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

On $ \epsilon $-escaping trajectories in homogeneous spaces

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 $.

Citation: Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete and 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.

[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.

[3]

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

[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.

[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.

[6]

T. DasL. FishmanD. 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.

[7]

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

[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.

[9]

A. EskinG. 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.

[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.

[11]

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

[12]

S. KadyrovD. KleinbockE. 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.

[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.

[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.

[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.

[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.

[17]

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

[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.

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.

[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.

[3]

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

[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.

[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.

[6]

T. DasL. FishmanD. 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.

[7]

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

[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.

[9]

A. EskinG. 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.

[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.

[11]

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

[12]

S. KadyrovD. KleinbockE. 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.

[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.

[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.

[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.

[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.

[17]

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

[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.

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