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SRB measures for partially hyperbolic systems whose central direction is mostly expanding

  • Chapter
The Theory of Chaotic Attractors

Abstract

We construct Sinai-Ruelle-Bowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms — the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting — under the assumption that the complementary subbundle is non-uniformly expanding. If the rate of expansion (Lyapunov exponents) is bounded away from zero, then there are only finitely many SRB measures. Our techniques extend to other situations, including certain maps with singularities or critical points, as well as diffeomorphisms having only a dominated splitting (and no uniformly hyperbolic subbundle).

Work carried out at the Laboratoire de Topologie, CNRS-UMR 5584, Dijon, and IMPA, Rio de Janeiro. Partially supported by IMPA and PRONEX-Dynamical Systems, Brazil, Université de Bourgogne, France, and Praxis XXI-Física Matemática and Centro de Matemática da Universidade do Porto, Portugal.

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Author information

Authors and Affiliations

  1. Dep. Matemática Pura, Faculdade de Ciências do Porto, P-4099-002, Porto, Portugal

    José F. Alves

  2. Lab. Topologie, CNRS—UMR 5584, Univ. Bourgogne, B.P. 47870, 21078, Dijon Cedex, France

    Christian Bonatti

  3. IMPA, Estrada D. Castorina 110, Jardim Botânico, 22460-320, Rio de Janeiro RJ, Brazil

    Marcelo Viana

Authors
  1. José F. Alves
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  2. Christian Bonatti
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  3. Marcelo Viana
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Editor information

Editors and Affiliations

  1. Institute for Physical Science and Technology, University of Maryland, 20742, College Park, MD, USA

    Brian R. Hunt

  2. Department of Mathematics, Michigan State University, 48824, East Lansing, Michigan, USA

    Tien-Yien Li

  3. Department of Mathematics, University of Delaware, 19716, Newark, Delaware, USA

    Judy A. Kennedy

  4. Department of Econometrics, University of Groningen, NL-9700 AV, Groningen, The Netherlands

    Helena E. Nusse

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Alves, J.F., Bonatti, C., Viana, M. (2000). SRB measures for partially hyperbolic systems whose central direction is mostly expanding. In: Hunt, B.R., Li, TY., Kennedy, J.A., Nusse, H.E. (eds) The Theory of Chaotic Attractors. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21830-4_24

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  • DOI: https://doi.org/10.1007/978-0-387-21830-4_24

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2330-1

  • Online ISBN: 978-0-387-21830-4

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