n this talk we will discuss the asymptotic entropy of packets of periodic torus orbits on locally homogeneous spaces. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adelic orbit and they are closely related to class groups of number fields. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer.
The distribution of packets of periodic torus orbit has been studied using dynamical methods in the pioneering work of Linnik in the rank 1 case (equidistribution on the 2-sphere). Recently Einsiedler, Lindenstrauss, Michel and Venkatesh have shown that Linnik’s “ergodic method†is closely related to metric entropy.
Inspired by Linnik’s ideas for bounding the separation of integral points on the sphere I will present a geometric expansion for the collision probability of torus packets in a general reductive group. This expansion is related to the double quotient of the ambient group by a maximal torus.
Studying the arithmetic geometry of the double quotient of inner forms of SLn and PGLn by a maximal torus allows us to significantly strengthen results towards the equidistribution of packets of periodic torus orbits on the associated higher rank S-arithmetic quotients. An important aspect of our method is that it applies to packets of periodic orbits of maximal tori which are only partially split at a fixed place.