Tori Approximation of Families of Diagonally Invariant Measures

Let A be the full diagonal group in SL n ( R ) . We study possible limits of Haar measures on periodic A -orbits in the space of unimodular lattices X n . We prove the existence of non-ergodic measures which are also weak limits of these compactly supported A -invariant measures. In fact, given any countably many A -invariant ergodic measures, we show that there exists a sequence of Haar measures on periodic A -orbits such that the ergodic decomposition of its weak limit has these measures as factors with positive weight. In particular, we prove that any compactly supported A -invariant and ergodic measure is the weak limit of the restriction of different compactly supported periodic measures to a fixed proportion of the time. In addition, for any c ∈ ( 0 , 1 ] we find a sequence of Haar measures on periodic A orbits that converges weakly to c m X n where m X n denotes the Haar measure on X n . In particular, we prove the existence of partial escape of mass for Haar measures on periodic A orbits. These results give affirmative answers to questions posed by Shapira in [ Sha16 ]. Our proofs are based on a modification of Shapira’s proof in [ Sha16 ] and on a generalization of a construction of Cassels, as well as on effective equidistribution estimates of Hecke neighbors by Clozel, Oh and Ullmo in [ COU01 ].