  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.

Dynamical Systems