2017, 11: 369-407. doi: 10.3934/jmd.2017015

Escape of mass in homogeneous dynamics in positive characteristic

1. 

Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark

2. 

Laboratoire de mathématiques déOrsay, UMR 8628 Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France

3. 

Mathematics Department, Technion, Israel Institute of Technology, Haifa, 32000 Israel

Received  December 20, 2015 Revised  December 05, 2016 Published  May 2017

We show that in positive characteristic the homogeneous probability measure supported on a periodic orbit of the diagonal group in the space of $2$-lattices, when varied along rays of Hecke trees, may behave in sharp contrast to the zero characteristic analogue: For a large set of rays, the measures fail to converge to the uniform probability measure on the space of $2$-lattices. More precisely, we prove that when the ray is rational there is uniform escape of mass, that there are uncountably many rays giving rise to escape of mass, and that there are rays along which the measures accumulate on measures which are not absolutely continuous with respect to the uniform measure on the space of $2$-lattices.

Citation: Alexander Kemarsky, Frédéric Paulin, Uri Shapira. Escape of mass in homogeneous dynamics in positive characteristic. Journal of Modern Dynamics, 2017, 11: 369-407. doi: 10.3934/jmd.2017015
References:
[1]

M. Aka and U. Shapira, On the evolution of continued fraction expansions in a fixed quadratic field, preprint, arXiv: 1201.1280, to appear in Journal d'Analyse Mathématique. Google Scholar

[2]

J. AthreyaA. Ghosh and A. Prasad, Ultrametric logarithm laws, Ⅱ, Monatsh. Math., 167 (2012), 333-356.  doi: 10.1007/s00605-012-0376-y.  Google Scholar

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Y. Benoist and H. Oh, Equidistribution of rational matrices in their conjugacy classes, Geom. Funct. Anal., 17 (2007), 1-32.  doi: 10.1007/s00039-006-0585-4.  Google Scholar

[4]

Y. Benoist and J.-F. Quint, Random walks on finite volume homogeneous spaces, Invent. Math., 187 (2012), 37-59.  doi: 10.1007/s00222-011-0328-5.  Google Scholar

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A. Broise-Alamichel and F. Paulin, Dynamique sur le rayon modulaire et fractions continues en caractéristique p, J. London Math. Soc., 76 (2007), 399-418.  doi: 10.1112/jlms/jdm063.  Google Scholar

[7]

L. ClozelH. Oh and E. Ullmo, Hecke operators and equidistribution of Hecke points, Invent. Math., 144 (2001), 327-351.  doi: 10.1007/s002220100126.  Google Scholar

[8]

L. Clozel and E. Ullmo, Equidistribution des points de Hecke, in Contribution to Automorphic Forms, Geometry and Number Theory (eds. H. Hida, D. Ramakrishnan and F. Shahidi), Johns Hopkins Univ. Press, 2004,193-254.  Google Scholar

[9]

S. G. Dani and G. Margulis, Limit distribution of orbits of unipotent flows and values of quadratic forms, Adv. Soviet Math., 16 (1993), 91-137.   Google Scholar

[10]

B. de Mathan and O. Teulié, Problèmes diophantiens simultanés, Monatsh. Math., 143 (2004), 229-245.  doi: 10.1007/s00605-003-0199-y.  Google Scholar

[11]

A. Eskin and G. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds, in Random Walks and Geometry, Walter de Gruiter, Berlin, 2004,431-444.  Google Scholar

[12]

A. Eskin and M. Mirzakani, Counting closed geodesics in moduli space, J. Mod. Dyn., 5 (2011), 71-105.  doi: 10.3934/jmd.2011.5.71.  Google Scholar

[13]

A. Eskin and H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems, 26 (2006), 163-167.  doi: 10.1017/S0143385705000428.  Google Scholar

[14]

H. Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Z., 33 (1931), 692-713.  doi: 10.1007/BF01174375.  Google Scholar

[15]

U. Hamenstädt, Dynamics of the Teichmüller flow on compact invariant sets, J. Mod. Dynamics, 4 (2010), 393-418.  doi: 10.3934/jmd.2010.4.393.  Google Scholar

[16]

A. Lasjaunias, A survey of Diophantine approximation in fields of power series, Monatsh. Math., 130 (2000), 211-229.  doi: 10.1007/s006050070036.  Google Scholar

[17]

A. Lubotzky, Lattices in rank one Lie groups over local fields, Geom. Funct. Anal., 1 (1991), 406-431.  doi: 10.1007/BF01895641.  Google Scholar

[18]

H. Nagao, On GL(2, K [x]), J. Inst. Polytech. Osaka City Univ. Ser. A, 10 (1959), 117-121.   Google Scholar

[19]

F. Paulin, Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p, Geom. Dedicata., 95 (2002), 65-85.  doi: 10.1023/A:1021270631563.  Google Scholar

[20]

F. Paulin and U. Shapira, High degree continued fraction expansions of quadratic irrationals in positive characteristic, in preparation. Google Scholar

[21]

G. Prasad and A. Rapinchuk, Subnormal subgroups of the groups of rational points of reductive algebraic groups, Proc. Amer. Math. Soc., 130 (2002), 2219-2227.  doi: 10.1090/S0002-9939-02-06514-0.  Google Scholar

[22]

I. Reiner, Maximal Orders, London Mathematical Society Monographs, No. 5, Academic Press, London-New York, 1975.  Google Scholar

[23]

M. Rosen, Number Theory in Function Fields, Graduate Texts in Mathematics, 210, SpringerVerlag, New York, 2002. doi: 10.1007/978-1-4757-6046-0.  Google Scholar

[24]

P. Sarnak, Reciprocal geodesics, Clay Math. Proc., 7 (2007), 217-237.   Google Scholar

[25]

W. Schmidt, On continued fractions and Diophantine approximation in power series fields, Acta Arith., 95 (2000), 139-166.   Google Scholar

[26]

J. -P. Serre, Local Fields, Graduate Texts in Mathematics, 67, Springer-Verlag, New York-Berlin, 1979.  Google Scholar

[27]

J. -P. Serre, Arbres, amalgames, SL2, 3ème éd. corr., Astérisque, 46, Soc. Math. France, Paris, 1977.  Google Scholar

[28]

U. Shapira, Full escape of mass for the diagonal group, preprint, arXiv: 1511.07251, to appear in International Mathematics Research Notices. doi: 10.1093/imrn/rnw144.  Google Scholar

[29]

J. Tits, Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups (Boulder, 1965), Proc. Symp. Pure Math. Ⅸ, Amer. Math. Soc., 1966, 33-62.  Google Scholar

[30]

J. Tits, Reductive groups over local fields, in Automorphic Forms, Representations and LFunctions (Corvallis, 1977), Part 1, Proc. Symp. Pure Math. ⅩⅩⅩⅢ, Amer. Math. Soc., 1979, 29-69.  Google Scholar

[31]

A. Weil, On the analogue of the modular group in characteristic p, in Functional Analysis and Related Fields (Chicago, 1968), Springer, 1970,211-223.  Google Scholar

[32] D. Witte Morris, Ratneros Theorems on Unipotent Flows, Chicago Univ. Press, Chicago, IL, 2005.   Google Scholar

show all references

References:
[1]

M. Aka and U. Shapira, On the evolution of continued fraction expansions in a fixed quadratic field, preprint, arXiv: 1201.1280, to appear in Journal d'Analyse Mathématique. Google Scholar

[2]

J. AthreyaA. Ghosh and A. Prasad, Ultrametric logarithm laws, Ⅱ, Monatsh. Math., 167 (2012), 333-356.  doi: 10.1007/s00605-012-0376-y.  Google Scholar

[3]

Y. Benoist and H. Oh, Equidistribution of rational matrices in their conjugacy classes, Geom. Funct. Anal., 17 (2007), 1-32.  doi: 10.1007/s00039-006-0585-4.  Google Scholar

[4]

Y. Benoist and J.-F. Quint, Random walks on finite volume homogeneous spaces, Invent. Math., 187 (2012), 37-59.  doi: 10.1007/s00222-011-0328-5.  Google Scholar

[5]

E. Breuillard, Equidistribution des orbites toriques sur les espaces homogènes (d'aprés M. Einsiedler, E. Lindenstrauss, Ph. Michel, A. Venkatesh), Séminaire Bourbaki, Exp. 1008, Astérisque, 332 (2010), 305–339.  Google Scholar

[6]

A. Broise-Alamichel and F. Paulin, Dynamique sur le rayon modulaire et fractions continues en caractéristique p, J. London Math. Soc., 76 (2007), 399-418.  doi: 10.1112/jlms/jdm063.  Google Scholar

[7]

L. ClozelH. Oh and E. Ullmo, Hecke operators and equidistribution of Hecke points, Invent. Math., 144 (2001), 327-351.  doi: 10.1007/s002220100126.  Google Scholar

[8]

L. Clozel and E. Ullmo, Equidistribution des points de Hecke, in Contribution to Automorphic Forms, Geometry and Number Theory (eds. H. Hida, D. Ramakrishnan and F. Shahidi), Johns Hopkins Univ. Press, 2004,193-254.  Google Scholar

[9]

S. G. Dani and G. Margulis, Limit distribution of orbits of unipotent flows and values of quadratic forms, Adv. Soviet Math., 16 (1993), 91-137.   Google Scholar

[10]

B. de Mathan and O. Teulié, Problèmes diophantiens simultanés, Monatsh. Math., 143 (2004), 229-245.  doi: 10.1007/s00605-003-0199-y.  Google Scholar

[11]

A. Eskin and G. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds, in Random Walks and Geometry, Walter de Gruiter, Berlin, 2004,431-444.  Google Scholar

[12]

A. Eskin and M. Mirzakani, Counting closed geodesics in moduli space, J. Mod. Dyn., 5 (2011), 71-105.  doi: 10.3934/jmd.2011.5.71.  Google Scholar

[13]

A. Eskin and H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems, 26 (2006), 163-167.  doi: 10.1017/S0143385705000428.  Google Scholar

[14]

H. Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Z., 33 (1931), 692-713.  doi: 10.1007/BF01174375.  Google Scholar

[15]

U. Hamenstädt, Dynamics of the Teichmüller flow on compact invariant sets, J. Mod. Dynamics, 4 (2010), 393-418.  doi: 10.3934/jmd.2010.4.393.  Google Scholar

[16]

A. Lasjaunias, A survey of Diophantine approximation in fields of power series, Monatsh. Math., 130 (2000), 211-229.  doi: 10.1007/s006050070036.  Google Scholar

[17]

A. Lubotzky, Lattices in rank one Lie groups over local fields, Geom. Funct. Anal., 1 (1991), 406-431.  doi: 10.1007/BF01895641.  Google Scholar

[18]

H. Nagao, On GL(2, K [x]), J. Inst. Polytech. Osaka City Univ. Ser. A, 10 (1959), 117-121.   Google Scholar

[19]

F. Paulin, Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p, Geom. Dedicata., 95 (2002), 65-85.  doi: 10.1023/A:1021270631563.  Google Scholar

[20]

F. Paulin and U. Shapira, High degree continued fraction expansions of quadratic irrationals in positive characteristic, in preparation. Google Scholar

[21]

G. Prasad and A. Rapinchuk, Subnormal subgroups of the groups of rational points of reductive algebraic groups, Proc. Amer. Math. Soc., 130 (2002), 2219-2227.  doi: 10.1090/S0002-9939-02-06514-0.  Google Scholar

[22]

I. Reiner, Maximal Orders, London Mathematical Society Monographs, No. 5, Academic Press, London-New York, 1975.  Google Scholar

[23]

M. Rosen, Number Theory in Function Fields, Graduate Texts in Mathematics, 210, SpringerVerlag, New York, 2002. doi: 10.1007/978-1-4757-6046-0.  Google Scholar

[24]

P. Sarnak, Reciprocal geodesics, Clay Math. Proc., 7 (2007), 217-237.   Google Scholar

[25]

W. Schmidt, On continued fractions and Diophantine approximation in power series fields, Acta Arith., 95 (2000), 139-166.   Google Scholar

[26]

J. -P. Serre, Local Fields, Graduate Texts in Mathematics, 67, Springer-Verlag, New York-Berlin, 1979.  Google Scholar

[27]

J. -P. Serre, Arbres, amalgames, SL2, 3ème éd. corr., Astérisque, 46, Soc. Math. France, Paris, 1977.  Google Scholar

[28]

U. Shapira, Full escape of mass for the diagonal group, preprint, arXiv: 1511.07251, to appear in International Mathematics Research Notices. doi: 10.1093/imrn/rnw144.  Google Scholar

[29]

J. Tits, Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups (Boulder, 1965), Proc. Symp. Pure Math. Ⅸ, Amer. Math. Soc., 1966, 33-62.  Google Scholar

[30]

J. Tits, Reductive groups over local fields, in Automorphic Forms, Representations and LFunctions (Corvallis, 1977), Part 1, Proc. Symp. Pure Math. ⅩⅩⅩⅢ, Amer. Math. Soc., 1979, 29-69.  Google Scholar

[31]

A. Weil, On the analogue of the modular group in characteristic p, in Functional Analysis and Related Fields (Chicago, 1968), Springer, 1970,211-223.  Google Scholar

[32] D. Witte Morris, Ratneros Theorems on Unipotent Flows, Chicago Univ. Press, Chicago, IL, 2005.   Google Scholar
Figure 1.  The modular ray ${\rm {PGL}}_2(k_\infty[Y])\backslash\!\backslash{\mathbb{T}}_\infty$
Figure 2.  Back and forth paths in cuspidal rays
Figure 3.  A partition of the Hecke sphere $S_\nu(n)$
Figure 4.  Rational Bruhat-Tits rays
Figure 5.  Sector-spheres in Hecke trees
Figure 6.  Iterated construction of nested sectors
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