Tori Approximation of Families of Diagonally Invariant Measures

Abstract: We approximate any portion of any orbit of the full diagonal group $A$ in the space of unimodular lattices in $\RR^n$ using a fixed proportion of a compact $A$-orbit. Using those approximations for the appropriate sequence of orbits, we prove the existence of non-ergodic measures which are also weak limits of compactly supported $A$-invariant measures. In fact, given any countably many $A$-invariant ergodic measures, our methods show that there exists a sequence of compactly supported periodic $A$-invariant measures such that the ergodic decomposition of its weak limit has these measures as factors with positive weight. Using the same methods, 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 $c\in (0,1]$ we find a sequence of compactly supported periodic $A$-invariant measures that converge weakly to $cm_{X_n}$ where $m_{X_n}$ denotes the Haar measure on $X_n$. In particular, we prove the existence of partial escape of mass for compact $A$-orbits. These results give affirmative answers to questions posed by Shapira in ~\cite{ShapiraEscape}. Our proofs are based on a modification of Shapira's proof in ~\cite{ShapiraEscape} and on a generalization of a construction of Cassels, as well as on effective equidistribution estimates of Hecke neighbors by Clozel, Oh and Ullmo, and a number theoretic construction of a special number field.