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Nonstationary normal forms and rigidity of group actions
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## Nonstationary normal forms and rigidity of group actions

### A. Katok and R. J. Spatzier

**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.

*Copyright 1997 Anatole Katok and Ralf J. Spatzier*

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#### Article Info

- ERA Amer. Math. Soc.
**02** (1996), pp. 124-133
- Publisher Identifier: S 1079-6762(96)00016-9
- 1991
*Mathematics Subject Classification*.Primary 58Fxx; Secondary 22E40, 28Dxx
- Received by the editors September 28, 1996
- Communicated by Gregory Margulis
- Comments (When Available)

**A. Katok**

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

*E-mail address:* `katok_a@math.psu.edu`

**R. J. Spatzier**

Department of Mathematics, University of Michigan, Ann Arbor, MI 48103

*E-mail address:* `spatzier@math.lsa.umich.edu`

The first author was partially supported by NSF grant DMS 9404061

The second author was partially supported by NSF grant DMS 9626173

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