Abstract
We construct a family of partially hyperbolic skew-product diffeomorphisms on $\mathbb{T}^3$ that are robustly transitive and admit two physical measures with intermingled basins. In particular, all these diffeomorphisms are not topologically mixing. Moreover, every such example exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.
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