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Abstract Quasigeodesic behavior of flow lines is a very useful property in the study of Anosov flows. Not every Anosov flow in dimension three is quasigeodesic. In fact, until recently, up to orbit equivalence, the only previously known examples of quasigeodesic Anosov flows were suspension flows. In a recent article, the second author proved that an Anosov flow on a hyperbolic 3-manifold is quasigeodesic if and only if it is non-$$\mathbb {R}$$-covered, and this result completes the classification of quasigeodesic Anosov flows on hyperbolic 3-manifolds. In this article, we prove that a new class of examples of Anosov flows are quasigeodesic. These are the first examples of quasigeodesic Anosov flows on 3-manifolds that are neither Seifert, nor solvable, nor hyperbolic. In general, it is very hard to show that a given flow is quasigeodesic and, in this article, we provide a new method to prove that an Anosov flow is quasigeodesic. 

We study 3–dimensional partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the geometry and dynamics of Burago and Ivanov’s center stable and center unstable branching foliations. This extends our previous study of the true foliations that appear in the dynamically coherent case. We complete the classification of such diffeomorphisms in Seifert fibered manifolds. In hyperbolic manifolds, we show that any such diffeomorphism is either dynamically coherent and has a power that is a discretized Anosov flow, or is of a new potential class called a double translation.