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SRB Measures and Young Towers for Surface Diffeomorphisms

Abstract

We give geometric conditions that are necessary and sufficient for the existence of Sinai–Ruelle–Bowen (SRB) measures for \(C^{1+\alpha }\) surface diffeomorphisms, thus proving a version of the Viana conjecture. As part of our argument we give an original method for constructing first return Young towers, proving that every hyperbolic measure, and in particular every SRB measure, can be lifted to such a tower. This method relies on a new general result on hyperbolic branches and shadowing for pseudo-orbits in non-uniformly hyperbolic sets which is of independent interest.

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Notes

  1. 1.

    These typically use specific geometric characteristics of the system under consideration.

  2. 2.

    Consider the identity map, for example.

  3. 3.

    If both Lyapunov exponents are negative or both are positive, then it can be shown that the corresponding ergodic component of the measure \( \mu \) is supported on an attracting or repelling periodic orbit, respectively; we exclude this trivial situation.

  4. 4.

    In fact, the existence of local stable and unstable curves can be proved under weaker conditions than those of nonzero Lyapunov exponents, see Definition 1.6 and Theorem 1.12.

  5. 5.

    The converse is not true: for example, if p is a hyperbolic fixed point whose stable and unstable curves form a figure-eight, then \(\delta _p\) is a hyperbolic physical measure which is not SRB [48, p. 140].

  6. 6.

    The converse is not true; the limits in the definition of nonzero Lyapunov exponents need not exist at every point (only almost every), even in uniform hyperbolicity. Although existence of these limits is not necessary for our results, the slow variation condition (H1) still plays a crucial role in Theorem 1.12, and it seems unlikely that it can be removed.

  7. 7.

    After this paper was completed we learned of recent work by Chen, Wang, and Zhang that uses a similar notion for systems with singularities; see Definition 9 in [25, §5.3].

  8. 8.

    One could imagine studying other equilibrium measures besides SRB by replacing Lebesgue measure here with a reference measure such as those studied in [35, 36], but we do not pursue this here.

  9. 9.

    As we will see in the next section, for surfaces Young’s tower conditions from [71] turn out to be necessary as well as sufficient, but this was not proved in that paper.

  10. 10.

    A more general inducing structure was introduced in [57]; it can be used to study the existence and ergodic properties of equilibrium measures, which include SRB measures.

  11. 11.

    We need the larger collection to guarantee that the rectangle we build is “saturated”, see Remark 11.10 and Sect. 11.5.

  12. 12.

    We could of course replace \( Q_{0} \) in the expression for \( \widehat{Q}\) by its explicit value but various calculations to be given below will be easier and clearer by keeping track of \( Q_{0} \) as an independent constant.

  13. 13.

    In [13] the Lyapunov change of coordinates \(L_x\) is required to be tempered, but we do not require this condition.

  14. 14.

    In [56] it is required that \(L_x\) is tempered, but this is not necessary for our formulation.

  15. 15.

    An elementary computation shows that the optimal lower bound is \((1-\omega )/\sqrt{2(1+\omega ^2)}\).

  16. 16.

    In fact one does not need the full strength of the hyperbolic branch property to make this definition; it suffices to have a hyperbolic branch associated with each (true) return.

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Acknowledgements

The authors wish to thank Stefano Bianchini who helped keep this project alive with many useful comments and suggestions when we had almost given up, Dima Dolgopyat for pointing out a missing argument in a previous version, and Vilton Pinheiro for helping us fill in the missing argument. We are also grateful to the anonymous referee for a very careful reading and many useful suggestions, which led to corrections and clarifications that have substantially improved the paper. V. C. is partially supported by NSF Grants DMS-1362838 and DMS-1554794. Ya. P. was partially supported by NSF Grant DMS-1400027.

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Appendix A: List of Terminology and Notation

Appendix A: List of Terminology and Notation

  1. (1)

    Almost returns, Definition 4.13 on page 23

  2. (2)

    Brackets, Definition 1.11 on page 7

  3. (3)

    Branch

    • \((C,\kappa )\)-hyperbolic, Definition 4.8 on page 22

    • \({\hat{\ell }}\)-regular, Definition 5.6 on page 29

    • \((C,\kappa )\)-hyperbolic branch property, Definition 4.14 on page 24

  4. (4)

    Overlapping charts, Definition 8.1 on page 40

  5. (5)

    Concatenation property, Definition 4.12 on page 23

  6. (6)

    Cones, Definition 4.2 on page 21

    • in regular neighbourhoods, Definition 5.10 on page 27

  7. (7)

    Conefield, Definition 4.3 on page 21

    • adapted, Definition 4.5 on page 21

  8. (8)

    Curves

    • local \((C,\lambda )\)-stable (unstable), Definition 1.10 on page 7

    • \({\mathcal {K}}\)-admissible, Definition 4.4 on page 21

    • stable and unstable admissible, Definition 4.6 on page 21

    • in regular neighbourhoods, Definition 5.2 on page 27

    • full length stable and unstable admissible, Definition 4.6 and Definition 5.2

  9. (9)

    \((\chi ,\varepsilon ,\ell ,r)\)-nice domain, Definition 1.16 on page 9

  10. (10)

    Measure

    • hyperbolic, Definition 1.2 on page 4

    • physical, Definition 1.1 on page 4

    • SRB, Definition 1.4 on page 5

  11. (11)

    Nice

    • domain, Definition 1.16 on page 9

    • regular set, Definition 1.19 on page 10

    • rectangle, Definition 2.3 on page 14

  12. (12)

    Pseudo-orbit

    • finite \(({\hat{\ell }},\delta ,\lambda )\)-, Definition 5.4 on page 28

    • bi-infinite \(({\hat{\ell }},\delta ,\lambda )\)-, Definition 5.9 on page 30

  13. (13)

    Rectangle

    • \((C,\lambda )\), Definition 2.1 on page 13

    • nice, Definition 2.3 on page 14

    • saturated, Definition 11.2 on page 62

  14. (14)

    Recurrence (recurrent)

    • recurrent and Lebesgue-strongly recurrent, Definition 1.21 on page 10

    • almost recurrent, Definition 4.13 on page 23

  15. (15)

    Regular

    • level sets, Definition 1.8 on page 7

    • \((\chi ,\varepsilon ,\ell )\)-regular set, Definition 1.9 on page 7

    • \((\chi ,\varepsilon ,\ell ,r)\)-nice regular set (nice regular), Definition 1.19 on page 10

    • \({\hat{\ell }}\)-regular branch, Definition 5.6 on page 29

  16. (16)

    Sequences, hyperbolic, Definition 11.8 on page 65

  17. (17)

    Set

    • fat, Definition 1.3 on page 5

    • \((\chi ,\varepsilon )\)-hyperbolic, Definition 1.6 on page 6

    • regular level, Definition 1.8 on page 7

    • \((\chi ,\varepsilon ,\ell )\)-regular, Definition 1.9 on page 7

    • s/u-subsets, Definition 2.4 on page 14

    • \((\chi ,\varepsilon ,\ell ,r)\)-nice regular (nice regular), Definition 1.19 on page 10

  18. (18)

    Stable and unstable strips,

    • in a nice domain, Definition 4.7 on page 21

    • in regular neighbourhoods, Definition 5.3 on page 28

  19. (19)

    T-returns time, Definition 2.5 on page 14

  20. (20)

    Tower

    • topological Young, Definition 2.6 on page 15

    • Young, Definition 2.10 on page 15

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Climenhaga, V., Luzzatto, S. & Pesin, Y. SRB Measures and Young Towers for Surface Diffeomorphisms. Ann. Henri Poincaré (2021). https://doi.org/10.1007/s00023-021-01113-5

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Mathematics Subject Classification

  • Primary 37D25
  • Secondary 37C40
  • 37E30