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A Few Remarks on Partially Hyperbolic Diffeomorphisms of \({\mathbb T}^3\) Isotopic to Anosov

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Abstract

We show that a strong partially hyperbolic diffeomorphism of \({\mathbb T}^3\) isotopic to Anosov has a unique quasi-attractor. Moreover, we study the entropy of the diffeomorphism restricted to this quasi-attractor.

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Notes

  1. In dimension 2, the fact that robust transitivity implies Anosov was obtained in [24] but the results in [25] which provide a different proof are important in the results of [4, 14].

  2. The direct implication in dimension 2 gives that a \(C^1\)-robustly chain-recurrent diffemorphism is Anosov. Again in dimension two, being Anosov is enough to show robust transitivity.

  3. Let us mention that among robustly chain-recurrent partially hyperbolic diffeomorphisms with one dimensional center those which are robustly transitive form a \(C^1\)-open and dense subset [2].

  4. Since this is the only notion considered here we remove the word strong from the definition. Beware in comparing with other literature.

  5. Indeed, it is reasonable to expect that it is possible to construct a diffeomorphism isotopic to a linear Anosov which has a partially hyperbolic attractor and a partially hyperbolic repeller which are disjoint and satisfy the properties predicted by the Addendum in the global partially hyperbolic setting. The main difficulty lies in understanding how to construct such and example while remaining partially hyperbolic in the wandering region.

  6. Presented at the 2nd Palis-Balzan Symposium on dynamical systems. Slides are available in the web page of the conference.

  7. Since \(A_f(\Gamma _1)\subset \Gamma _1\) it follows that if \(m < k\) then \(A_f^{-m}(\Gamma _1) \subset A_f^k(\Gamma _1)\).

  8. This means that it has a unique rotation vector which is totally irrational, see [29].

  9. The case where \(A_f\) has two eigenvalues smaller than one is considerably easier and very similar to the proof above, but the other case is a little bit more complicated since the induced dynamics is no longer semiconjugate to a rigid translation.

  10. The question of wether a non-resonant torus homeomorphism may have more than one minimal set is, I believe, open. Even in the case where it is semiconjugate to a rigid translation.

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Acknowledgments

The Author benefited from discussions with C. Bonatti, S. Crovisier, N. Gourmelon and M.Sambarino. The author was partially supported by CSIC group 618, FCE-3-2011-1-6749 and the Palis-Balzan research project

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  1. CMAT, Facultad de Ciencias, Universidad de la República, Uruguay, Montevideo, Uruguay

    Rafael Potrie

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Potrie, R. A Few Remarks on Partially Hyperbolic Diffeomorphisms of \({\mathbb T}^3\) Isotopic to Anosov. J Dyn Diff Equat 26, 805–815 (2014). https://doi.org/10.1007/s10884-014-9362-5

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