Let $M$ be a closed smooth manifold and let $f:M\to M$ be a diffeomorphism. $C^1$-generically, a continuum-wise expansive satisfies Axiom A without cycles. Moreover, there is a partially hyperbolic diffeomorphism $f$ such that it is not continuum-wise expansive.
Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms
Chain-transitivity of partially hyperbolic diffeomorphisms on $\mathbb{T}^3$ isotopic to Anosov
Symbolic extensions for partially hyperbolic diffeomorphisms
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