-
Previous Article
Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces
- DCDS Home
- This Issue
-
Next Article
A topological study of planar vector field singularities
Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices
1. | Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA |
2. | College of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China |
In this paper, we study an analytic curve $ \varphi: I = [a, b]\rightarrow \mathrm{M}(m\times n, \mathbb{R}) $ in the space of $ m $ by $ n $ real matrices, and show that if $ \varphi $ satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into a homogeneous space $ G/\Gamma $, and prove that under the action of some expanding diagonal subgroup $ A = \{a(t): t \in \mathbb{R}\} $, the translates of the curve tend to be equidistributed in $ G/\Gamma $, as $ t \rightarrow +\infty $. The proof relies on the linearization technique and representation theory.
References:
[1] |
M. Aka, E. Breuillard, L. Rosenzweig and N. de Saxcé,
On metric diophantine approximation in matrices and lie groups, C. R. Math. Acad. Sci. Paris, 353 (2015), 185-189.
doi: 10.1016/j.crma.2014.12.007. |
[2] |
M. Aka, E. Breuillard, L. Rosenzweig and N. de Saxcé,
Diophantine approximation on matrices and lie groups, Geom. Funct. Anal., 28 (2018), 1-57.
doi: 10.1007/s00039-018-0436-0. |
[3] |
R. C. Baker,
Dirichlet's theorem on Diophantine approximation, Math. Proc. Cambridge Philos. Soc., 83 (1978), 37-59.
doi: 10.1017/S030500410005427X. |
[4] |
Y. Bugeaud,
Approximation by algebraic integers and hausdorff dimension, Journal of the London Mathematical Society (2), 65 (2002), 547-559.
doi: 10.1112/S0024610702003137. |
[5] |
S. G. Dani,
On orbits of unipotent flows on homogeneous spaces, Ergodic Theory and Dynamical Systems, 4 (1984), 25-34.
doi: 10.1017/S0143385700002248. |
[6] |
S. G. Dani and G. A. Margulis,
Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), 1-17.
doi: 10.1007/BF02872005. |
[7] |
H. Davenport and W. M. Schmidt,
Dirichlet's theorem on diophantine approximation. Ⅱ, Acta Arithmetica, 16 (1969/70), 413-424.
doi: 10.4064/aa-16-4-413-424. |
[8] |
W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics, 129. Readings in Mathematics, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-0979-9. |
[9] |
D. Kleinbock, G. Margulis and J. B. Wang,
Metric diophantine approximation for systems of linear forms via dynamics, International Journal of Number Theory, 6 (2010), 1139-1168.
doi: 10.1142/S1793042110003423. |
[10] |
D. Y. Kleinbock and G. A. Margulis,
Flows on homogeneous spaces and diophantine approximation on manifolds, Annals of Mathematics (2), 148 (1998), 339-360.
doi: 10.2307/120997. |
[11] |
D. Kleinbock and B. Weiss,
Dirichlet's theorem on diophantine approximation and homogeneous flows, Journal of Modern Dynamics, 2 (2008), 43-62.
doi: 10.3934/jmd.2008.2.43. |
[12] |
S. Mozes and N. Shah,
On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems, 15 (1995), 149-159.
doi: 10.1017/S0143385700008282. |
[13] |
M. Ratner,
On raghunathan's measure conjecture, Annals of Mathematics (2), 134 (1991), 545-607.
doi: 10.2307/2944357. |
[14] |
M. Ratner,
Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280.
doi: 10.1215/S0012-7094-91-06311-8. |
[15] |
N. A. Shah,
Expanding translates of curves and dirichlet-minkowski theorem on linear forms, Journal of the American Mathematical Society, 23 (2010), 563-589.
doi: 10.1090/S0894-0347-09-00657-2. |
[16] |
N. A Shah,
Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proceedings Mathematical Sciences, 106 (1996), 105-125.
doi: 10.1007/BF02837164. |
[17] |
N. A. Shah,
Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds, Duke Mathematical Journal, 148 (2009), 281-304.
doi: 10.1215/00127094-2009-027. |
[18] |
N. A. Shah, Equidistribution of expanding translates of curves and dirichlets theorem on diophantine approximation, Inventiones Mathematicae, 177 (2009), 509-532. Google Scholar |
[19] |
N. A. Shah,
Limiting distributions of curves under geodesic flow on hyperbolic manifolds, Duke Mathematical Journal, 148 (2009), 251-279.
doi: 10.1215/00127094-2009-026. |
[20] |
L. Yang,
Equidistribution of expanding curves in homogeneous spaces and diophantine approximation for square matrices, Proc. Amer. Math. Soc., 144 (2016), 5291-5308.
doi: 10.1090/proc/13170. |
[21] |
L. Yang,
Expanding curves in $\mathrm{T}^1(\mathbb{H}^n)$ under geodesic flow and equidistribution in homogeneous spaces, Israel J. Math., 216 (2016), 389-413.
doi: 10.1007/s11856-016-1414-6. |
show all references
References:
[1] |
M. Aka, E. Breuillard, L. Rosenzweig and N. de Saxcé,
On metric diophantine approximation in matrices and lie groups, C. R. Math. Acad. Sci. Paris, 353 (2015), 185-189.
doi: 10.1016/j.crma.2014.12.007. |
[2] |
M. Aka, E. Breuillard, L. Rosenzweig and N. de Saxcé,
Diophantine approximation on matrices and lie groups, Geom. Funct. Anal., 28 (2018), 1-57.
doi: 10.1007/s00039-018-0436-0. |
[3] |
R. C. Baker,
Dirichlet's theorem on Diophantine approximation, Math. Proc. Cambridge Philos. Soc., 83 (1978), 37-59.
doi: 10.1017/S030500410005427X. |
[4] |
Y. Bugeaud,
Approximation by algebraic integers and hausdorff dimension, Journal of the London Mathematical Society (2), 65 (2002), 547-559.
doi: 10.1112/S0024610702003137. |
[5] |
S. G. Dani,
On orbits of unipotent flows on homogeneous spaces, Ergodic Theory and Dynamical Systems, 4 (1984), 25-34.
doi: 10.1017/S0143385700002248. |
[6] |
S. G. Dani and G. A. Margulis,
Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), 1-17.
doi: 10.1007/BF02872005. |
[7] |
H. Davenport and W. M. Schmidt,
Dirichlet's theorem on diophantine approximation. Ⅱ, Acta Arithmetica, 16 (1969/70), 413-424.
doi: 10.4064/aa-16-4-413-424. |
[8] |
W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics, 129. Readings in Mathematics, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-0979-9. |
[9] |
D. Kleinbock, G. Margulis and J. B. Wang,
Metric diophantine approximation for systems of linear forms via dynamics, International Journal of Number Theory, 6 (2010), 1139-1168.
doi: 10.1142/S1793042110003423. |
[10] |
D. Y. Kleinbock and G. A. Margulis,
Flows on homogeneous spaces and diophantine approximation on manifolds, Annals of Mathematics (2), 148 (1998), 339-360.
doi: 10.2307/120997. |
[11] |
D. Kleinbock and B. Weiss,
Dirichlet's theorem on diophantine approximation and homogeneous flows, Journal of Modern Dynamics, 2 (2008), 43-62.
doi: 10.3934/jmd.2008.2.43. |
[12] |
S. Mozes and N. Shah,
On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems, 15 (1995), 149-159.
doi: 10.1017/S0143385700008282. |
[13] |
M. Ratner,
On raghunathan's measure conjecture, Annals of Mathematics (2), 134 (1991), 545-607.
doi: 10.2307/2944357. |
[14] |
M. Ratner,
Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280.
doi: 10.1215/S0012-7094-91-06311-8. |
[15] |
N. A. Shah,
Expanding translates of curves and dirichlet-minkowski theorem on linear forms, Journal of the American Mathematical Society, 23 (2010), 563-589.
doi: 10.1090/S0894-0347-09-00657-2. |
[16] |
N. A Shah,
Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proceedings Mathematical Sciences, 106 (1996), 105-125.
doi: 10.1007/BF02837164. |
[17] |
N. A. Shah,
Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds, Duke Mathematical Journal, 148 (2009), 281-304.
doi: 10.1215/00127094-2009-027. |
[18] |
N. A. Shah, Equidistribution of expanding translates of curves and dirichlets theorem on diophantine approximation, Inventiones Mathematicae, 177 (2009), 509-532. Google Scholar |
[19] |
N. A. Shah,
Limiting distributions of curves under geodesic flow on hyperbolic manifolds, Duke Mathematical Journal, 148 (2009), 251-279.
doi: 10.1215/00127094-2009-026. |
[20] |
L. Yang,
Equidistribution of expanding curves in homogeneous spaces and diophantine approximation for square matrices, Proc. Amer. Math. Soc., 144 (2016), 5291-5308.
doi: 10.1090/proc/13170. |
[21] |
L. Yang,
Expanding curves in $\mathrm{T}^1(\mathbb{H}^n)$ under geodesic flow and equidistribution in homogeneous spaces, Israel J. Math., 216 (2016), 389-413.
doi: 10.1007/s11856-016-1414-6. |
[1] |
Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145. |
[2] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[3] |
Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623 |
[4] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
[5] |
Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 |
[6] |
Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021004 |
[7] |
Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 |
[8] |
Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265 |
[9] |
Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
[10] |
Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 |
[11] |
Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321 |
[12] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[13] |
Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 |
[14] |
Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 |
[15] |
Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119 |
[16] |
Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065 |
[17] |
Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 |
[18] |
Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115 |
[19] |
Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]