Uniqueness of almost periodic outer flows on the hyperfinite type $\mathrm{II}_1$ factor

We show that any almost periodic outer flow $\alpha : \mathbb R \curvearrowright R$ on the hyperfinite type $\mathrm{II}_1$ factor with Connes' spectrum $\Gamma(\alpha) = \mathbb R$ satisfies the Rokhlin property and thus is unique up to cocycle conjugacy. The proof relies on a key cocycle perturbation result for type $\mathrm{III}$ amenable equivalence relations. As a byproduct of our methods, we also show that every almost periodic factor of type $\mathrm{III}_1$ with separable predual has an extremal almost periodic faithful normal state.