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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