the shift map's uniqueness

Abstract

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 $\mathbb N^*$.