Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Electronic Research Announcements
Electronic Research Announcements
ISSN 1079-6762

Nonstationary normal forms and rigidity of group actions


Authors: A. Katok and R. J. Spatzier
Journal: Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 124-133
MSC (1991): Primary 58Fxx; Secondary 22E40, 28Dxx
MathSciNet review: 1426721
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We develop a proper ``nonstationary'' generalization of the classical theory of normal forms for local contractions. In particular, it is shown under some assumptions that the centralizer of a contraction in an extension is a particular Lie group, determined by the spectrum of the linear part of the contractions. We show that most homogeneous Anosov actions of higher rank abelian groups are locally $C^{\infty }$ rigid (up to an automorphism). This result is the main part in the proof of local $C^{\infty }$ rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the actions of cocompact lattices on Furstenberg boundaries, in particular projective spaces, and (ii) the actions by automorphisms of tori and nilmanifolds. The main new technical ingredient in the proofs is the centralizer result mentioned above.


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

  • 1. Kuo-Tsai Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math. 85 (1963), 693–722. MR 0160010 (28 #3224)
  • 2. John Franks, Anosov diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 61–93. MR 0271990 (42 #6871)
  • 3. Étienne Ghys, Rigidité différentiable des groupes fuchsiens, Inst. Hautes Études Sci. Publ. Math. 78 (1993), 163–185 (1994) (French). MR 1259430 (95d:57009)
  • 4. M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin, 1977. MR 0501173 (58 #18595)
  • 5. M. Kanai, A new approach to the rigidity of discrete group actions, preprint 1995.
  • 6. A. Katok, Hyperbolic measures for actions of higher rank abelian groups, preprint 1996.
  • 7. A. Katok, Normal forms and invariant geometric structures on transverse contracting foliations, preprint 1996.
  • 8. Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374 (96c:58055)
  • 9. A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms, Israel J. Math. 75 (1991), no. 2-3, 203–241. MR 1164591 (93g:58076), http://dx.doi.org/10.1007/BF02776025
  • 10. A. Katok, J. Lewis and R. J. Zimmer, Cocycle superrigidity and rigidity for lattice actions on tori, Topology 35 (1996), 27-38. CMP 96:06
  • 11. Anatole Katok and Ralf J. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 131–156. MR 1307298 (96c:58132)
  • 12. A. Katok and R. J. Spatzier, Invariant measures for higher rank hyperbolic abelian actions, Erg. Th. and Dynam. Syst. 16 (1996), no. 4, 751-778. CMP 96:17
  • 13. A. Katok and R. J. Spatzier, Differential rigidity of hyperbolic abelian actions, preprint 1992.
  • 14. A. Katok and R. J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, preprint.
  • 15. G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825 (92h:22021)
  • 16. Anthony Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math. 96 (1974), 422–429. MR 0358865 (50 #11324)
  • 17. William Parry, Ergodic properties of affine transformations and flows on nilmanifolds., Amer. J. Math. 91 (1969), 757–771. MR 0260975 (41 #5595)
  • 18. N. Qian, Tangential flatness and global rigidity of higher rank lattice actions, preprint.
  • 19. N. Qian, Smooth conjugacy for Anosov diffeomorphisms and rigidity of Anosov actions of higher rank lattices, preprint.
  • 20. N. Qian and C. Yue, Local rigidity of Anosov higher rank lattice actions, preprint 1996.
  • 21. N. Qian and R. J. Zimmer, Entropy rigidity for semisimple group actions, preprint.
  • 22. Shlomo Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 809–824. MR 0096853 (20 #3335)
  • 23. Cheng Bo Yue, Smooth rigidity of rank-1 lattice actions on the sphere at infinity, Math. Res. Lett. 2 (1995), no. 3, 327–338. MR 1338792 (96e:58122)
  • 24. Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417 (86j:22014)

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (1991): 58Fxx, 22E40, 28Dxx

Retrieve articles in all journals with MSC (1991): 58Fxx, 22E40, 28Dxx


Additional Information

A. Katok
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
Email: katok_a@math.psu.edu

R. J. Spatzier
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, MI 48103
Email: spatzier@math.lsa.umich.edu

DOI: http://dx.doi.org/10.1090/S1079-6762-96-00016-9
PII: S 1079-6762(96)00016-9
Received by editor(s): September 28, 1996
Additional Notes: The first author was partially supported by NSF grant DMS 9404061
The second author was partially supported by NSF grant DMS 9626173
Communicated by: Gregory Margulis
Article copyright: © Copyright 1997 Anatole Katok and Ralf J. Spatzier