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
These typically use specific geometric characteristics of the system under consideration.
Consider the identity map, for example.
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.
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].
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.
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].
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.
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.
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.
In [13] the Lyapunov change of coordinates \(L_x\) is required to be tempered, but we do not require this condition.
In [56] it is required that \(L_x\) is tempered, but this is not necessary for our formulation.
An elementary computation shows that the optimal lower bound is \((1-\omega )/\sqrt{2(1+\omega ^2)}\).
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.
References
Alves, J.F., Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2), 351–398 (2000)
Alves, J.F., Carvalho, M., Milhazes, J.M.: Statistical stability and continuity of SRB entropy for systems with Gibbs-Markov structures. Commun. Math. Phys. 296(3), 739–767 (2010)
Alves, J.F., Carvalho, M., Freitas, J.M.: Statistical stability for Hénon maps of the Benedicks–Carleson type. Ann. Inst. H. Poincaré Anal. NonLinéaire 27(2), 595–637 (2010)
Alves, J.F., Dias, C.L., Luzzatto, S.: Geometry of expanding absolutely continuous invariant measures and the liftability problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(1), 101–120 (2013)
Alves, J.F., Dias, C.L., Luzzatto, S., Pinheiro, V.: SRB measures for partially hyperbolic systems whose central direction is weakly expanding. J. Eur. Math. Soc. 19(10), 2911–2946 (2017)
Alves, J.F., Freitas, J.M., Luzzatto, S., Vaienti, S.: From rates of mixing to recurrence times via large deviations. Adv. Math. 228(2), 1203–1236 (2011)
Alves, J.F., Li, X.: Gibbs–Markov–Young structures with (stretched) exponential tail for partially hyperbolic attractors. Adv. Math. 279, 405–437 (2015)
Alves, J.F., Luzzatto, S., Pinheiro, V.: Markov structures and decay of correlations for non-uniformly expanding dynamical systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(6), 817–839 (2005)
Alves, J.F., Pinheiro, V.: Slow rates of mixing for dynamical systems with hyperbolic structures. J. Stat. Phys. 131(3), 505–534 (2008)
Alves, J.F., Pinheiro, V.: Gibbs–Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction. Adv. Math. 223(5), 1706–1730 (2010)
Bálint, P., Tóth, I.P.: Exponential decay of correlations in multi-dimensional dispersing billiards. Ann. Henri Poincaré 9(7), 1309–1369 (2008)
Barreira, L., Pesin, Y.: Lectures on Lyapunov exponents and smooth ergodic theory. Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, Appendix A by M. Brin and Appendix B by D. Dolgopyat, H. Hu and Pesin, pp. 3–106
Barreira, L., Pesin, Y.: Nonuniform hyperbolicity. Encyclopedia of Mathematics and its Applications, vol. 115, Cambridge University Press, Cambridge. Dynamics of systems with nonzero Lyapunov exponents (2007)
Ben Ovadia, S.: Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. J. Mod. Dyn. 13, 43–113 (2018)
Ben Ovadia, S.: Hyperbolic SRB measures and the leaf condition. Commun. Math. Phys. (2021). https://doi.org/10.1007/s00220-021-04208-6
Benedicks, M., Young, L.-S.: Sinaĭ–Bowen–Ruelle measures for certain Hénon maps. Invent. Math. 112(3), 541–576 (1993)
Benedicks, M., Young, L.-S.: Markov extensions and decay of correlations for certain Hénon maps. Astérisque (2000), no. 261, xi, 13–56, Géométrie complexe et systèmes dynamiques (Orsay, 1995)
Bonatti, C., Díaz, L.J., Viana, M.: Dynamics beyond uniform hyperbolicity. Encyclopaedia of Mathematical Sciences, vol. 102, Springer, Berlin. A global geometric and probabilistic perspective. Mathematical Physics, III (2005)
Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Isr. J. Math. 115, 157–193 (2000)
Brin, M., Kifer, Yu.: Dynamics of Markov chains and stable manifolds for random diffeomorphisms. Ergod. Theory Dyn. Syst. 7(3), 351–374 (1987)
Bruin, H., Luzzatto, S., Van Strien, S.: Decay of correlations in one-dimensional dynamics. Ann. Sci. École Norm. Sup. (4) 36(4), 621–646 (2003)
Burguet, D.: Entropy of physical measures for \(C^\infty \) smooth systems. Commun. Math. Phys 375(2), 1201–1222 (2020)
Burns, K., Climenhaga, V., Fisher, T., Thompson, D.J.: Unique equilibrium states for geodesic flows in nonpositive curvature. Geom. Funct. Anal. 28(5), 1209–1259 (2018)
Burns, K., Dolgopyat, D., Pesin, Y., Pollicott, M.: Stable ergodicity for partially hyperbolic attractors with negative central exponents. J. Mod. Dyn. 2(1), 63–81 (2008)
Chen, J., Wang, F., Zhang, H.-K.: Markov partition and thermodynamic formalism for hyperbolic systems with singularities, arXiv:1709.00527 (2017)
Chernov, N.: Decay of correlations and dispersing billiards. J. Stat. Phys. 94(3–4), 513–556 (1999)
Chernov, N., Zhang, H.-K.: Billiards with polynomial mixing rates. Nonlinearity 18(4), 1527–1553 (2005)
Chung, Y.M.: Large deviations on Markov towers. Nonlinearity 24(4), 1229–1252 (2011)
Climenhaga, V.: Specification and towers in shift spaces. Commun. Math. Phys. 364(2), 441–504 (2018)
Climenhaga, V., Dolgopyat, D., Pesin, Y.: Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps. Commun. Math. Phys. 346(2), 553–602 (2016)
Climenhaga, V., Fisher, T., Thompson, D.J.: Unique equilibrium states for Bonatti–Viana diffeomorphisms. Nonlinearity 31(6), 2532–2570 (2018)
Climenhaga, V., Fisher, T., Thompson, D.J.: Equilibrium states for Mañé diffeomorphisms. Ergod. Theory Dyn. Syst. 39(9), 2433–2455 (2019)
Climenhaga, V., Luzzatto, S., Pesin, Y.: The geometric approach for constructing Sinai–Ruelle–Bowen measures. J. Stat. Phys. 166(3–4), 467–493 (2017)
Climenhaga, V., Pesin, Y.: Building thermodynamics for non-uniformly hyperbolic maps. Arnold Math. J. 3(1), 37–82 (2017)
Climenhaga, V., Pesin, Y., Zelerowicz, A.: Equilibrium states in dynamical systems via geometric measure theory. Bull. Am. Math. Soc. (N.S.) 56(4), 569–610 (2019)
Climenhaga, V., Pesin, Y., Zelerowicz, A.: Equilibrium measures for some partially hyperbolic systems. J. Mod. Dyn. 16, 155–205 (2020)
Cyr, V., Sarig, O.: Spectral gap and transience for Ruelle operators on countable Markov shifts. Commun. Math. Phys. 292(3), 637–666 (2009)
Demers, M.F.: Functional norms for Young towers. Ergod. Theory Dyn. Syst. 30(5), 1371–1398 (2010)
Díaz-Ordaz, K., Holland, M.P., Luzzatto, S.: Statistical properties of one-dimensional maps with critical points and singularities. Stoch. Dyn. 6(4), 423–458 (2006)
Gouëzel, S.: Sharp polynomial estimates for the decay of correlations. Isr. J. Math. 139, 29–65 (2004)
Gouëzel, S.: Decay of correlations for nonuniformly expanding systems. Bull. Soc. Math. France 134(1), 1–31 (2006)
Gupta, C., Holland, M., Nicol, M.: Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps. Ergod. Theory Dyn. Syst. 31(5), 1363–1390 (2011)
Haydn, N.T.A., Psiloyenis, Y.: Return times distribution for Markov towers with decay of correlations. Nonlinearity 27(6), 1323–1349 (2014)
Hirayama, M.: Periodic probability measures are dense in the set of invariant measures. Discrete Contin. Dyn. Syst. 9(5), 1185–1192 (2003)
Hirsch, M.W.: Differential Topology. Graduate Texts in Mathematics, vol. 33, Springer, New York (1994). Corrected reprint of the 1976 original
Holland, M.: Slowly mixing systems and intermittency maps. Ergod. Theory Dyn. Syst. 25(1), 133–159 (2005)
Holland, M., Nicol, M., Török, A.: Extreme value theory for non-uniformly expanding dynamical systems. Trans. Am. Math. Soc. 364(2), 661–688 (2012)
Katok, A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes Études Sci. Publ. Math. 51, 137–173 (1980)
Ledrappier, F., Young, L.-S.: The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula. Ann. Math. (2) 122(3), 509–539 (1985)
Lima, Y., Matheus, C.: Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities. Ann. Sci. Éc. Norm. Supér. (4) 51(1), 1–38 (2018)
Lima, Y.: Sarig, Omri: Symbolic dynamics for three-dimensional flows with positive topological entropy. J. Eur. Math. Soc. (JEMS) 21(1), 199–256 (2019)
Markarian, R.: Billiards with polynomial decay of correlations. Ergod. Theory Dyn. Syst. 24(1), 177–197 (2004)
Maume-Deschamps, V.: Projective metrics and mixing properties on towers. Trans. Am. Math. Soc. 353(8), 3371–3389 (2001)
Melbourne, I., Nicol, M.: Almost sure invariance principle for nonuniformly hyperbolic systems. Commun. Math. Phys. 260(1), 131–146 (2005)
Melbourne, I., Nicol, M.: Large deviations for nonuniformly hyperbolic systems. Trans. Am. Math. Soc. 360(12), 6661–6676 (2008)
Pesin, Y.B.: Families of invariant manifolds that correspond to nonzero characteristic exponents. Izv. Akad. Nauk SSSR Ser. Mat. 40(6), 1332–1379, 1440 (1976)
Pesin, Y., Senti, S., Zhang, K.: Thermodynamics of towers of hyperbolic type. Trans. Am. Math. Soc. 368(12), 8519–8552 (2016)
Pesin, Y.B., Senti, S., Zhang, K.: Lifting measures to inducing schemes. Ergod. Theory Dyn. Syst. 28(2), 553–574 (2008)
Pesin, Y.B., Sinaĭ, Y.G.: Gibbs measures for partially hyperbolic attractors. Ergod. Theory Dyn. Syst. 2(3-4) (1982), 417–438 (1983)
Pilyugin, S.Y.: Shadowing in dynamical systems. Lecture Notes in Mathematics, vol. 1706. Springer, Berlin (1999)
Pinheiro, V.: Expanding measures. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(6), 889–939 (2011)
Pinheiro, V.: Lift and synchronization, work in progress (2019)
Rey-Bellet, L., Young, L.-S.: Large deviations in non-uniformly hyperbolic dynamical systems. Ergod. Theory Dyn. Syst. 28(2), 587–612 (2008)
Sarig, O.: Subexponential decay of correlations. Invent. Math. 150(3), 629–653 (2002)
Sarig, O.M.: Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Am. Math. Soc. 26(2), 341–426 (2013)
Shahidi, F., Zelerowicz, A.: Thermodynamics via inducing. J. Stat. Phys. 175(2), 351–383 (2019)
Shub, M.: Stabilité globale des systèmes dynamiques. Astérisque, vol. 56, Société Mathématique de France, Paris, 1978, With an English preface and summary
Tsujii, M.: Regular points for ergodic Sinaĭ measures. Trans. Am. Math. Soc. 328(2), 747–766 (1991)
Viana, M.: Dynamics: a probabilistic and geometric perspective, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), no. Extra Vol. I, 1998, pp. 557–578
Wang, Q., Young, L.-S.: Toward a theory of rank one attractors. Ann. Math. (2) 167(2), 349–480 (2008)
Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. (2) 147(3), 585–650 (1998)
Young, L.-S.: Recurrence times and rates of mixing. Isr. J. Math. 110, 153–188 (1999)
Young, L.-S.: What are SRB measures, and which dynamical systems have them? J. Stat. Phys. 108, 733–754. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays.
Zweimüller, R.: Invariant measures for general(ized) induced transformations. Proc. Am. Math. Soc. 133(8), 2283–2295 (2005)
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
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(1)
Almost returns, Definition 4.13 on page 23
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(2)
Brackets, Definition 1.11 on page 7
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(3)
Branch
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(4)
Overlapping charts, Definition 8.1 on page 40
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(5)
Concatenation property, Definition 4.12 on page 23
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(6)
Cones, Definition 4.2 on page 21
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in regular neighbourhoods, Definition 5.10 on page 27
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(7)
Conefield, Definition 4.3 on page 21
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adapted, Definition 4.5 on page 21
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(8)
Curves
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local \((C,\lambda )\)-stable (unstable), Definition 1.10 on page 7
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\({\mathcal {K}}\)-admissible, Definition 4.4 on page 21
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stable and unstable admissible, Definition 4.6 on page 21
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in regular neighbourhoods, Definition 5.2 on page 27
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full length stable and unstable admissible, Definition 4.6 and Definition 5.2
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(9)
\((\chi ,\varepsilon ,\ell ,r)\)-nice domain, Definition 1.16 on page 9
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(10)
Measure
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(11)
Nice
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(12)
Pseudo-orbit
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(13)
Rectangle
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(14)
Recurrence (recurrent)
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(15)
Regular
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(16)
Sequences, hyperbolic, Definition 11.8 on page 65
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(17)
Set
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fat, Definition 1.3 on page 5
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\((\chi ,\varepsilon )\)-hyperbolic, Definition 1.6 on page 6
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regular level, Definition 1.8 on page 7
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\((\chi ,\varepsilon ,\ell )\)-regular, Definition 1.9 on page 7
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s/u-subsets, Definition 2.4 on page 14
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\((\chi ,\varepsilon ,\ell ,r)\)-nice regular (nice regular), Definition 1.19 on page 10
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(18)
Stable and unstable strips,
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(19)
T-returns time, Definition 2.5 on page 14
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(20)
Tower
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Climenhaga, V., Luzzatto, S. & Pesin, Y. SRB Measures and Young Towers for Surface Diffeomorphisms. Ann. Henri Poincaré 23, 973–1059 (2022). https://doi.org/10.1007/s00023-021-01113-5
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DOI: https://doi.org/10.1007/s00023-021-01113-5