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Local rigidity of higher rank non-abelian action on torus

Published online by Cambridge University Press:  25 September 2017

ZHENQI JENNY WANG
ZHENQI JENNY WANG*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA email wangzq@math.msu.edu
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Abstract

In this paper, we show local smooth rigidity for higher rank ergodic nilpotent action by toral automorphisms. In former papers all examples for actions enjoying the local smooth rigidity phenomenon are higher rank and have no rank-one factors. In this paper we give examples of smooth rigidity of actions having rank-one factors. The method is a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 6 , June 2019 , pp. 1668 - 1709
Copyright
© Cambridge University Press, 2017 

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References

Brown, L. D.. Representation of finitely generated nilpotent groups. Pacific J. Math. 45(1) (1973), 13–26.Google Scholar
Damjanovic, D. and Katok, A.. Local rigidity of partially hyperbolic actions I. KAM method and ℤ k actions on the torus. Ann. of Math. (2) 172 (2010), 1805–1858.Google Scholar
Damjanovic, D. and Katok, A.. Local rigidity of homogeneous parabolic actions: I. A model case. J. Mod. Dyn. 172 (2011), 203–235.Google Scholar
De La Llave, R.. A tutorial on KAM theory. Smooth Ergodic Theory and its Applications (Seattle, WA, 1999) (Proceedings of Symposia in Pure Mathematics, 69) . American Mathematical Society, Providence, RI, 2001, pp. 175–292.Google Scholar
Fisher, D., Kalinin, B. and Spatzier, R.. Global rigidity of higher rank Anosov actions on tori and nilmanifolds. J. Amer. Math. Soc. 26(1) (2013), 167–198 (with an appendix by James F. Davis).Google Scholar
Fisher, D. and Margulis, G.. Local rigidity of affine actions of higher rank groups and lattices. Ann. of Math. (2) 170 (2009), 67–122.Google Scholar
Katok, A. and Spatzier, R.. First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Publ. Math. Inst. Hautes Études Sci. 79 (1994), 131–156.Google Scholar
Katok, A. and Spatzier, R.. Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions. Math. Res. Lett. 1 (1994), 193–202.Google Scholar
Katok, A. and Spatzier, R.. Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions. Proc. Steklov Inst. Math. 216 (1997), 287–314.Google Scholar
Katznelson, Y.. Ergodic automorphisms of T n are Bernoulli shifts. Israel J. Math. 10 (1971), 186–195.Google Scholar
Kronecker, L.. Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten. J. Reine Angew. Math. 53 (1857), 173–175.Google Scholar
Lazutkin, V. F.. KAM Theory and Semiclassical Approximations to Eigenfunctions (Ergebnisse der Mathematik und ihrer Grenzgebiete, 24) . Springer, New York, 1993.Google Scholar
Starkov, A. N.. The first cohomology group, mixing, and minimal sets of the commutative group of algebraic actions on a torus. J. Math. Sci. New York 95 (1999), 2576–2582.Google Scholar
Hu, H., Shi, E. and Wang, Z. J.. Heisenberg group actions and non-chaotic properties of central element actions, submitted.Google Scholar