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January  2021, 41(1): 329-357. doi: 10.3934/dcds.2020365

On $ \epsilon $-escaping trajectories in homogeneous spaces

Federico Rodriguez Hertz and  Zhiren Wang

Pennsylvania State University, State College, PA 16802, USA

Received  January 2020 Revised  August 2020 Published  January 2021 Early access  October 2020

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

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 $.

Keywords: Escaping trajectories, Hausdorff dimension, homogeneous spaces, diagonal flow, homogeneous dynamics.
Mathematics Subject Classification: 37A17, 37D40, 11L55.
Citation: Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365
References:
[1]

Y. Benoist and J.-F. Quint, Random walks on finite volume homogeneous spaces, Invent. Math., 187 (2012), 37-59.  doi: 10.1007/s00222-011-0328-5.  Google Scholar

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Y. Cheung, Hausdorff dimension of the set of singular pairs, Ann. of Math. (2), 173 (2011), 127-167.  doi: 10.4007/annals.2011.173.1.4.  Google Scholar

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Y. Cheung and N. Chevallier, Hausdorff dimension of singular vectors, Duke Math. J., 165 (2016), 2273-2329.  doi: 10.1215/00127094-3477021.  Google Scholar

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S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55-89.  doi: 10.1515/crll.1985.359.55.  Google Scholar

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S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel'fand Seminar, Adv. Soviet Math., Amer. Math. Soc., Providence, RI. 16 (1993), 91-137.  Google Scholar

[6]

T. Das, L. Fishman, D. Simmons and M. Urbański, A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation, C. R. Math. Acad. Sci. Paris, 355 (2017), 835-846.  doi: 10.1016/j.crma.2017.07.007.  Google Scholar

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T. Das, F. Fishman, D. Simmons and M. Urbański, A variational principle in the parametric geometry of numbers, arXiv: 1901.06602, 2019. Google Scholar

[8]

A. Eskin and G. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds, Random walks and geometry, Walter de Gruyter, Berlin, (2004), 431-444. doi: 10.1016/s0169-5983(04)00042-5.  Google Scholar

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A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141.  doi: 10.2307/120984.  Google Scholar

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H. Garland and M. S. Raghunathan, Fundamental domains for lattices in ($ \mathbb{R}$-)rank $1$ semisimple Lie groups, Ann. of Math. (2), 92 (1970), 279-326.  doi: 10.2307/1970838.  Google Scholar

[11]

S. Kadyrov, Entropy and escape of mass for Hilbert modular spaces, J. Lie Theory, 22 (2012), 701-722.   Google Scholar

[12]

S. Kadyrov, D. Kleinbock, E. Lindenstrauss and G. A. Margulis, Singular systems of linear forms and non-escape of mass in the space of lattices, J. Anal. Math., 133 (2017), 253-277.  doi: 10.1007/s11854-017-0033-4.  Google Scholar

[13]

O. Khalil, Bounded and divergent trajectories and expanding curves on homogeneous spaces, Trans. Amer. Math. Soc., 373 (2020), 7473-7525.  doi: 10.1090/tran/8161.  Google Scholar

[14]

D. Kleinbock, An extension of quantitative nondivergence and applications to Diophantine exponents, Trans. Amer. Math. Soc., 360 (2008), 6497-6523.  doi: 10.1090/S0002-9947-08-04592-3.  Google Scholar

[15]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Sinaǐ 's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, 171 (1996), 141-172. doi: 10.1090/trans2/171/11.  Google Scholar

[16]

D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2), 148 (1998), 339-360.  doi: 10.2307/120997.  Google Scholar

[17]

F. M. Malyšhev, Decompositions of nilpotent Lie algebras, Mat. Zametki, 23 (1978), 27-30.   Google Scholar

[18]

G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 17, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-51445-6.  Google Scholar

show all references

References:
[1]

Y. Benoist and J.-F. Quint, Random walks on finite volume homogeneous spaces, Invent. Math., 187 (2012), 37-59.  doi: 10.1007/s00222-011-0328-5.  Google Scholar

[2]

Y. Cheung, Hausdorff dimension of the set of singular pairs, Ann. of Math. (2), 173 (2011), 127-167.  doi: 10.4007/annals.2011.173.1.4.  Google Scholar

[3]

Y. Cheung and N. Chevallier, Hausdorff dimension of singular vectors, Duke Math. J., 165 (2016), 2273-2329.  doi: 10.1215/00127094-3477021.  Google Scholar

[4]

S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55-89.  doi: 10.1515/crll.1985.359.55.  Google Scholar

[5]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel'fand Seminar, Adv. Soviet Math., Amer. Math. Soc., Providence, RI. 16 (1993), 91-137.  Google Scholar

[6]

T. Das, L. Fishman, D. Simmons and M. Urbański, A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation, C. R. Math. Acad. Sci. Paris, 355 (2017), 835-846.  doi: 10.1016/j.crma.2017.07.007.  Google Scholar

[7]

T. Das, F. Fishman, D. Simmons and M. Urbański, A variational principle in the parametric geometry of numbers, arXiv: 1901.06602, 2019. Google Scholar

[8]

A. Eskin and G. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds, Random walks and geometry, Walter de Gruyter, Berlin, (2004), 431-444. doi: 10.1016/s0169-5983(04)00042-5.  Google Scholar

[9]

A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141.  doi: 10.2307/120984.  Google Scholar

[10]

H. Garland and M. S. Raghunathan, Fundamental domains for lattices in ($ \mathbb{R}$-)rank $1$ semisimple Lie groups, Ann. of Math. (2), 92 (1970), 279-326.  doi: 10.2307/1970838.  Google Scholar

[11]

S. Kadyrov, Entropy and escape of mass for Hilbert modular spaces, J. Lie Theory, 22 (2012), 701-722.   Google Scholar

[12]

S. Kadyrov, D. Kleinbock, E. Lindenstrauss and G. A. Margulis, Singular systems of linear forms and non-escape of mass in the space of lattices, J. Anal. Math., 133 (2017), 253-277.  doi: 10.1007/s11854-017-0033-4.  Google Scholar

[13]

O. Khalil, Bounded and divergent trajectories and expanding curves on homogeneous spaces, Trans. Amer. Math. Soc., 373 (2020), 7473-7525.  doi: 10.1090/tran/8161.  Google Scholar

[14]

D. Kleinbock, An extension of quantitative nondivergence and applications to Diophantine exponents, Trans. Amer. Math. Soc., 360 (2008), 6497-6523.  doi: 10.1090/S0002-9947-08-04592-3.  Google Scholar

[15]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Sinaǐ 's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, 171 (1996), 141-172. doi: 10.1090/trans2/171/11.  Google Scholar

[16]

D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2), 148 (1998), 339-360.  doi: 10.2307/120997.  Google Scholar

[17]

F. M. Malyšhev, Decompositions of nilpotent Lie algebras, Mat. Zametki, 23 (1978), 27-30.   Google Scholar

[18]

G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 17, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-51445-6.  Google Scholar

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