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On $ \epsilon $-escaping trajectories in homogeneous spaces

F.R.H. was supported by NSF grant DMS-1900778.Z.W. was supported by NSF grant DMS-1753042

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

    Mathematics Subject Classification: 37A17, 37D40, 11L55.


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