Archive | 2021
On \\begin{document}$ \\epsilon $\\end{document} -escaping trajectories in homogeneous spaces
Abstract
Let \\begin{document}$ G/\\Gamma $\\end{document} be a finite volume homogeneous space of a semisimple Lie group \\begin{document}$ G $\\end{document} , and \\begin{document}$ \\{\\exp(tD)\\} $\\end{document} be a one-parameter \\begin{document}$ \\operatorname{Ad} $\\end{document} -diagonalizable subgroup inside a simple Lie subgroup \\begin{document}$ G_0 $\\end{document} of \\begin{document}$ G $\\end{document} . Denote by \\begin{document}$ Z_{\\epsilon,D} $\\end{document} the set of points \\begin{document}$ x\\in G/\\Gamma $\\end{document} whose \\begin{document}$ \\{\\exp(tD)\\} $\\end{document} -trajectory has an escape for at least an \\begin{document}$ \\epsilon $\\end{document} -portion of mass along some subsequence. We prove that the Hausdorff codimension of \\begin{document}$ Z_{\\epsilon,D} $\\end{document} is at least \\begin{document}$ c\\epsilon $\\end{document} , where \\begin{document}$ c $\\end{document} depends only on \\begin{document}$ G $\\end{document} , \\begin{document}$ G_0 $\\end{document} and \\begin{document}$ \\Gamma $\\end{document} .