We study the limit of a sequence of probability measures on a non-compact homogeneous spaces invariant under diagonalizable flow. In this context the limit measure may not be probability. Our particular interest is to study how much mass could be left in the limit if we additionally assume that our measures have high entropy. This is a part of the project on generalizing a theorem of M. Einsiedler, E. Lindenstrauss, Ph. Michel, and A. Venkatesh. They prove that for any sequence (mui) of probability measure on SL2( Z )SL2( R ) invariant under the time-one-map T of geodesic flow with entropies hmi (T) ≥ c one has that any weak* limit mu of (mui) has at least mu( X) ≥ 2c -- 1 mass left. We first consider the homogeneous space SL3( Z )SL3( R ) with an action T of a particular diagonal element diag(e1/2, e 1/2, e--1) and prove a generalization. Next, by constructing T-invariant probability measure with high entropy we show that our result is sharp. We also consider the Hilbert Modular space type quotient spaces and again obtain the a generalization by studying any diagonal element. As an application one can calculate an upper bound for the Hausdorff dimension of the set of points that lie on divergent trajectories with respect to the diagonal element considered, giving an alternative proof to a result of Y. Cheung. The work regarding SL3( Z )SL3( R ) is joint work with my co-adviser M. Einsiedler.
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