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Anomalous partially hyperbolic diffeomorphisms II: stably ergodic examples

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Abstract

We construct examples of robustly transitive and stably ergodic partially hyperbolic diffeomorphisms f on compact 3-manifolds with fundamental groups of exponential growth such that \(f^n\) is not homotopic to identity for all \(n>0\). These provide counterexamples to a classification conjecture of Pujals.

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Notes

  1. The conjecture was formalized in [9], see also [20, Section 2.2] and [13].

  2. As remarked in [9] it might be better to consider \(W^c\)-conjugacy to a topologically Anosov flow for technical reasons. See [9, Conjecture 1].

  3. It is plausible that all our new examples have this property.

  4. [16] presents special examples of a higher dimensional dynamically coherent partially hyperbolic diffeomorphisms with non-smooth center foliation. It is plausible that these examples cannot be homotoped to ones with smooth center foliation. However they are \(W^c\) conjugate to partially hyperbolic diffeomorphisms with smooth center foliation.

  5. The fact that F has to be \(C^2\) is for technical reasons which we shall not explain here.

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Acknowledgments

The second author would like to thank Dmitry Scheglov for many enlightening discussions and Anton Petrunin for a very useful communication. The third author thanks the hospitality of the Institut de Matématiques de Bourgogne, Dijon.

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Authors and Affiliations

  1. Institut de Mathématiques de Bourgogne CNRS-URM 5584, Université de Bourgogne, 21004, Dijon, France

    Christian Bonatti

  2. Department of Mathematical Sciences, Binghamton University, Binghamton, NY, 13902, USA

    Andrey Gogolev

  3. CMAT, Facultad de Ciencias, Universidad de la República, 11400, Montevideo, Uruguay

    Rafael Potrie

Authors
  1. Christian Bonatti
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  2. Andrey Gogolev
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  3. Rafael Potrie
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Corresponding author

Correspondence to Rafael Potrie.

Additional information

A. G. was partially supported by NSF grant DMS-1266282; R. P. was partially supported by CSIC group 618 and the Palis Balzan project.

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Bonatti, C., Gogolev, A. & Potrie, R. Anomalous partially hyperbolic diffeomorphisms II: stably ergodic examples. Invent. math. 206, 801–836 (2016). https://doi.org/10.1007/s00222-016-0663-7

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