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Positivity of the Top Lyapunov Exponent for Cocycles on Semisimple Lie Groups over Hyperbolic Bases

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Abstract

A theorem of Viana says that almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents. In this note we extend this result to cocycles on any noncompact classical semisimple Lie group.

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Notes

  1. We need \(C^{1+\alpha }\) regularity in order to apply the so-called Pesin’s theory (providing a measurable family of invariant manifolds with many good properties). In fact, it is known (see Bonatti et al. 2013 for instance) that Pesin’s theory may fail in \(C^1\) regularity.

  2. I.e., a semi-algebraic set is an element of the smallest Boolean ring of subsets of \(\mathbb {R}^n\) containing all subsets of the form \(\{(x_1,\ldots ,x_n)\in \mathbb {R}^n: P(x_1,\ldots , x_n)>0\}\) with \(P\in \mathbb {R}[X_1,\ldots ,X_n]\).

  3. These implications fail in the complex case; for example the compact group \(\mathrm {SU}(d)\) is Zariski-dense in \(\mathrm {SL}(d,\mathbb {C})\).

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Acknowledgements

The authors are grateful to the referee for a careful reading of the manuscript and for useful suggestions that helped to improve the presentation of the paper.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade da Beira Interior, Rua Marquês d’Ávila e Bolama, 6201-001, Covilhã, Portugal

    Mário Bessa

  2. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna, 4860, Santiago, Chile

    Jairo Bochi

  3. Departamento de Matemática, Universidade Federal do Estado do Rio de Janeiro, Av. Pasteur 458, Rio de Janeiro, 22290-240, Brazil

    Michel Cambrainha

  4. Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 93439, Villetaneuse, France

    Carlos Matheus

  5. Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, Salvador, 40170-110, Brazil

    Paulo Varandas

  6. Université Paris Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Université Paris 06, 75013, Paris, France

    Disheng Xu

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  1. Mário Bessa
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  2. Jairo Bochi
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  4. Carlos Matheus
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Corresponding author

Correspondence to Paulo Varandas.

Additional information

Bessa is partially supported by FCT—‘Fundação para a Ciência e a Tecnologia’, through Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, project UID/MAT/00212/2013. Bochi is partially supported by project Fondecyt 1140202 (Chile). Varandas is partially supported by CNPq-Brazil.

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Bessa, M., Bochi, J., Cambrainha, M. et al. Positivity of the Top Lyapunov Exponent for Cocycles on Semisimple Lie Groups over Hyperbolic Bases. Bull Braz Math Soc, New Series 49, 73–87 (2018). https://doi.org/10.1007/s00574-017-0048-6

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