American Institute of Mathematical Sciences

Advanced
August  2017, 37(8): 4231-4238. doi: 10.3934/dcds.2017180

Separated nets arising from certain higher rank $\mathbb{R}^k$ actions on homogeneous spaces

Changguang Dong

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received  December 2016 Revised  March 2017 Published  April 2017

Fund Project: Research was partially supported by NSF grant DMS 1304830.

We prove that separated net arising from certain higher rank $\mathbb R.k$ actions on homogeneous spaces is bi-Lipschitz equivalent to a lattice.

Keywords: Separated net, abelian action, equidistribution.
Mathematics Subject Classification: Primary: 37C85, 37A17.
Citation: Changguang Dong. Separated nets arising from certain higher rank $\mathbb{R}^k$ actions on homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4231-4238. doi: 10.3934/dcds.2017180
References:
[1]

D. Burago and B. Kleiner, Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps, Geom. Funct. Anal., 8 (1998), 273-282.  doi: 10.1007/s000390050056.  Google Scholar

[2]

A. Haynes, M. Kelly and B. Weiss, Equivalence relations on separated nets arising from linear toral flows, Proceedings of the London Mathematical Society, 109 (2014), 1203-1228.  doi: 10.1112/plms/pdu036.  Google Scholar

[3]

A. -P. José, D. Coronel and J. -M. Gambaudo, Linearly repetitive Delone sets are rectifiable, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 30 (2013), 275-290.  doi: 10.1016/j.anihpc.2012.07.006.  Google Scholar

[4]

A. Katok, The special representation theorem for multi-dimensional group actions, Asterisque, 49 (1977), 117-140.   Google Scholar

[5]

C. McMullen, Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal., 8 (1998), 304-314.  doi: 10.1007/s000390050058.  Google Scholar

[6]

M. Ratner, Horocycle flows are loosely Bernoulli, Israel Journal of Mathematics, 31 (1978), 122-132.  doi: 10.1007/BF02760543.  Google Scholar

[7]

C. Salvatore and L. Flaminio, Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds, Journal of Modern Dynamics, 9 (2015), 305-353.  doi: 10.3934/jmd.2015.9.305.  Google Scholar

[8]

J. Tanis, Effective equidistribution for some unipotent flows in PSL ${\left({2, \mathbb{R}} \right)^k}$ mod cocompact irreducible lattice, preprint, arXiv: 1412.5353v3 (2015). Google Scholar

show all references

References:
[1]

D. Burago and B. Kleiner, Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps, Geom. Funct. Anal., 8 (1998), 273-282.  doi: 10.1007/s000390050056.  Google Scholar

[2]

A. Haynes, M. Kelly and B. Weiss, Equivalence relations on separated nets arising from linear toral flows, Proceedings of the London Mathematical Society, 109 (2014), 1203-1228.  doi: 10.1112/plms/pdu036.  Google Scholar

[3]

A. -P. José, D. Coronel and J. -M. Gambaudo, Linearly repetitive Delone sets are rectifiable, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 30 (2013), 275-290.  doi: 10.1016/j.anihpc.2012.07.006.  Google Scholar

[4]

A. Katok, The special representation theorem for multi-dimensional group actions, Asterisque, 49 (1977), 117-140.   Google Scholar

[5]

C. McMullen, Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal., 8 (1998), 304-314.  doi: 10.1007/s000390050058.  Google Scholar

[6]

M. Ratner, Horocycle flows are loosely Bernoulli, Israel Journal of Mathematics, 31 (1978), 122-132.  doi: 10.1007/BF02760543.  Google Scholar

[7]

C. Salvatore and L. Flaminio, Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds, Journal of Modern Dynamics, 9 (2015), 305-353.  doi: 10.3934/jmd.2015.9.305.  Google Scholar

[8]

J. Tanis, Effective equidistribution for some unipotent flows in PSL ${\left({2, \mathbb{R}} \right)^k}$ mod cocompact irreducible lattice, preprint, arXiv: 1412.5353v3 (2015). Google Scholar

[1]

Salvatore Cosentino, Livio Flaminio. Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds. Journal of Modern Dynamics, 2015, 9: 305-353. doi: 10.3934/jmd.2015.9.305
[2]

Stefano Cosenza, Paolo Crucitti, Luigi Fortuna, Mattia Frasca, Manuela La Rosa, Cecilia Stagni, Lisa Usai. From Net Topology to Synchronization in HR Neuron Grids. Mathematical Biosciences & Engineering, 2005, 2 (1) : 53-77. doi: 10.3934/mbe.2005.2.53
[3]

J. Douglas Wright. On the spectrum of the superposition of separated potentials.. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 273-281. doi: 10.3934/dcdsb.2013.18.273
[4]

Benjamin Dozier. Equidistribution of saddle connections on translation surfaces. Journal of Modern Dynamics, 2019, 14: 87-120. doi: 10.3934/jmd.2019004
[5]

Brandon Seward. Every action of a nonamenable group is the factor of a small action. Journal of Modern Dynamics, 2014, 8 (2) : 251-270. doi: 10.3934/jmd.2014.8.251
[6]

Michael Hutchings. Mean action and the Calabi invariant. Journal of Modern Dynamics, 2016, 10: 511-539. doi: 10.3934/jmd.2016.10.511
[7]

Wenyu Pan. Effective equidistribution of circles in the limit sets of Kleinian groups. Journal of Modern Dynamics, 2017, 11: 189-217. doi: 10.3934/jmd.2017009
[8]

V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511
[9]

John Franks, Michael Handel, Kamlesh Parwani. Fixed points of Abelian actions. Journal of Modern Dynamics, 2007, 1 (3) : 443-464. doi: 10.3934/jmd.2007.1.443
[10]

Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492
[11]

Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020  doi: 10.3934/mfc.2020013
[12]

Chady Ghnatios, Guangtao Xu, Adrien Leygue, Michel Visonneau, Francisco Chinesta, Alain Cimetiere. On the space separated representation when addressing the solution of PDE in complex domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 475-500. doi: 10.3934/dcdss.2016008
[13]

Carlos Munuera, Wanderson Tenório, Fernando Torres. Locally recoverable codes from algebraic curves with separated variables. Advances in Mathematics of Communications, 2020, 14 (2) : 265-278. doi: 10.3934/amc.2020019
[14]

Sandra Carillo, Mauro Lo Schiavo, Cornelia Schiebold. Abelian versus non-Abelian Bäcklund charts: Some remarks. Evolution Equations & Control Theory, 2019, 8 (1) : 43-55. doi: 10.3934/eect.2019003
[15]

Magdalena Czubak, Robert L. Jerrard. Topological defects in the abelian Higgs model. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1933-1968. doi: 10.3934/dcds.2015.35.1933
[16]

Eldho K. Thomas, Nadya Markin, Frédérique Oggier. On Abelian group representability of finite groups. Advances in Mathematics of Communications, 2014, 8 (2) : 139-152. doi: 10.3934/amc.2014.8.139
[17]

S. Eigen, V. S. Prasad. Tiling Abelian groups with a single tile. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 361-365. doi: 10.3934/dcds.2006.16.361
[18]

Sanghoon Kwon, Seonhee Lim. Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 169-186. doi: 10.3934/dcds.2018008
[19]

Kathryn Dabbs, Michael Kelly, Han Li. Effective equidistribution of translates of maximal horospherical measures in the space of lattices. Journal of Modern Dynamics, 2016, 10: 229-254. doi: 10.3934/jmd.2016.10.229
[20]

Nimish Shah, Lei Yang. Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5247-5287. doi: 10.3934/dcds.2020227

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (36)
  • HTML views (40)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences