# American Institue of Mathematical Sciences

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 5, 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. [2] J. Athreya, A. Ghosh, A. Prasad, Ultrametric logarithm laws, Ⅱ, Monatsh. Math., 167 (2012), 333-356. [3] Y. Benoist, H. Oh, Equidistribution of rational matrices in their conjugacy classes, Geom. Funct. Anal., 17 (2007), 1-32. [4] Y. Benoist, J.-F. Quint, Random walks on finite volume homogeneous spaces, Invent. Math., 187 (2012), 37-59. [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. [6] A. Broise-Alamichel, F. Paulin, Dynamique sur le rayon modulaire et fractions continues en caractéristique p, J. London Math. Soc., 76 (2007), 399-418. [7] L. Clozel, H. Oh, E. Ullmo, Hecke operators and equidistribution of Hecke points, Invent. Math., 144 (2001), 327-351. [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. [9] S. G. Dani, G. Margulis, Limit distribution of orbits of unipotent flows and values of quadratic forms, Adv. Soviet Math., 16 (1993), 91-137. [10] B. de Mathan, O. Teulié, Problèmes diophantiens simultanés, Monatsh. Math., 143 (2004), 229-245. [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. [12] A. Eskin, M. Mirzakani, Counting closed geodesics in moduli space, J. Mod. Dyn., 5 (2011), 71-105. [13] A. Eskin, H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems, 26 (2006), 163-167. [14] H. Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Z., 33 (1931), 692-713. [15] U. Hamenstädt, Dynamics of the Teichmüller flow on compact invariant sets, J. Mod. Dynamics, 4 (2010), 393-418. [16] A. Lasjaunias, A survey of Diophantine approximation in fields of power series, Monatsh. Math., 130 (2000), 211-229. [17] A. Lubotzky, Lattices in rank one Lie groups over local fields, Geom. Funct. Anal., 1 (1991), 406-431. [18] H. Nagao, On GL(2, K [x]), J. Inst. Polytech. Osaka City Univ. Ser. A, 10 (1959), 117-121. [19] F. Paulin, Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p, Geom. Dedicata., 95 (2002), 65-85. [20] F. Paulin and U. Shapira, High degree continued fraction expansions of quadratic irrationals in positive characteristic, in preparation. [21] G. Prasad, A. Rapinchuk, Subnormal subgroups of the groups of rational points of reductive algebraic groups, Proc. Amer. Math. Soc., 130 (2002), 2219-2227. [22] I. Reiner, Maximal Orders, London Mathematical Society Monographs, No. 5, Academic Press, London-New York, 1975. [23] M. Rosen, Number Theory in Function Fields, Graduate Texts in Mathematics, 210, SpringerVerlag, New York, 2002. [24] P. Sarnak, Reciprocal geodesics, Clay Math. Proc., 7 (2007), 217-237. [25] W. Schmidt, On continued fractions and Diophantine approximation in power series fields, Acta Arith., 95 (2000), 139-166. [26] J. -P. Serre, Local Fields, Graduate Texts in Mathematics, 67, Springer-Verlag, New York-Berlin, 1979. [27] J. -P. Serre, Arbres, amalgames, SL2, 3ème éd. corr. , Astérisque, 46, Soc. Math. France, Paris, 1977. [28] U. Shapira, Full escape of mass for the diagonal group, preprint, arXiv: 1511.07251, to appear in International Mathematics Research Notices. [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. [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. [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. [32] D. Witte Morris, Ratneros Theorems on Unipotent Flows, Chicago Univ. Press, Chicago, IL, 2005.

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##### 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. [2] J. Athreya, A. Ghosh, A. Prasad, Ultrametric logarithm laws, Ⅱ, Monatsh. Math., 167 (2012), 333-356. [3] Y. Benoist, H. Oh, Equidistribution of rational matrices in their conjugacy classes, Geom. Funct. Anal., 17 (2007), 1-32. [4] Y. Benoist, J.-F. Quint, Random walks on finite volume homogeneous spaces, Invent. Math., 187 (2012), 37-59. [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. [6] A. Broise-Alamichel, F. Paulin, Dynamique sur le rayon modulaire et fractions continues en caractéristique p, J. London Math. Soc., 76 (2007), 399-418. [7] L. Clozel, H. Oh, E. Ullmo, Hecke operators and equidistribution of Hecke points, Invent. Math., 144 (2001), 327-351. [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. [9] S. G. Dani, G. Margulis, Limit distribution of orbits of unipotent flows and values of quadratic forms, Adv. Soviet Math., 16 (1993), 91-137. [10] B. de Mathan, O. Teulié, Problèmes diophantiens simultanés, Monatsh. Math., 143 (2004), 229-245. [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. [12] A. Eskin, M. Mirzakani, Counting closed geodesics in moduli space, J. Mod. Dyn., 5 (2011), 71-105. [13] A. Eskin, H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems, 26 (2006), 163-167. [14] H. Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Z., 33 (1931), 692-713. [15] U. Hamenstädt, Dynamics of the Teichmüller flow on compact invariant sets, J. Mod. Dynamics, 4 (2010), 393-418. [16] A. Lasjaunias, A survey of Diophantine approximation in fields of power series, Monatsh. Math., 130 (2000), 211-229. [17] A. Lubotzky, Lattices in rank one Lie groups over local fields, Geom. Funct. Anal., 1 (1991), 406-431. [18] H. Nagao, On GL(2, K [x]), J. Inst. Polytech. Osaka City Univ. Ser. A, 10 (1959), 117-121. [19] F. Paulin, Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p, Geom. Dedicata., 95 (2002), 65-85. [20] F. Paulin and U. Shapira, High degree continued fraction expansions of quadratic irrationals in positive characteristic, in preparation. [21] G. Prasad, A. Rapinchuk, Subnormal subgroups of the groups of rational points of reductive algebraic groups, Proc. Amer. Math. Soc., 130 (2002), 2219-2227. [22] I. Reiner, Maximal Orders, London Mathematical Society Monographs, No. 5, Academic Press, London-New York, 1975. [23] M. Rosen, Number Theory in Function Fields, Graduate Texts in Mathematics, 210, SpringerVerlag, New York, 2002. [24] P. Sarnak, Reciprocal geodesics, Clay Math. Proc., 7 (2007), 217-237. [25] W. Schmidt, On continued fractions and Diophantine approximation in power series fields, Acta Arith., 95 (2000), 139-166. [26] J. -P. Serre, Local Fields, Graduate Texts in Mathematics, 67, Springer-Verlag, New York-Berlin, 1979. [27] J. -P. Serre, Arbres, amalgames, SL2, 3ème éd. corr. , Astérisque, 46, Soc. Math. France, Paris, 1977. [28] U. Shapira, Full escape of mass for the diagonal group, preprint, arXiv: 1511.07251, to appear in International Mathematics Research Notices. [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. [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. [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. [32] D. Witte Morris, Ratneros Theorems on Unipotent Flows, Chicago Univ. Press, Chicago, IL, 2005.
The modular ray ${\rm {PGL}}_2(k_\infty[Y])\backslash\!\backslash{\mathbb{T}}_\infty$
Back and forth paths in cuspidal rays
A partition of the Hecke sphere $S_\nu(n)$
Rational Bruhat-Tits rays
Sector-spheres in Hecke trees
Iterated construction of nested sectors
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