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
DOI:
https://doi.org/10.1090/S1079-6762-96-00016-9
MathSciNet review:
1426721
Full-text PDF Free Access
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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.
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- M. Kanai, A new approach to the rigidity of discrete group actions, preprint 1995.
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- A. Katok, Normal forms and invariant geometric structures on transverse contracting foliations, preprint 1996.
- A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, 1995.
- A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms, Israel J. of Math. 75 (1991), 203–241.
- A. Katok, J. Lewis and R. J. Zimmer, Cocycle superrigidity and rigidity for lattice actions on tori, Topology 35 ,(1996), 27–38.
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- 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.
- A. Katok and R. J. Spatzier, Differential rigidity of hyperbolic abelian actions, preprint 1992.
- A. Katok and R. J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, preprint.
- G. A. Margulis, Discrete subgroups of semisimple Lie groups, Springer Verlag, Berlin 1991.
- A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. of Math. 96 (1974), 422–429.
- W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771.
- N. Qian, Tangential flatness and global rigidity of higher rank lattice actions, preprint.
- N. Qian, Smooth conjugacy for Anosov diffeomorphisms and rigidity of Anosov actions of higher rank lattices, preprint.
- N. Qian and C. Yue, Local rigidity of Anosov higher rank lattice actions, preprint 1996.
- N. Qian and R. J. Zimmer, Entropy rigidity for semisimple group actions, preprint.
- S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. of Math. 79 (1957), 809–824.
- C. Yue, Smooth rigidity of rank 1 lattice actions on the sphere at infinity, Math. Res. L. 2 (1995), 327–338.
- R. J. Zimmer, Ergodic theory and semisimple groups, Boston, Birkhäuser, 1984.
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Additional Information
A. Katok
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
MR Author ID:
99105
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
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
