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Dominated Splitting from Constant Periodic Data and Global Rigidity of Anosov Automorphisms

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Abstract

We show that a \(\operatorname{GL}(d,\mathbb{R})\) cocycle over a hyperbolic system with constant periodic data has a dominated splitting whenever the periodic data indicates it should. This implies global periodic data rigidity of generic Anosov automorphisms of \(\mathbb{T}^{d}\). Further, our approach also works when the periodic data is narrow, that is, sufficiently close to constant. We can show global periodic data rigidity for certain non-linear Anosov diffeomorphisms in a neighborhood of an irreducible Anosov automorphism with simple spectrum.

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Acknowledgements

The authors are grateful to Daniel Mitsutani for many helpful discussions. The authors are also grateful to Dmitry Dolgopyat and Amie Wilkinson for helpful comments on the manuscript. The authors are grateful to the referee for many valuable comments and suggestions.

After the paper was finished, the authors learned from Jairo Bochi during a visit to Penn State that Misha Guysinsky announced a similar result some years ago for \(\operatorname{SL}(2,\mathbb{R})\) cocycles [Sad13, Thm. 1.2]. We are grateful to Misha Guysinsky for subsequent discussion, which revealed that he has some different techniques that yield similar results and may be used to address a related question of Velozo Ruiz [Vel20] about cocycles that are close to fiber bunched [Guy23].

Funding

The first author was supported by the National Science Foundation under Award No. DMS-2202967. The second author was partially supported by NSF award DMS-2247747.

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Authors and Affiliations

  1. Department of Mathematics, The University of Maryland, College Park, MD, 20742, USA

    Jonathan DeWitt

  2. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    Andrey Gogolev

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  1. Jonathan DeWitt
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Correspondence to Jonathan DeWitt.

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DeWitt, J., Gogolev, A. Dominated Splitting from Constant Periodic Data and Global Rigidity of Anosov Automorphisms. Geom. Funct. Anal. (2024). https://doi.org/10.1007/s00039-024-00680-z

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