If a partially hyperbolic diffeomorphism on a torus of dimension d greater than 3 has stable and unstable foliations which are quasi-isometric on the universal cover, and its center 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.

If a partially hyperbolic diffeomorphism on a torus of dimension d greater than 3 has stable and unstable foliations which are quasi-isometric on the universal cover, and its center 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.