American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcds.2021183
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification

Runlin Zhang

Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road Haidian District, Beijing, 100871, China

Received  June 2021 Revised  September 2021 Early access November 2021

Let $ \mathit{\boldsymbol{\mathrm{G}}} $ be a semisimple linear algebraic group defined over rational numbers, $ \mathrm{K} $ be a maximal compact subgroup of its real points and $ \Gamma $ be an arithmetic lattice. One can associate a probability measure $ \mu_{ \mathrm{H}} $ on $ \Gamma \backslash \mathrm{G} $ for each subgroup $ \mathit{\boldsymbol{\mathrm{H}}} $ of $ \mathit{\boldsymbol{\mathrm{G}}} $ defined over $ \mathbb{Q} $ with no non-trivial rational characters. As G acts on $ \Gamma \backslash \mathrm{G} $ from the right, we can push forward this measure by elements from $ \mathrm{G} $. By pushing down these measures to $ \Gamma \backslash \mathrm{G}/ \mathrm{K} $, we call them homogeneous. It is a natural question to ask what are the possible weak-$ * $ limits of homogeneous measures. In the non-divergent case this has been answered by Eskin–Mozes–Shah. In the divergent case Daw–Gorodnik–Ullmo prove a refined version in some non-trivial compactifications of $ \Gamma \backslash \mathrm{G}/ \mathrm{K} $ for $ \mathit{\boldsymbol{\mathrm{H}}} $ generated by real unipotents. In the present article we build on their work and generalize the theorem to the case of general $ \mathit{\boldsymbol{\mathrm{H}}} $ with no non-trivial rational characters. Our results rely on (1) a non-divergent criterion on $ {\text{SL}}_n $ proved by geometry of numbers and a theorem of Kleinbock–Margulis; (2) relations between partial Borel–Serre compactifications associated with different groups proved by geometric invariant theory and reduction theory. 193 words.

Keywords: Homogeneous dynamics, Borel–Serre compactification, equidistribution, non-divergence criterion, measure rigidity.
Mathematics Subject Classification: Primary: 37A17; Secondary: 22E46.
Citation: Runlin Zhang. Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021183
References:
[1]

A. Borel, Introduction to Arithmetic Groups, University Lectures Series, vol. 73, American Mathematical Soc., 2019. doi: 10.1090/ulect/073.  Google Scholar

[2]

A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006.  Google Scholar

[3]

A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv., 48 (1973), 436-491.  doi: 10.1007/BF02566134.  Google Scholar

[4]

S. G. Dani and G. A. Margulis, Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), 1-17.  doi: 10.1007/BF02872005.  Google Scholar

[5]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel$\overset{'}{\mathop{f}}$and Seminar, Adv. Soviet Math., Amer. Math. Soc., Providence, RI, 16 (1993), 91-137.   Google Scholar

[6]

C. Daw, A. Gorodnik and E. Ullmo, Convergence of measures on compactifications of locally symmetric spaces, Math. Z., 297 (2021), 1293-1328.  doi: 10.1007/s00209-020-02558-w.  Google Scholar

[7]

C. Daw, A. Gorodnik, E. Ullmo and J. Li, The Space of Homogeneous Probability Measures on $\overline{\Gamma\backslash X}^S_{ \rm max }$ is Compact, e-prints, 2019, arXiv: 1910.04568. Google Scholar

[8]

W. Duke, Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J., 71 (1993), 143-179.  doi: 10.1215/S0012-7094-93-07107-4.  Google Scholar

[9]

A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J., 71 (1993), 181-209.  doi: 10.1215/S0012-7094-93-07108-6.  Google Scholar

[10]

A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., 147 (1998), 93-141.  doi: 10.2307/120984.  Google Scholar

[11]

A. Eskin, S. Mozes and N. Shah, Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math., 143 (1996), 253-299.  doi: 10.2307/2118644.  Google Scholar

[12]

A. Eskin, S. Mozes and N. Shah, Non-divergence of translates of certain algebraic measures, Geom. Funct. Anal., 7 (1997), 48-80.  doi: 10.1007/PL00001616.  Google Scholar

[13]

D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math., 148 (1998), 339-360.  doi: 10.2307/120997.  Google Scholar

[14]

B. Klingler and A. Yafaev, The André-Oort conjecture, Ann. of Math., 180 (2014), 867-925.  doi: 10.4007/annals.2014.180.3.2.  Google Scholar

[15]

G. D. Mostow, Self-adjoint groups, Ann. of Math., 62 (1955), 44-55.  doi: 10.2307/2007099.  Google Scholar

[16]

S. Mozes and N. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems, 15 (1995), 149-159.  doi: 10.1017/S0143385700008282.  Google Scholar

[17]

D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, 3$^{rd}$ edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994.  Google Scholar

[18]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607.  doi: 10.2307/2944357.  Google Scholar

[19]

T. A. Springer, Linear Algebraic Groups, 2$^{nd}$ edition, Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. doi: 10.1007/978-0-8176-4840-4.  Google Scholar

[20]

E. Ullmo and A. Yafaev, Galois orbits and equidistribution of special subvarieties: Towards the André-Oort conjecture, Ann. of Math., 180 (2014), 823-865.  doi: 10.4007/annals.2014.180.3.1.  Google Scholar

[21]

R. Zhang, Translates of homogeneous measures associated with Observable Subgroups on some homogeneous spaces, arXiv e-prints, 2020, arXiv: 1909.02666. Google Scholar

show all references

References:
[1]

A. Borel, Introduction to Arithmetic Groups, University Lectures Series, vol. 73, American Mathematical Soc., 2019. doi: 10.1090/ulect/073.  Google Scholar

[2]

A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006.  Google Scholar

[3]

A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv., 48 (1973), 436-491.  doi: 10.1007/BF02566134.  Google Scholar

[4]

S. G. Dani and G. A. Margulis, Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), 1-17.  doi: 10.1007/BF02872005.  Google Scholar

[5]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel$\overset{'}{\mathop{f}}$and Seminar, Adv. Soviet Math., Amer. Math. Soc., Providence, RI, 16 (1993), 91-137.   Google Scholar

[6]

C. Daw, A. Gorodnik and E. Ullmo, Convergence of measures on compactifications of locally symmetric spaces, Math. Z., 297 (2021), 1293-1328.  doi: 10.1007/s00209-020-02558-w.  Google Scholar

[7]

C. Daw, A. Gorodnik, E. Ullmo and J. Li, The Space of Homogeneous Probability Measures on $\overline{\Gamma\backslash X}^S_{ \rm max }$ is Compact, e-prints, 2019, arXiv: 1910.04568. Google Scholar

[8]

W. Duke, Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J., 71 (1993), 143-179.  doi: 10.1215/S0012-7094-93-07107-4.  Google Scholar

[9]

A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J., 71 (1993), 181-209.  doi: 10.1215/S0012-7094-93-07108-6.  Google Scholar

[10]

A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., 147 (1998), 93-141.  doi: 10.2307/120984.  Google Scholar

[11]

A. Eskin, S. Mozes and N. Shah, Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math., 143 (1996), 253-299.  doi: 10.2307/2118644.  Google Scholar

[12]

A. Eskin, S. Mozes and N. Shah, Non-divergence of translates of certain algebraic measures, Geom. Funct. Anal., 7 (1997), 48-80.  doi: 10.1007/PL00001616.  Google Scholar

[13]

D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math., 148 (1998), 339-360.  doi: 10.2307/120997.  Google Scholar

[14]

B. Klingler and A. Yafaev, The André-Oort conjecture, Ann. of Math., 180 (2014), 867-925.  doi: 10.4007/annals.2014.180.3.2.  Google Scholar

[15]

G. D. Mostow, Self-adjoint groups, Ann. of Math., 62 (1955), 44-55.  doi: 10.2307/2007099.  Google Scholar

[16]

S. Mozes and N. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems, 15 (1995), 149-159.  doi: 10.1017/S0143385700008282.  Google Scholar

[17]

D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, 3$^{rd}$ edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994.  Google Scholar

[18]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607.  doi: 10.2307/2944357.  Google Scholar

[19]

T. A. Springer, Linear Algebraic Groups, 2$^{nd}$ edition, Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. doi: 10.1007/978-0-8176-4840-4.  Google Scholar

[20]

E. Ullmo and A. Yafaev, Galois orbits and equidistribution of special subvarieties: Towards the André-Oort conjecture, Ann. of Math., 180 (2014), 823-865.  doi: 10.4007/annals.2014.180.3.1.  Google Scholar

[21]

R. Zhang, Translates of homogeneous measures associated with Observable Subgroups on some homogeneous spaces, arXiv e-prints, 2020, arXiv: 1909.02666. Google Scholar

[1]

Amal Attouchi, Eero Ruosteenoja. Gradient regularity for a singular parabolic equation in non-divergence form. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5955-5972. doi: 10.3934/dcds.2020254
[2]

Serena Dipierro, Aram Karakhanyan, Enrico Valdinoci. Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6073-6090. doi: 10.3934/dcds.2018262
[3]

Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[4]

Jingxue Yin, Chunhua Jin. Critical exponents and traveling wavefronts of a degenerate-singular parabolic equation in non-divergence form. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 213-227. doi: 10.3934/dcdsb.2010.13.213
[5]

Andrea Bonfiglioli, Ermanno Lanconelli and Francesco Uguzzoni. Levi's parametrix for some sub-elliptic non-divergence form operators. Electronic Research Announcements, 2003, 9: 10-18.

[6]

Teresa Alberico, Costantino Capozzoli, Luigi D'Onofrio, Roberta Schiattarella. $G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 129-137. doi: 10.3934/dcdss.2019009
[7]

Mitsuru Shibayama. Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3707-3719. doi: 10.3934/dcds.2015.35.3707
[8]

Nimish Shah, Lei Yang. Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5247-5287. doi: 10.3934/dcds.2020227
[9]

Agnieszka Badeńska. Measure rigidity for some transcendental meromorphic functions. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2375-2402. doi: 10.3934/dcds.2012.32.2375
[10]

Boris Kalinin, Anatole Katok and Federico Rodriguez Hertz. New progress in nonuniform measure and cocycle rigidity. Electronic Research Announcements, 2008, 15: 79-92. doi: 10.3934/era.2008.15.79
[11]

Mike Boyle. The work of Mike Hochman on multidimensional symbolic dynamics and Borel dynamics. Journal of Modern Dynamics, 2019, 15: 427-435. doi: 10.3934/jmd.2019026
[12]

Raphaël Danchin, Piotr B. Mucha. Divergence. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1163-1172. doi: 10.3934/dcdss.2013.6.1163
[13]

Ralf Spatzier. On the work of Rodriguez Hertz on rigidity in dynamics. Journal of Modern Dynamics, 2016, 10: 191-207. doi: 10.3934/jmd.2016.10.191
[14]

Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1
[15]

Danijela Damjanovic, Anatole Katok. Local rigidity of homogeneous parabolic actions: I. A model case. Journal of Modern Dynamics, 2011, 5 (2) : 203-235. doi: 10.3934/jmd.2011.5.203
[16]

Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123
[17]

Maria Rosaria Lancia, Valerio Regis Durante, Paola Vernole. Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1493-1520. doi: 10.3934/dcdss.2016060
[18]

Tien-Cuong Dinh, Nessim Sibony. Rigidity of Julia sets for Hénon type maps. Journal of Modern Dynamics, 2014, 8 (3&4) : 499-548. doi: 10.3934/jmd.2014.8.499
[19]

F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures. A criterion for ergodicity for non-uniformly hyperbolic diffeomorphisms. Electronic Research Announcements, 2007, 14: 74-81. doi: 10.3934/era.2007.14.74
[20]

Masayuki Asaoka. Local rigidity of homogeneous actions of parabolic subgroups of rank-one Lie groups. Journal of Modern Dynamics, 2015, 9: 191-201. doi: 10.3934/jmd.2015.9.191

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (84)
  • HTML views (75)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy