-
Previous Article
Mean equicontinuity, complexity and applications
- DCDS Home
- This Issue
-
Next Article
$ 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 $.
References:
| [1] |
Y. Benoist and J.-F. Quint,
Random walks on finite volume homogeneous spaces, Invent. Math., 187 (2012), 37-59.
doi: 10.1007/s00222-011-0328-5. |
| [2] |
Y. Cheung,
Hausdorff dimension of the set of singular pairs, Ann. of Math. (2), 173 (2011), 127-167.
doi: 10.4007/annals.2011.173.1.4. |
| [3] |
Y. Cheung and N. Chevallier,
Hausdorff dimension of singular vectors, Duke Math. J., 165 (2016), 2273-2329.
doi: 10.1215/00127094-3477021. |
| [4] |
S. G. Dani,
Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55-89.
doi: 10.1515/crll.1985.359.55. |
| [5] |
S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel'fand Seminar, Adv. Soviet Math., Amer. Math. Soc., Providence, RI. 16 (1993), 91-137. |
| [6] |
T. Das, L. Fishman, D. Simmons and M. Urbański,
A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation, C. R. Math. Acad. Sci. Paris, 355 (2017), 835-846.
doi: 10.1016/j.crma.2017.07.007. |
| [7] |
T. Das, F. Fishman, D. Simmons and M. Urbański, A variational principle in the parametric geometry of numbers, arXiv: 1901.06602, 2019. |
| [8] |
A. Eskin and G. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds, Random walks and geometry, Walter de Gruyter, Berlin, (2004), 431-444.
doi: 10.1016/s0169-5983(04)00042-5. |
| [9] |
A. Eskin, G. Margulis and S. Mozes,
Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141.
doi: 10.2307/120984. |
| [10] |
H. Garland and M. S. Raghunathan,
Fundamental domains for lattices in ($ \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. |
| [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. |
| [1] |
Alexander Gorodnik, Frédéric Paulin. Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows. Journal of Modern Dynamics, 2014, 8 (1) : 25-59. doi: 10.3934/jmd.2014.8.25 |
| [2] |
Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405 |
| [3] |
Daniel Guan. Classification of compact complex homogeneous spaces with invariant volumes. Electronic Research Announcements, 1997, 3: 90-92. |
| [4] |
Danijela Damjanovic, James Tanis, Zhenqi Jenny Wang. On globally hypoelliptic abelian actions and their existence on homogeneous spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6747-6766. doi: 10.3934/dcds.2020164 |
| [5] |
Daniel Guan. Classification of compact homogeneous spaces with invariant symplectic structures. Electronic Research Announcements, 1997, 3: 52-54. |
| [6] |
Maxim Sølund Kirsebom. Extreme value theory for random walks on homogeneous spaces. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4689-4717. doi: 10.3934/dcds.2014.34.4689 |
| [7] |
Alexander Kemarsky, Frédéric Paulin, Uri Shapira. Escape of mass in homogeneous dynamics in positive characteristic. Journal of Modern Dynamics, 2017, 11: 369-407. doi: 10.3934/jmd.2017015 |
| [8] |
Steve Hofmann, Dorina Mitrea, Marius Mitrea, Andrew J. Morris. Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets. Electronic Research Announcements, 2014, 21: 8-18. doi: 10.3934/era.2014.21.8 |
| [9] |
Nimish Shah, Lei Yang. Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5247-5287. doi: 10.3934/dcds.2020227 |
| [10] |
T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665 |
| [11] |
Pierre Magal. Global stability for differential equations with homogeneous nonlinearity and application to population dynamics. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 541-560. doi: 10.3934/dcdsb.2002.2.541 |
| [12] |
Hongyan Zhang, Siyu Liu, Yue Zhang. Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1149-1164. doi: 10.3934/dcdss.2017062 |
| [13] |
Alessandro Fonda, Rafael Ortega. Positively homogeneous equations in the plane. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 475-482. doi: 10.3934/dcds.2000.6.475 |
| [14] |
Changguang Dong. Separated nets arising from certain higher rank $\mathbb{R}^k$ actions on homogeneous spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4231-4238. doi: 10.3934/dcds.2017180 |
| [15] |
María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks and Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012 |
| [16] |
Yilei Tang. Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2029-2046. doi: 10.3934/dcds.2018082 |
| [17] |
Alexander Krasnosel'skii, Alexei Pokrovskii. On subharmonics bifurcation in equations with homogeneous nonlinearities. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 747-762. doi: 10.3934/dcds.2001.7.747 |
| [18] |
Mohammad Shafiee. A note on $2$-plectic homogeneous manifolds. Journal of Geometric Mechanics, 2015, 7 (3) : 389-394. doi: 10.3934/jgm.2015.7.389 |
| [19] |
Matthew B. Rudd. Statistical exponential formulas for homogeneous diffusion. Communications on Pure and Applied Analysis, 2015, 14 (1) : 269-284. doi: 10.3934/cpaa.2015.14.269 |
| [20] |
Li Ma, De Tang. Evolution of dispersal in advective homogeneous environments. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5815-5830. doi: 10.3934/dcds.2020247 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]