Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Equidistribution of dilations of polynomial curves in nilmanifolds

Authors: Michael Björklund and Alexander Fish
Journal: Proc. Amer. Math. Soc. 137 (2009), 2111-2123
MSC (2000): Primary 60B15
Published electronically: January 27, 2009
MathSciNet review: 2480293
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the asymptotic behaviour under dilations of probability measures supported on polynomial curves in nilmanifolds. We prove, under some mild conditions, the effective equidistribution of such measures to the Haar measure. We also formulate a mean ergodic theorem for $ \mathbb{R}^n$-representations on Hilbert spaces, restricted to a moving phase of low dimension. Furthermore, we bound the necessary dilation of a given smooth curve in $ \mathbb{R}^n$ so that the canonical projection onto $ \mathbb{T}^n $ is $ \varepsilon$-dense.

References [Enhancements On Off] (What's this?)

  • 1. Deninger, Ch.; Singhof, W. The $ e$-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups. Invent. Math. 78 (1984), no. 1, 101-112. MR 762355 (86i:58133)
  • 2. Diestel, J. and Uhl, J. J., Jr. Vector measures. With a foreword by B. J. Pettis. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977. MR 0453964 (56:12216)
  • 3. Folland, G. B. and Stein, E. M. Hardy spaces on homogeneous groups. Mathematical Notes, 28. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581 (84h:43027)
  • 4. Grafakos, L. Classical and modern Fourier analysis. Pearson Education, Inc., Prentice Hall, Upper Saddle River, N.J., 2004. MR 2449250
  • 5. Leibman, A. Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergodic Theory and Dynamical Systems 25 (2005), no. 1, 201-213. MR 2122919 (2006j:37004)
  • 6. Monod, N. Continuous bounded cohomology of locally compact groups. Lecture Notes in Mathematics, 1758. Springer-Verlag, Berlin, 2001. MR 1840942 (2002h:46121)
  • 7. Randol, B. The behavior under projection of dilating sets in a covering space. Trans. Amer. Math. Soc. 285 (1984), no. 2, 855-859. MR 752507 (86g:58023)
  • 8. Samoılenko, Yu. S. Spectral theory of families of selfadjoint operators. Kluwer, Dordrecht, 1991. MR 1135325 (92j:47038)
  • 9. Schmidt, W. Badly approximable systems of linear forms. J. Number Theory 1 (1969), 139-154. MR 0248090 (40:1344)
  • 10. Shah, N. Limiting distributions of evolution of smooth curves under geodesic flow on hyperbolic manifolds-II. Preprint.
  • 11. Taylor, M. Noncommutative harmonic analysis. Mathematical Surveys and Monographs, 22. American Mathematical Society, Providence, R.I., 1986. MR 852988 (88a:22021)
  • 12. Warner, F. W. Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983. MR 722297 (84k:58001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 60B15

Retrieve articles in all journals with MSC (2000): 60B15

Additional Information

Michael Björklund
Affiliation: Department of Mathematics, KTH - Royal Institute of Technology, SE-100 44 Stockholm, Sweden

Alexander Fish
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210

Received by editor(s): September 5, 2008
Published electronically: January 27, 2009
Additional Notes: The research of the second author was partly done during his visit to MSRI, Berkeley
Communicated by: Bryna Kra
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society