September  2016, 36(9): 4723-4738. doi: 10.3934/dcds.2016005

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

1. 

Institute of Computer Languages Theory and Logic Group, Technische Universität Wien, Favoritenstrasse 9-11, A-1040 Vienna, Austria

2. 

Department of Mathematics and Computer Science “Ulisse Dini", University of Florence, Viale Morgagni 67/a, I-50134 Florence, Italy

Received  July 2015 Revised  March 2016 Published  May 2016

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 $F$.
Citation: Leonardo Manuel Cabrer, Daniele Mundici. Classifying GL$(n,\mathbb{Z})$-orbits of points and rational subspaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4723-4738. doi: 10.3934/dcds.2016005
References:
[1]

L. M. Cabrer and D. Mundici, Classifying orbits of the affine group over the integers,, Ergodic Theory Dynam. Systems, (2015). doi: 10.1017/etds.2015.45. Google Scholar

[2]

J. S. Dani, Density properties of orbits under discrete groups,, J. Indian Math. Soc., 39 (1975), 189. Google Scholar

[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. doi: 10.1016/j.jnt.2008.12.010. Google Scholar

[4]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992). Google Scholar

[5]

G. Ewald, Combinatorial Convexity and Algebraic Geometry,, Grad. Texts in Math., (1996). doi: 10.1007/978-1-4612-4044-0. Google Scholar

[6]

H. Federer, Geometric Measure Theory,, Springer, (1969). Google Scholar

[7]

A. Guilloux, A brief remark on orbits of $\mathsf{SL}(2,\mathbbZ)$ in the Euclidean plane},, Ergodic Theory Dynam. Systems, 30 (2010), 1101. doi: 10.1017/S0143385709000315. Google Scholar

[8]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers,, Fifth edition, (1979). Google Scholar

[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. doi: 10.1112/jlms/jdr061. Google Scholar

[10]

R. Morelli, The birational geometry of toric varieties,, J. Algebraic Geom., 5 (1996), 751. Google Scholar

[11]

D. Mundici, The Haar theorem for lattice-ordered abelian groups with order-unit,, Discrete Contin. Dyn. Syst., 21 (2008), 537. doi: 10.3934/dcds.2008.21.537. Google Scholar

[12]

D. Mundici, Invariant measure under the affine group over $\mathbbZ$,, Combin. Probab. Comput., 23 (2014), 248. doi: 10.1017/S096354831300062X. Google Scholar

[13]

A. Nogueira, Orbit distribution on $\mathbbR^2$ under the natural action of $ SL(2,\mathbbZ)$,, Indag. Math. (N.S.), 13 (2002), 103. doi: 10.1016/S0019-3577(02)90009-1. Google Scholar

[14]

A. Nogueira, Lattice orbit distribution on $\mathbbR^2$,, Ergodic Theory Dynam. Systems, 30 (2010), 1201. doi: 10.1017/S0143385709000558. Google Scholar

[15]

T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties,, A Series of Modern Surveys in Mathematics, (1988). Google Scholar

[16]

J. R. Stallings, Lectures on Polyhedral Topology,, Tata Inst. Fund. Res., (1967). Google Scholar

[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, 2 (2005), 1173. Google Scholar

[18]

J. Włodarczyk, Decompositions of birational toric maps in blow-ups and blow-downs,, Trans. Amer. Math. Soc., 349 (1997), 373. doi: 10.1090/S0002-9947-97-01701-7. Google Scholar

show all references

References:
[1]

L. M. Cabrer and D. Mundici, Classifying orbits of the affine group over the integers,, Ergodic Theory Dynam. Systems, (2015). doi: 10.1017/etds.2015.45. Google Scholar

[2]

J. S. Dani, Density properties of orbits under discrete groups,, J. Indian Math. Soc., 39 (1975), 189. Google Scholar

[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. doi: 10.1016/j.jnt.2008.12.010. Google Scholar

[4]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992). Google Scholar

[5]

G. Ewald, Combinatorial Convexity and Algebraic Geometry,, Grad. Texts in Math., (1996). doi: 10.1007/978-1-4612-4044-0. Google Scholar

[6]

H. Federer, Geometric Measure Theory,, Springer, (1969). Google Scholar

[7]

A. Guilloux, A brief remark on orbits of $\mathsf{SL}(2,\mathbbZ)$ in the Euclidean plane},, Ergodic Theory Dynam. Systems, 30 (2010), 1101. doi: 10.1017/S0143385709000315. Google Scholar

[8]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers,, Fifth edition, (1979). Google Scholar

[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. doi: 10.1112/jlms/jdr061. Google Scholar

[10]

R. Morelli, The birational geometry of toric varieties,, J. Algebraic Geom., 5 (1996), 751. Google Scholar

[11]

D. Mundici, The Haar theorem for lattice-ordered abelian groups with order-unit,, Discrete Contin. Dyn. Syst., 21 (2008), 537. doi: 10.3934/dcds.2008.21.537. Google Scholar

[12]

D. Mundici, Invariant measure under the affine group over $\mathbbZ$,, Combin. Probab. Comput., 23 (2014), 248. doi: 10.1017/S096354831300062X. Google Scholar

[13]

A. Nogueira, Orbit distribution on $\mathbbR^2$ under the natural action of $ SL(2,\mathbbZ)$,, Indag. Math. (N.S.), 13 (2002), 103. doi: 10.1016/S0019-3577(02)90009-1. Google Scholar

[14]

A. Nogueira, Lattice orbit distribution on $\mathbbR^2$,, Ergodic Theory Dynam. Systems, 30 (2010), 1201. doi: 10.1017/S0143385709000558. Google Scholar

[15]

T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties,, A Series of Modern Surveys in Mathematics, (1988). Google Scholar

[16]

J. R. Stallings, Lectures on Polyhedral Topology,, Tata Inst. Fund. Res., (1967). Google Scholar

[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, 2 (2005), 1173. Google Scholar

[18]

J. Włodarczyk, Decompositions of birational toric maps in blow-ups and blow-downs,, Trans. Amer. Math. Soc., 349 (1997), 373. doi: 10.1090/S0002-9947-97-01701-7. Google Scholar

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