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Sinai’s Work on Markov Partitions and SRB Measures

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The Abel Prize 2013-2017

Part of the book series: The Abel Prize ((AP))

Abstract

Some principal contributions of Ya. Sinai to hyperbolic theory of dynamical systems, focusing mainly on constructions of Markov partitions and of Sinai–Ruelle–Bowen measures, are discussed. Some further developments in these directions stemming from Sinai’s work, are described.

The author was partially supported by NSF grant DMS–1400027.

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Notes

  1. 1.

    After Alekseev’s untimely death in 1980, the seminar was run by Sinai only.

  2. 2.

    This term was first used in works of Chirikov, Ford and Yorke.

  3. 3.

    In fact, the dependence in x is Hölder continuous.

  4. 4.

    One can show that the dependence in x is Hölder continuous.

  5. 5.

    We use here the fact that the set Λ is locally maximal.

  6. 6.

    In other words, θ identifies the rectangle R with the product R ∩ V (s)(x) × R ∩ V (u)(x).

  7. 7.

    In [14], Bowen used a method similar to the original Sinai method known as the method of successive approximations. In [15] he used a different approach based on pseudo-orbits.

  8. 8.

    Both μ u(x) and m u(x) are probability measures.

  9. 9.

    In this paper we use the definition of SRB measure that requires that it is hyperbolic. One can weaken the hyperbolicity requirement by assuming that some (but not necessarily all) Lyapunov exponents are non-zero (with at least one positive). It was proved by Ledrappier and Young [43, 44] that within the class of such measures, SRB measures are the only ones that satisfy the entropy formula.

  10. 10.

    This is a two dimensional smooth map with a hyperbolic fixed point whose stable and unstable separatrices form the eight figure. Inside each of the two loops there is a repelling fixed point.

  11. 11.

    Clearly, the set Λ is locally maximal.

  12. 12.

    The matrix A is assumed to be hyperbolic having one positive and two negative eigenvalues.

  13. 13.

    Recall that given x ∈ M, a subspace E(x) ⊂ T x M, and θ(x) > 0, the cone at x around E(x) with angle θ(x) is defined by \(K(x,E(x),\theta (x))=\{v\in T_x M\colon \angle (v,E(x))<\theta (x)\}\).

  14. 14.

    We stress that the subspaces E 1(x) and E 2(x) do note have to be invariant under df.

  15. 15.

    Indeed, consider F = f 1 × f 2, where f 1 is a topologically transitive Anosov diffeomorphism and f 2 a diffeomorphism close to the identity. Then any measure μ = μ 1 × μ 2, where μ 1 is the unique SRB-measure for f 1 and μ 2 any f 2-invariant measure, is a u-measure for F. Thus, F has a unique u-measure if and only if f 2 is uniquely ergodic. On the other hand, F is topologically mixing if and only if f 2 is topologically mixing.

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Acknowledgements

I would like to thank B. Gurevich and O. Sarig who read the paper and gave me many valuable comments. I also would like to thank B. Weiss and A. Katok for their remarks on the paper. Part of the work was done when I visited ICERM (Brown University, Providence) and the Erwin Schrödinger Institute (Vienna). I would like to thank them for their hospitality.

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  1. Department of Mathematics, Pennsylvania State University, University Park, PA, USA

    Yakov Pesin

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  1. Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, Trondheim, Norway

    Helge Holden

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Ragni Piene

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Pesin, Y. (2019). Sinai’s Work on Markov Partitions and SRB Measures. In: Holden, H., Piene, R. (eds) The Abel Prize 2013-2017. The Abel Prize. Springer, Cham. https://doi.org/10.1007/978-3-319-99028-6_11

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