### Stable ergodicity of skew products

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Answering a question of Smale, we prove that the space of C 1 diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.

We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns–Wilkinson and Avila–Santamaria–Viana. Combining this new technique with other constructions we prove that ${C}^{1}$-generic partially hyperbolic symplectomorphisms are ergodic. We also construct new examples of stably ergodic partially hyperbolic diffeomorphisms.

We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.

Given any compact manifold $M$, we construct a non-empty open subset $\mathcal{O}$ of the space ${\mathrm{Diff}}^{1}\left(M\right)$ of ${C}^{1}$-diffeomorphisms and a dense subset $\mathcal{D}\subset \mathcal{O}$ such that the centralizer of every diffeomorphism in $\mathcal{D}$ is uncountable, hence non-trivial.

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