Universal autohomeomorphisms of $\mathbb{N}^*$

TL;DR

This paper investigates the existence of universal autohomeomorphisms of the Stone-Čech remainder of the natural numbers, demonstrating their existence under CH and non-existence in models with only trivial autohomeomorphisms.

Contribution

It establishes the set-theoretic conditions under which universal autohomeomorphisms of * exist or do not exist, linking topology with set theory.

Findings

Under CH, universal autohomeomorphisms exist.

In models with only trivial autohomeomorphisms, none exist.

Set-theoretic assumptions determine the existence of such automorphisms.

Abstract

We study the existence of universal autohomeomorphisms of $\mathbb{N}^*$. We prove that $\mathsf{CH}$ implies there is such an autohomeomorphism and show that there are none in any model where all autohomeomorphisms of $\mathbb{N}^*$ are trivial.