Classifying GL$(n,\mathbb{Z})$-orbits of points and rational subspaces

Leonardo Manuel  Cabrer; Daniele  Mundici

doi:10.3934/dcds.2016005

Article Contents

2016, Volume 36, Issue 9: 4723-4738. Doi: 10.3934/dcds.2016005

This issue Previous Article Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems Next Article Dominated splitting, partial hyperbolicity and positive entropy

Classifying GL$(n,\mathbb{Z})$-orbits of points and rational subspaces

Leonardo Manuel Cabrer^1, and
Daniele Mundici^2,

1.
Institute of Computer Languages Theory and Logic Group, Technische Universität Wien, Favoritenstrasse 9-11, A-1040 Vienna
2.
Department of Mathematics and Computer Science “Ulisse Dini", University of Florence, Viale Morgagni 67/a, I-50134 Florence

Received: July 2015

Revised: March 2016

Published: May 2017

Abstract Related Papers Cited by

Abstract

We first show that the subgroup of the abelian real group $\mathbb{R}$ generated by the coordinates of a point in $x\in\mathbb{R}^n$ completely classifies the GL$(n,\mathbb{Z})$-orbit of $x$. This yields a short proof of J.S. Dani's theorem: the GL$(n,\mathbb{Z})$-orbit of $x\in\mathbb{R}^n$ is dense iff $x_i/x_j\in \mathbb{R}\setminus \mathbb{Q}$ for some $i,j=1,\dots,n$. We then classify GL$(n,\mathbb{Z})$-orbits of rational affine subspaces $F$ of $\mathbb{R}^n$. We prove that the dimension of $F$ together with the volume of a special parallelotope associated to $F$ yields a complete classifier of the GL$(n,\mathbb{Z})$-orbit of $F$.

Keywords:

Mathematics Subject Classification: Primary: 37C85; Secondary: 11B57, 22F05, 37A45.

Citation:

Related Papers

Cited by

References

[1]	L. M. Cabrer and D. Mundici, Classifying orbits of the affine group over the integers, Ergodic Theory Dynam. Systems, in press, (2015), 14pp.doi: 10.1017/etds.2015.45.
[2]	J. S. Dani, Density properties of orbits under discrete groups, J. Indian Math. Soc., 39 (1975), 189-217.
[3]	S. G. Dani and A. Nogueira, On $SL(n,\mathbbZ)_+$-orbits on $\mathbbR^n$ and positive integral solutions of linear inequalities, J. of Number Theory, 129 (2009), 2526-2529.doi: 10.1016/j.jnt.2008.12.010.
[4]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
[5]	G. Ewald, Combinatorial Convexity and Algebraic Geometry, Grad. Texts in Math., Vol. 168, Springer-Verlag, New York, 1996.doi: 10.1007/978-1-4612-4044-0.
[6]	H. Federer, Geometric Measure Theory, Springer, New York, 1969.
[7]	A. Guilloux, A brief remark on orbits of $\mathsf{SL}(2,\mathbbZ)$ in the Euclidean plane}, Ergodic Theory Dynam. Systems, 30 (2010), 1101-1109.doi: 10.1017/S0143385709000315.
[8]	G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition, Clarendon Press, Oxford, 1979.
[9]	M. Laurent and A. Nogueira, Approximation to points in the plane by $\mathsf{SL}(2, \mathbbZ)$-orbits, J. London Math. Soc., 85 (2012), 409-429.doi: 10.1112/jlms/jdr061.
[10]	R. Morelli, The birational geometry of toric varieties, J. Algebraic Geom., 5 (1996), 751-782.
[11]	D. Mundici, The Haar theorem for lattice-ordered abelian groups with order-unit, Discrete Contin. Dyn. Syst., 21 (2008), 537-549.doi: 10.3934/dcds.2008.21.537.
[12]	D. Mundici, Invariant measure under the affine group over $\mathbbZ$, Combin. Probab. Comput., 23 (2014), 248-268.doi: 10.1017/S096354831300062X.
[13]	A. Nogueira, Orbit distribution on $\mathbbR^2$ under the natural action of $ SL(2,\mathbbZ)$, Indag. Math. (N.S.), 13 (2002), 103-124.doi: 10.1016/S0019-3577(02)90009-1.
[14]	A. Nogueira, Lattice orbit distribution on $\mathbbR^2$, Ergodic Theory Dynam. Systems, 30 (2010), 1201-1214. Erratum, ibid., p. 1215.doi: 10.1017/S0143385709000558.
[15]	T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties, A Series of Modern Surveys in Mathematics, Vol. 3, Springer-Verlag, New York, 1988.
[16]	J. R. Stallings, Lectures on Polyhedral Topology, Tata Inst. Fund. Res., Lectures in Mathematics, Vol. 43, Mumbay, 1967.
[17]	E. Witten, $\mathsf{SL}(2, \mathbbZ)$ action on three-dimensional conformal field theories with abelian symmetry, In: From Fields to Strings: Circumnavigating Theoretical Physics, Ian Kogan Memorial Collection (in 3 volumes), M. Shifman et al., Eds., World Scientific, Singapore, 2 (2005), 1173-1200.
[18]	J. Włodarczyk, Decompositions of birational toric maps in blow-ups and blow-downs, Trans. Amer. Math. Soc., 349 (1997), 373-411.doi: 10.1090/S0002-9947-97-01701-7.