In this talk, we shall introduce the notion of the pointwise partially hyperbolic diffeomorphism on an open region. Here “pointwise” means the expansion and contraction depend on points. Under the pointwise dominated splitting and boundary control conditions, we show that such a system admits unstable and stable manifolds, and admits a finite or an infinite u-Gibbs measure. If the system is pointwise hyperbolic, then the u-Gibbs measure is in fact an Sinai-Ruelle-Bowen (SRB) measure or an infinite SRB measure. Applications are some almost Anosov diffeomorphisms and gentle perturbations of Katok's map.
This is a joint work with Huyi Hu and Yunhua Zhou.

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