Joint equidistribution of maximal flat cylinders and holonomies for Anosov homogeneous spaces

Let $G$ be a connected semisimple real algebraic group and $P<G$ be a minimal parabolic subgroup with Langlands decomposition $P=MAN$. Let $\mathrm{\Gamma }<G$ be a Zariski dense Anosov subgroup with respect to $P$. Since $\mathrm{\Gamma }$ is Anosov, the set of conjugacy classes of primitive elements of $\mathrm{\Gamma }$ is in one-to-one correspondence with the set of (positively oriented) maximal flat cylinders in $\mathrm{\Gamma }\mathrm{\backslash }G\mathrm{/}M$. We describe the joint equidistribution of maximal flat cylinders and their holonomies as their circumferences tend to infinity. This result can be viewed as the Anosov analogue of the joint equidistribution result of closed geodesics and holonomies in rank one by Margulis–Mohammadi–Oh.