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The Geometric Approach for Constructing Sinai–Ruelle–Bowen Measures

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Abstract

An important class of ‘physically relevant’ measures for dynamical systems with hyperbolic behavior is given by Sinai–Ruelle–Bowen (SRB) measures. We survey various techniques for constructing SRB measures and studying their properties, paying special attention to the geometric ‘push-forward’ approach. After describing this approach in the uniformly hyperbolic setting, we review recent work that extends it to non-uniformly hyperbolic systems.

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Notes

  1. Our definition of SRB measure includes the requirement that the measure is hyperbolic. In fact, one can extend the notion of SRB measures to those that have some or even all Lyapunov exponents zero (in the latter case we take \(W^u(x)=\{x\}\)). It was proved by Ledrappier and Young [39] that a measure satisfies the entropy formula if and only if it is an SRB measure in this more general sense, but we stress that such an SRB measure may not be physical, i.e., its basin may be of zero Lebesgue measure.

  2. Note that the subspaces \(E^s(x)\) and \(E^u(x)\) for \(x\in \tilde{U}\) need not be invariant under df.

  3. This requires a version of the Besicovitch covering lemma, which is usually formulated for geometrical balls, so one must choose the \(W_i\) in such a way that each \(f^n(W_i)\) is ‘sufficiently close’ to being a ball in \(f^n(W)\).

  4. Although note that the ‘mostly expanding’ condition of Theorem 6.3 is slightly more restrictive than saying that all Lyapunov exponents in \(E^{cu}\) are positive.

  5. We stress that non-uniform hyperbolicity on an invariant set does not require presence of any invariant measure. On the other hand if f preserves a hyperbolic measure then there is an invariant set \(\Lambda \) on which f is non-uniformly hyperbolic.

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Acknowledgments

V.C. was partially supported by NSF Grant DMS-1362838. Ya.P. was partially supported by NSF Grant DMS-1400027. V.C. and Ya.P. would like to thank Erwin Schrödinger Institute and ICERM where the part of the work was done for their hospitality.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX 77204, USA

    Vaughn Climenhaga

  2. Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, Trieste, Italy

    Stefano Luzzatto

  3. Department of Mathematics, Pennsylvania State University, McAllister Building, University Park, PA, 16802, USA

    Yakov Pesin

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  1. Vaughn Climenhaga
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  2. Stefano Luzzatto
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  3. Yakov Pesin
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Correspondence to Yakov Pesin.

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It is our great pleasure to have this survey included in this special issue dedicated to the 80th birthday of the great dynamicists D. Ruelle and Y. Sinai. We take this opportunity to acknowledge the tremendous impact that their work has had and continues to have in this field of research.

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Climenhaga, V., Luzzatto, S. & Pesin, Y. The Geometric Approach for Constructing Sinai–Ruelle–Bowen Measures. J Stat Phys 166, 467–493 (2017). https://doi.org/10.1007/s10955-016-1608-7

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