Locally unipotent invariant measures and limit distribution of a sequence of polynomial trajectories on homogeneous spaces

Abstract

Let $ G $ be a Lie group, and let $ \Gamma $ be a lattice in $ G $. We introduce the notion of locally unipotent invariant measures on $ G/\Gamma $. We then prove that under some conditions, the limit measure supported on the image of polynomial trajectories on $ G/\Gamma $ is locally unipotent invariant, thus give a partial answer to an equidistribution problem for higher dimensional polynomial trajectories on homogeneous spaces, which was raised by Shah in [15].The proof relies on Ratner's measure classification theorem, a linearization technique for polynomial trajectories near singular sets, and a twisting technique of Shah.