On $ \epsilon $-escaping trajectories in homogeneous spaces

Hertz Federico Rodriguez; Wang Zhiren

doi:10.3934/dcds.2020365

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2021, Volume 41, Issue 1: 329-357. Doi: 10.3934/dcds.2020365

This issue Previous Article $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization Next Article Mean equicontinuity, complexity and applications

On $ \epsilon $-escaping trajectories in homogeneous spaces

Pennsylvania State University, State College, PA 16802, USA

Received: January 2020

Revised: August 2020

Early access: October 2020

Published: January 2021

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

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Abstract

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 $.

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

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