We prove that a class of weakly partially hyperbolic endomorphisms on $\mathbb{T}^{2}$ are dynamically coherent and leaf conjugate to linear toral endomorphisms. Moreover, we give an example of a partially hyperbolic endomorphism on $\mathbb{T}^{2}$ which does not admit a centre foliation.

This paper surveys recent results on classifying partially hyperbolic diffeomorphisms. This includes the construction of branching foliations and leaf conjugacies on three-dimensional manifolds with solvable fundamental group. Classification results in higher-dimensional settings are also discussed. The paper concludes with an overview of the construction of new partially hyperbolic examples derived from Anosov flows.

If a partially hyperbolic diffeomorphism on a torus of dimension $d\geq 3$ has stable and unstable foliations which are quasi-isometric on the universal cover, and its centre direction is one-dimensional, then the diffeomorphism is leaf conjugate to a linear toral automorphism. In other words, the hyperbolic structure of the diffeomorphism is exactly that of a linear, and thus simple to understand, example. In particular, every partially hyperbolic diffeomorphism on the 3-torus is leaf conjugate to a linear toral automorphism.

We show that a partially hyperbolic diffeomorphism is plaque expansive (a form of structural stability for its center foliation) if the strong stable and unstable foliations are quasi-isometric in the universal cover. In particular, all partially hyperbolic diffeomorphisms on the 3-torus are plaque expansive.