Hostname: page-component-7bb8b95d7b-qxsvm Total loading time: 0 Render date: 2024-10-06T23:32:24.974Z Has data issue: false hasContentIssue false
  • English
  • Français

Hyperbolic geometry and pointwise ergodic theorems

Published online by Cambridge University Press:  12 December 2017

LEWIS BOWEN  and
AMOS NEVO
LEWIS BOWEN
Affiliation:
Mathematics Department, 1 University Station C1200, University of Texas at Austin, Austin, TX 78712, USA email lpbowen@math.utexas.edu
AMOS NEVO
Affiliation:
Mathematics Department, Technion – Israel Institute of Technology, Haifa 32000, Israel email anevo@tx.technion.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real rank one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 10 , October 2019 , pp. 2689 - 2716
Copyright
© Cambridge University Press, 2017 

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.)

References

Becker, H. and Kechris, A. S.. The Descriptive Set Theory of Polish Group Actions (London Mathematical Society Lecture Note Series, 232) . Cambridge University Press, Cambridge, 1996.Google Scholar
Bowen, L. and Nevo, A.. Geometric covering arguments and ergodic theorems for free groups. Enseign. Math. 59 (2013), 133–164.Google Scholar
Bridson, M. and Haefliger, A.. Metric Spaces of Non-Positive Curvature (Grundlehren der mathematischen Wissenschaften, 319) . Springer, Berlin, 1999.Google Scholar
Casselman, W. and Milicic, D.. Asymptotic behavior of matrix coefficients of admissible representations. Duke Math. J. 49(4) (1982), 869–930.Google Scholar
Cowling, M.. Sur les coefficients des representations unitaires des groupes de Lie simples. Analyse harmonique sur les groupes de Lie (Séminaire, Nancy-Strasbourg, 1976–1978) (Lecture Notes in Mathematics, 739) . Springer, Berlin, 1979, pp. 132–178.Google Scholar
Fava, N. A.. Weak type inequalities for product operators. Studia Math. 42 (1972), 271–288.Google Scholar
Gorodnik, A. and Nevo, A.. The Ergodic Theory of Lattice Subgroups (Annals of Mathematics Studies, 172) . Princeton University Press, Princeton, NJ, 2010.Google Scholar
Helgason, S.. Groups and Geometric Analysis. Elsevier, Burlington, MA, 1984.Google Scholar
Helgason, S.. Geometric Analysis on Symmetric Spaces (Mathematical Surveys and Monographs, 39) . American Mathematical Society, Providence, RI, 1994.Google Scholar
Howe, R. and Moore, C. C.. Asymptotic properties of unitary representations. J. Funct. Anal. 32(1) (1979), 72–96.Google Scholar
Kechris, A. S.. Global Aspects of Ergodic Group Actions (Mathematical Surveys and Monographs, 160) . American Mathematical Society, Providence, RI, 2010.Google Scholar
Knapp, A. W.. Lie Groups Beyond An Introduction (Progress in Mathematics, 140) . Birkhäuser, Boston, 1996.Google Scholar
Koornwinder, T. H.. Jacobi functions and analysis on noncompact semisimple Lie groups. Special Functions: Group Theoretical Aspects and Applications, 185 (Mathematics and Its Applications, 18) . Reidel, Dordrecht, 1984.Google Scholar
Nevo, A.. Pointwise ergodic theorems for radial averages on simple Lie groups I. Duke Math. J. 76 (1994), 113–140.Google Scholar
Nevo, A.. Pointwise ergodic theorems for radial averages on simple Lie groups II. Duke Math. J. 86 (1997), 239–259.Google Scholar
Nevo, A.. Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups. Math. Res. Lett. 5(3) (1998), 305–325.Google Scholar
Nevo, A.. Pointwise ergodic theorems for actions of groups. Handbook of Dynamical Systems. Vol. 1B Eds. Hasselblatt, B. and Katok, A.. Elsevier, Amsterdam, 2006, pp. 871–982.Google Scholar
Nevo, A.. Equidistribution in measure-preserving actions of semisimple groups: case of $\text{SL}_{2}(\mathbb{R})$ . Preprint, 2017, arXiv:1708.03886.Google Scholar
Nevo, A and Stein, E. M.. Analogs of Wiener’s ergodic theorems for semi-simple Lie groups I. Ann. of Math. (2) 145 (1997), 565–595.Google Scholar
Pitt, H. R.. Some generalizations of the ergodic theorem. Math. Proc. Cambridge Philos. Soc. 38 (1942), 325–343.Google Scholar
Wiener, N.. The ergodic theorem. Duke Math. J. 5 (1939), 1–18.Google Scholar