Margulis–Mohammadi–Oh (‘14) described the asymptotic joint equdistribution of the closed geodesics and holonomies of a geometrically finite rank one locally symmetric manifold as their lengths grow to infinity.
We obtain an Anosov analogue of this result. More precisely, in the quotient of a higher rank semisimple Lie group by an Anosov subgroup, non-trivial closed orbits under a maximal real-split torus give rise to maximal flat cylinders. We prove an analogous joint equidistribution result of the maximal flat cylinders and holonomies with circumferences growing to infinity.
This is joint work with Elijah Fromm.