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

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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.
[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.
[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.
[4]	A. Katok, The special representation theorem for multi-dimensional group actions, Asterisque, 49 (1977), 117-140.
[5]	C. McMullen, Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal., 8 (1998), 304-314. doi: 10.1007/s000390050058.
[6]	M. Ratner, Horocycle flows are loosely Bernoulli, Israel Journal of Mathematics, 31 (1978), 122-132. doi: 10.1007/BF02760543.
[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.
[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).

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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.
[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.
[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.
[4]	A. Katok, The special representation theorem for multi-dimensional group actions, Asterisque, 49 (1977), 117-140.
[5]	C. McMullen, Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal., 8 (1998), 304-314. doi: 10.1007/s000390050058.
[6]	M. Ratner, Horocycle flows are loosely Bernoulli, Israel Journal of Mathematics, 31 (1978), 122-132. doi: 10.1007/BF02760543.
[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.
[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).

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