# American Institute of Mathematical Sciences

September  2020, 40(9): 5247-5287. doi: 10.3934/dcds.2020227

## 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

* Corresponding author: Lei Yang

Received  September 2019 Revised  February 2020 Published  June 2020

Fund Project: This material is based upon work supported by the National Science Foundation under Grant Nos. 1301715 and 1700394.
The second author is supported in part by ISF grant 2095/15, ERC grant AdG 267259, NSFC grant 11743006, 11801384 and the Fundamental Research Funds for the Central Universities YJ201769

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.

Citation: 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
##### 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. [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. [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] Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43 [2] Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109 [3] Mateusz Krukowski. Arzelà-Ascoli's theorem in uniform spaces. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 283-294. doi: 10.3934/dcdsb.2018020 [4] John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367 [5] Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics and Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011 [6] Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313 [7] Betseygail Rand, Lorenzo Sadun. An approximation theorem for maps between tiling spaces. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 323-326. doi: 10.3934/dcds.2011.29.323 [8] Sanghoon Kwon, Seonhee Lim. Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 169-186. doi: 10.3934/dcds.2018008 [9] Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067 [10] V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413 [11] Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555 [12] Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945 [13] Fatiha Alabau-Boussouira, Piermarco Cannarsa. A constructive proof of Gibson's stability theorem. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 611-617. doi: 10.3934/dcdss.2013.6.611 [14] Koray Karabina, Edward Knapp, Alfred Menezes. Generalizations of Verheul's theorem to asymmetric pairings. Advances in Mathematics of Communications, 2013, 7 (1) : 103-111. doi: 10.3934/amc.2013.7.103 [15] Shalosh B. Ekhad and Doron Zeilberger. Proof of Conway's lost cosmological theorem. Electronic Research Announcements, 1997, 3: 78-82. [16] Florian Wagener. A parametrised version of Moser's modifying terms theorem. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 719-768. doi: 10.3934/dcdss.2010.3.719 [17] Cristina Stoica. An approximation theorem in classical mechanics. Journal of Geometric Mechanics, 2016, 8 (3) : 359-374. doi: 10.3934/jgm.2016011 [18] Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990 [19] Brandon Seward. Krieger's finite generator theorem for actions of countable groups Ⅱ. Journal of Modern Dynamics, 2019, 15: 1-39. doi: 10.3934/jmd.2019012 [20] Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491

2021 Impact Factor: 1.588