Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T23:22:16.519Z Has data issue: false hasContentIssue false
  • English
  • Français

Hausdorff dimension of divergent trajectories on homogeneous spaces

Part of: Ergodic theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  19 December 2019

Lifan Guan  and
Ronggang Shi
Lifan Guan
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK email guanlifan@gmail.com
Ronggang Shi
Affiliation:
Shanghai Center for Mathematical Sciences, Fudan University, Shanghai200433, PR China email ronggang@fudan.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

For a one-parameter subgroup action on a finite-volume homogeneous space, we consider the set of points admitting divergent-on-average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves a conjecture of Cheung [Hausdorff dimension of the set of singular pairs, Ann. of Math. (2) 173 (2011), 127–167].

Keywords

homogeneous dynamicsdivergent trajectoryHausdorff dimension

MSC classification

Primary: 37A17: Homogeneous flows
Secondary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 2 , February 2020 , pp. 340 - 359
Copyright
© The Authors 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

1

Current address: Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstrasse 3-5, D-37073 Gottingen, Germany

L. G. is supported by EPSRC Programme Grant EP/J018260/1 and R. S. is supported by NSFC 11871158.

References

Benoist, Y. and Quint, J.-F., Random walks on finite volume homogeneous spaces, Invent. Math. 187 (2012), 37–59.CrossRefGoogle Scholar
Bernik, V., Kleinbock, D. and Margulis, G. A., Khintchine type theorems on manifolds: the convergence case for standard multiplicative versions, Int. Math. Res. Not. IMRN 9 (2001), 453–486.CrossRefGoogle Scholar
Bishop, C. and Peres, Y., Fractals in probability and analysis, Cambridge Studies in Advanced Mathematics, vol. 162 (Cambridge University Press, Cambridge, 2017).CrossRefGoogle Scholar
Borel, A., Linear algebraic groups, Graduate Texts in Mathematics, vol. 126, second edition (Springer, New York, NY, 1991).CrossRefGoogle Scholar
Cheung, Y., Hausdorff dimension of the set of points on divergent trajectories of a homogeneous flow on a product space, Ergodic Theory Dynam. Systems 27 (2007), 65–85.CrossRefGoogle Scholar
Cheung, Y., Hausdorff dimension of the set of singular pairs, Ann. of Math. (2) 173 (2011), 127–167.CrossRefGoogle Scholar
Cheung, Y. and Chevallier, N., Hausdorff dimension of singular vectors, Duke Math. J. 165 (2016), 2273–2329.CrossRefGoogle Scholar
Corwin, L. and Greenleaf, F., Representations of nilpotent Lie groups and their applications, Part I: Basic theory and examples, Cambridge Studies in Advanced Mathematics, vol. 18 (Cambridge University Press, Cambridge, 1990).Google Scholar
Dani, S. G., On orbits of unipotent flows on homogeneous spaces, Ergodic Theory Dynam. Systems 4 (1984), 25–34.CrossRefGoogle Scholar
Dani, S. G., Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math. 359 (1985), 55–89.Google Scholar
Dani, S. G., On orbits of unipotent flows on homogeneous spaces II, Ergodic Theory Dynam. Systems 6 (1986), 167–182.CrossRefGoogle Scholar
Das, T., Fishman, L., Simmons, D. and Urbanski, M., A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation, C. R. Math. Acad. Sci. Paris 355 (2017), 835–846.CrossRefGoogle Scholar
Einsiedler, M. and Kadyrov, S., Entropy and escape of mass for SL3(ℝ)/SL3(ℤ), Israel J. Math. 190 (2012), 253–288.CrossRefGoogle Scholar
Einsiedler, M., Lindenstrauss, E., Michel, P. and Venkatesh, A., The distribution of closed geodesics on the modular surface, and Duke’s theorem, Enseign. Math. (2) 58 (2012), 249–313.CrossRefGoogle Scholar
Eskin, A. and Margulis, G. A., Recurrence properties of random walks on finite volume homogeneous manifolds, in Random walks and geometry (de Gruyter, Berlin, 2004), 431–444.Google Scholar
Eskin, A., Margulis, G. A. and Mozes, S., Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2) 147 (1998), 93–141.CrossRefGoogle Scholar
Kadyrov, S., Entropy and escape of mass for Hilbert modular spaces, J. Lie Theory 22 (2012), 701–722.Google Scholar
Kadyrov, S., Kleinbock, D., Lindenstrauss, E. and Margulis, G. A., Singular systems of linear forms and non-escape of mass in the space of lattices, J. Anal. Math. 133 (2017), 253–277.CrossRefGoogle Scholar
Kadyrov, S. and Pohl, A., Amount of failure of upper-semicontinuity of entropy in noncompact rank one situations, and Hausdorff dimension, Ergodic Theory Dynam. Systems 37 (2017), 539–563.CrossRefGoogle Scholar
Kleinbock, D. and Margulis, G. A., Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinai’s Moscow seminar on dynamical systems, American Mathematical Society Translations Series 2, vol. 171, Advances in the Mathematical Sciences, vol. 28 (American Mathematical Society, Providence, RI, 1996), 141–172.Google Scholar
Kleinbock, D. and Weiss, B., Modified Schmidt games and a conjecture of Margulis, J. Mod. Dyn. 7 (2013), 429–460.CrossRefGoogle Scholar
Liao, L., Shi, R., Solan, O. N. and Tamam, N., Hausdorff dimension of weighted singular vectors, J. Eur. Math. Soc. (JEMS), doi:10.4171/JEMS/934 (2019), to appear in print.CrossRefGoogle Scholar
Margulis, G. A., On the action of unipotent groups in the space of lattices, in Lie groups and their representations, Proc. summer school in group representations, Bolyai János Mathematical Society, Budapest, 1971 (Halsted, New York, NY, 1975), 365–370.Google Scholar
Margulis, G. A. and Tomanov, G. M., Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math. 116 (1994), 347–392.CrossRefGoogle Scholar
Mattila, P., Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44 (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
Shi, R., Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space, Trans. Amer. Math. Soc., to appear. Preprint (2014), arXiv:1405.2067.Google Scholar
Weiss, B., Divergent trajectories on noncompact parameter spaces, Geom. Funct. Anal. 14 (2004), 94–149.CrossRefGoogle Scholar
Yang, L., Hausdorff dimension of divergent diagonal geodesics on product of finite-volume hyperbolic spaces, Ergodic Theory Dynam. Systems 39 (2019), 1401–1439.CrossRefGoogle Scholar