We introduce a new method of constructing Birkhoff sections for pseudo-Anosov flows, which uses the connection between pseudo-Anosov flows and veering triangulations. This method allows for explicit constructions, as well as control over the Birkhoff section in terms of its Euler characteristic and the complexity of the boundary orbits. In particular, we show that any transitive pseudo-Anosov flow has a Birkhoff section with two boundary components.

We construct one-dimensional foliations which are subfoliations of two-dimensional foliations in $3$ -manifolds. The subfoliation is by quasigeodesics in each two-dimensional leaf, but it is not funnel: not all quasigeodesics share a common ideal point in most leaves.

For $k \geq 2$ , we prove that in a $C^{1}$ -open and $C^{k}$ -dense set of some classes of $C^{k}$ -Anosov flows, all Lyapunov exponents have multiplicity one with respect to appropriate measures. The classes are geodesic flows with equilibrium states of Holder-continuous potentials, volume-preserving flows, and all fiber-bunched Anosov flows with equilibrium states of Holder-continuous potentials.

We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .