On classification of higher rank Anosov actions on compact manifold
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ON CLASSIFICATION OF HIGHER RANK ANOSOV ACTIONS ON COMPACT MANIFOLD
BY
´∗ Danijela Damjanovic Department of Mathematics, Kungliga Tekniska H¨ ogskolan Lindstedtsv¨ agen 25, SE-100 44 Stockholm, Sweden e-mail: [email protected]
AND
Disheng Xu Department of Mathematics, The University of Chicago, Chicago, IL 60637, US e-mail: [email protected] Dedicated to the memory of Anatole Katok
ABSTRACT
We prove global smooth classification results for Anosov Zk actions on general compact manifolds, under certain irreduciblity conditions and the presence of sufficiently many Anosov elements. In particular, we remove all the uniform control assumptions which were used in all the previous results towards the Katok–Spatzier global rigidity conjecture on general manifolds. The main idea is to create a new mechanism labelled nonuniform redefining argument, to prove continuity of certain dynamicallydefined objects. This leads to uniform control for higher-rank actions and should apply to more general rigidity problems in dynamical systems.
∗ The first author is supported by the Swedish Research Council grant 2015-04644.
Received May 19, 2018 and in revised form June 30, 2019
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´ AND D. XU D. DAMJANOVIC
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Isr. J. Math.
Contents
1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Setting and statements . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . Subexponential growth for elements on a Lyapunov hyperplane . . . . . . . . . . . . . . . . . 5. A specific splitting . . . . . . . . . . . . . . . . . . . . . 6. Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . 7. The pairwise jointly integrable case and the Lyapunov pinching case . . . . . . . . . . . . . . . 8. Resonance-free case . . . . . . . . . . . . . . . . . . . . Appendix A. Proof of Case 2 of Lemma 6.10 . . . . . . . . Appendix B. Proof of Lemma 6.12 . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Anosov diffeomorphisms are a well-studied class of systems, which in many respects either have or are expected to have rigid dynamical features. For example, it is a well known result of Anosov that the complete topological orbit structure of an Anosov diffeomorphism is preserved under C 1 -small perturbations. Principal examples of Anosov diffeomorphisms are hyperbolic affine maps on nilmanifolds (in particular on tori), and manifolds finitely covered by nilmanifolds (infranilmanifolds). A long outstanding global topological rigidity conjecture says that all Anosov diffeomorphisms are topologically conjugate to affine maps on infranilmanifolds. For actions on nilmanifolds, the conjecture has been proved by Franks and Manning [18], [41], but only a few results are known for general manifolds, (cf. [19], [9] and the references therein). These topological local and global rigidity results for Anosov diffeomorphisms cannot be improved in general to smooth rigidity results. It is eas
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