Cocycle perturbations and ergodicity for actions on type III factors

We study cocycle perturbations of state preserving actions on type ${\mathrm{I}\mathrm{I}\mathrm{I}}_1$ factors. Extending the theorem of Marrakchi and Vaes for type ${\mathrm{I}\mathrm{I}}_1$ factors, we show that a state preserving outer $\mathbb{Z}$-action on a type ${\mathrm{I}\mathrm{I}\mathrm{I}}_1$ factor with trivial bicentralizer admits a unitary cocycle whose perturbation becomes an ergodic action. This partially answers a question of Marrakchi and Vaes. A major difference from the type $\mathrm{I}\mathrm{I}$ case is that the modular automorphism group naturally appears as part of the action, making the construction of the required cocycle more delicate.