Counting closed orbits of Anosov flows in free homotopy classes

This is joint work with Thomas Barthelme of Penn State University. There are Anosov and pseudo-Anosov flows so that some orbits are freely homotopic to infinitely many other orbits. An Anosov flow is $R$-covered if either the stable or unstable foliations lift to foliations in the universal cover with leaf space homeomorphic to the reals. These are extremely common. A free homotopy class is a maximal collection of closed orbits of the flow that are pairwise freely homotopic to each other. We first construct explicit examples of Anosov flows with some infinite free homotopy classes. Then we mention the result that if an $R$-covered Anosov flow has all free homotopy classes that are finite, then up to a finite cover the flow is topologically conjugate to either a suspension or a geodesic flow. This is a strong rigidity result that says that infinite free homotopy classes are extremely common amongst Anosov flows in 3-manifolds. The second part of the talk is about analyzing growth of length of orbits in a fixed infinite free homotopy class. We analyse the interaction of such a free homotopy class with the torus decomposition of the manifold: for examples whether all orbits in the infinite free homotopy classes are contained in a Seifert piece or atoroidal piece. There is a natural ordering of an infinite subset of such a collection, indexed as $(\gamma_i)$. We analyse the growth of the length of $\gamma_i$ as a function of $i$. We obtain several inequalities: for example if the manifold is hyperbolic then the growth of length of $\gamma_i$ is exponential. These inequalities have consequences for the ergodic theory of the Anosov flow.