# American Institue of Mathematical Sciences

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

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

 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.
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
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##### References:
 [1] D. Burago, B. Kleiner, Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps, Geom. Funct. Anal., 8 (1998), 273-282. [2] A. Haynes, M. Kelly, B. Weiss, Equivalence relations on separated nets arising from linear toral flows, Proceedings of the London Mathematical Society, 109 (2014), 1203-1228. [3] A. -P. José, D. Coronel, J. -M. Gambaudo, Linearly repetitive Delone sets are rectifiable, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 30 (2013), 275-290. [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. [6] M. Ratner, Horocycle flows are loosely Bernoulli, Israel Journal of Mathematics, 31 (1978), 122-132. [7] C. Salvatore, L. Flaminio, Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds, Journal of Modern Dynamics, 9 (2015), 305-353. [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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