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$ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization
On $ \epsilon $-escaping trajectories in homogeneous spaces
Pennsylvania State University, State College, PA 16802, USA |
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 $.
References:
[1] |
Y. Benoist and J.-F. Quint,
Random walks on finite volume homogeneous spaces, Invent. Math., 187 (2012), 37-59.
doi: 10.1007/s00222-011-0328-5. |
[2] |
Y. Cheung,
Hausdorff dimension of the set of singular pairs, Ann. of Math. (2), 173 (2011), 127-167.
doi: 10.4007/annals.2011.173.1.4. |
[3] |
Y. Cheung and N. Chevallier,
Hausdorff dimension of singular vectors, Duke Math. J., 165 (2016), 2273-2329.
doi: 10.1215/00127094-3477021. |
[4] |
S. G. Dani,
Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55-89.
doi: 10.1515/crll.1985.359.55. |
[5] |
S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel'fand Seminar, Adv. Soviet Math., Amer. Math. Soc., Providence, RI. 16 (1993), 91-137. |
[6] |
T. Das, L. Fishman, D. Simmons and M. Urbański,
A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation, C. R. Math. Acad. Sci. Paris, 355 (2017), 835-846.
doi: 10.1016/j.crma.2017.07.007. |
[7] |
T. Das, F. Fishman, D. Simmons and M. Urbański, A variational principle in the parametric geometry of numbers, arXiv: 1901.06602, 2019. Google Scholar |
[8] |
A. Eskin and G. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds, Random walks and geometry, Walter de Gruyter, Berlin, (2004), 431-444.
doi: 10.1016/s0169-5983(04)00042-5. |
[9] |
A. Eskin, G. Margulis and S. Mozes,
Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141.
doi: 10.2307/120984. |
[10] |
H. Garland and M. S. Raghunathan,
Fundamental domains for lattices in ($ \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. 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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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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