×

zbMATH — the first resource for mathematics

Nonstationary normal forms and rigidity of group actions. (English) Zbl 0871.58073
Summary: 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.

MSC:
37G05 Normal forms for dynamical systems
22E40 Discrete subgroups of Lie groups
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Kuo-Tsai Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math. 85 (1963), 693 – 722. · Zbl 0119.07505 · doi:10.2307/2373115 · doi.org
[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. · Zbl 0207.54304
[3] Étienne Ghys, Rigidité différentiable des groupes fuchsiens, Inst. Hautes Études Sci. Publ. Math. 78 (1993), 163 – 185 (1994) (French). · Zbl 0812.58066
[4] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. · Zbl 0355.58009
[5] M. Kanai, A new approach to the rigidity of discrete group actions, preprint 1995. · Zbl 0874.58006
[6] A. Katok, Hyperbolic measures for actions of higher rank abelian groups, preprint 1996. · Zbl 0859.58021
[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. · Zbl 0878.58020
[9] A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms, Israel J. Math. 75 (1991), no. 2-3, 203 – 241. · Zbl 0785.22012 · doi:10.1007/BF02776025 · doi.org
[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 · Zbl 0857.57037
[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. · Zbl 0819.58027
[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 · Zbl 0859.58021
[13] A. Katok and R. J. Spatzier, Differential rigidity of hyperbolic abelian actions, preprint 1992. · Zbl 0938.37010
[14] A. Katok and R. J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, preprint. · Zbl 0938.37010
[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. · Zbl 0732.22008
[16] Anthony Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math. 96 (1974), 422 – 429. · Zbl 0242.58003 · doi:10.2307/2373551 · doi.org
[17] William Parry, Ergodic properties of affine transformations and flows on nilmanifolds., Amer. J. Math. 91 (1969), 757 – 771. · Zbl 0183.51503 · doi:10.2307/2373350 · doi.org
[18] N. Qian, Tangential flatness and global rigidity of higher rank lattice actions, preprint. · Zbl 0877.22003
[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. · Zbl 0890.57051
[22] Shlomo Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 809 – 824. · Zbl 0080.29902 · doi:10.2307/2372437 · doi.org
[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. · Zbl 0844.22017 · doi:10.4310/MRL.1995.v2.n3.a10 · doi.org
[24] Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. · Zbl 0571.58015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.