Invariant measures for actions of unipotent groups over local fields on homogeneous spaces

Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

“…The origins of this method can be traced back to the work of S.G.Dani and Smillie [15] in the context of Fuchsian groups. General results and techniques are developed in [11,12,14,38,39,40,42,53,54,55,56].…”

Rigidity of measures—The high entropy case and non-commuting foliations

“…Theorem 2.4, 2.7, and 2.8 all rely heavily on Theorem 8.5 and in part also on the relation between the conditional measures and entropy. The latter we recall from [25] and slightly extend in Section 9, where we will also prove the results presented in this section.…”

“…a product of real and p-adic algebraic groups) would imply an S-arithmetic analogue of the Oppenheim conjecture, and they proved a result in that direction. The S-algebraic Raghunathan's conjecture was proved in full independently by G. Margulis and G. Tomanov [25] and by M. Ratner [31].…”

“…Instead of that we can also obtain a stronger conclusion by assuming ergodicity for α σ and combining our result with measure rigidity for groups generated by unipotent subgroups, see [25] or [31].…”

“…In this section we give a formula relating entropy and conditional measures. We will use and slightly reformulate the results from [25,Sect. 9].…”