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

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