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

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Abstract

We consider partially hyperbolic diffeomorphisms preserving a splitting of the tangent bundle into a strong-unstable subbundleE uu (uniformly expanding) and a subbundleE c, dominated byE uu.

We prove that if the central directionE c is mostly contracting for the diffeomorphism (negative Lyapunov exponents), then the ergodic Gibbsu-states are the Sinai-Ruelle-Bowen measures, there are finitely many of them, and their basins cover a full measure subset. If the strong-unstable leaves are dense, there is a unique Sinai-Ruelle-Bowen measure.

We describe some applications of these results, and we also introduce a construction of robustly transitive diffeomorphisms in dimension larger than three, having no uniformly hyperbolic (neither contracting nor expanding) invariant subbundles.

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Authors and Affiliations

  1. Lab. Topologie, CNRS-UMR 5584, Dep. Mathématiques, Univ. Bourgogne, B.P. 47870, 21078, Dijon Cedex, France

    Christian Bonatti

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

    Marcelo Viana

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  1. Christian Bonatti
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  2. Marcelo Viana
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Correspondence to Christian Bonatti.

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Work partially supported by CNRS and CNPq/PRONEX-Dynamical Systems, and carried out at Laboratoire de Topologie, Dijon, and IMPA, Rio de Janeiro.

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Bonatti, C., Viana, M. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Isr. J. Math. 115, 157–193 (2000). https://doi.org/10.1007/BF02810585

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