Aperiodic automorphisms of the Razak--Jacelon algebra

We completely classify aperiodic automorphisms of the Razak–Jacelon algebra $\mathcal{W}$ up to outer conjugacy. As a technical ingredient, based on the works of Connes, Herman–Ocneanu, Kishimoto, and Sato, we introduce a notion of weak stability for automorphisms of simple separable monotracial C${}^{\ast }$-algebras. We also consider automorphisms of UHF algebras of infinite type and the Jiang–Su algebra.