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Robust Exponential Mixing and Convergence to Equilibrium for Singular-Hyperbolic Attracting Sets

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Abstract

We extend results on robust exponential mixing for geometric Lorenz attractors, with a dense orbit and a unique singularity, to singular-hyperbolic attracting sets with any number of (either Lorenz- or non-Lorenz-like) singularities and finitely many ergodic physical/SRB invariant probability measures, whose basins cover a full Lebesgue measure subset of the trapping region of the attracting set. We obtain exponential mixing for any physical probability measure supported in the trapping region and also exponential convergence to equilibrium, for a \(C^2\) open subset of vector fields in any d-dimensional compact manifold (\(d\ge 3\)).

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Notes

  1. We write \(\overline{A}\) to denote the topological closure of a set A.

  2. That is the same as singular-hyperbolicity, but allowing \(\dim E^{cu}>2\) and demanding that volume expansion holds along every two-dimensional subspace of \(E^{cu}\).

  3. Note that here we are assuming that the return time to the base of the semiflow is constant on stable leaves.

  4. This definition was first given in [11] but its statement was only valid for 3-flows. We present here a corrected proof for completeness.

  5. We write \(A+B\) the union of the disjoint subsets A and B.

  6. We also use the term curve to denote the image of the curve.

  7. See [34] for the definition of p-variation.

  8. See Proposition 2.3 and [17, Section 6.1] where it is shown how to get \((\pi _\gamma )_*\mu ^i_F=\mu ^i_f\).

  9. The subset \({{\mathcal {S}}}\) can be identified with \(h(\Gamma _0)\) while \({{\mathcal {D}}}\setminus {{\mathcal {S}}}\) can be identified with \(h(\Gamma _1)\)

  10. Recall that \(\tau \) is constant on stable leaves.

References

  1. Alves, J.F.: Nonuniformly Hyperbolic Attractors: Geometric and Probabilistic Aspects. Springer (2020)

  2. 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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alves, J.F., Luzzatto, S., Pinheiro, V.: Lyapunov exponents and rates of mixing for one-dimensional maps. Ergod. Theory Dyn. Syst. 24(3), 637–657 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Anosov, D.V.: Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Math. Inst. 90, 1–235 (1967)

    MathSciNet  Google Scholar 

  6. Araujo, V.: Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability. Ergod. Theory Dyn. Syst. (to appear), 1–28 (2020)

  7. Araujo, V., Arbieto, A., Salgado, L.: Dominated splittings for flows with singularities. Nonlinearity 26(8), 2391 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Araujo, V., Butterley, O., Varandas, P.: Open sets of Axiom A flows with exponentially mixing attractors. Proc. Am. Math. Soc. 144, 2971–2984 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Araujo, V., Galatolo, S., Pacifico, M.J.: Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors. Mathematische Zeitschrift 276(3–4), 1001–1048 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Araujo, V., Melbourne, I.: Exponential decay of correlations for nonuniformly hyperbolic flows with a \(C^{1+\alpha }\) stable foliation, including the classical Lorenz attractor. Annales Henri Poincaré, 2975–3004 (2016)

  11. Araujo, V., Melbourne, I.: Existence and smoothness of the stable foliation for sectional hyperbolic attractors. Bull. Lond. Math. Soc. 49(2), 351–367 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Araujo, V., Melbourne, I.: Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation. Adv. Math. 349, 212–245 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Araujo, V., Melbourne, I., Varandas, P.: Rapid mixing for the lorenz attractor and statistical limit laws for their time-1 maps. Commun. Math. Phys. 340(3), 901–938 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Araujo, V., Pacifico, M.J.: Three Dimensional Flows. XXV Brazillian Mathematical Colloquium. IMPA, Rio de Janeiro (2007)

  15. Araujo, V., Pacifico, M.J.: Three-dimensional flows, volume 53 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Heidelberg, (2010). With a foreword by Marcelo Viana

  16. Araujo, V., Pacifico, M.J., Pinheiro, M.: Adapted random perturbations for non-uniformly expanding maps. Stoch. Dyn. 14(04), 1450007 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Araujo, V., Pacifico, M.J., Pujals, E.R., Viana, M.: Singular-hyperbolic attractors are chaotic. Trans. A.M.S. 361, 2431–2485 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Araujo, V., Souza, A., Trindade, E.: Upper large deviations bound for singular-hyperbolic attracting sets. J. Dyn. Differ. Equ. 31(2), 601–652 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Araujo, V., Varandas, P.: Robust exponential decay of correlations for singular-flows. Commun. Math. Phys. 311, 215–246 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Araujo, V., Varandas, P.: Erratum to: robust exponential decay of correlations for singular-flows. Commun. Math. Phys., 1–3 (2015)

  21. Arbieto, A.: Sectional lyapunov exponents. Proc. Am. Math. Soc. 138, 3171–3178 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Avila, A., Gouëzel, S., Yoccoz, J.-C.: Exponential mixing for the Teichmüller flow. Publ. Math. Inst. Hautes Études Sci. 104, 143–211 (2006)

    Article  MATH  Google Scholar 

  23. Bonatti, C., Pumariño, A., Viana, M.: Lorenz attractors with arbitrary expanding dimension. C. R. Acad. Sci. Paris Sér. I Math. 325(8), 883–888 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. Invent. Math. 29, 181–202 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  25. Butterley, O., Melbourne, I.: Disintegration of invariant measures for hyperbolic skew products. Israel J. Math. 219(1), 171–188 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Butterley, O., War, K.: Open sets of exponentially mixing anosov flows. J. Eur. Math. Soc. 22(7), 2253–2285 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Daltro, D., Varandas, P.: Exponential decay of correlations for gibbs measures and semiflows over \(c^{1+\alpha }\) piecewise expanding maps. Annales Henri Poincaré (2021)

  28. Daltro, D., Varandas, P.: Exponential decay of correlations for Gibbs measures on attractors of Axiom A flows. E-prints arXiv:2104.11839 (2021)

  29. Dolgopyat, D.: On decay of correlations in Anosov flows. Ann. Math. 147(2), 357–390 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Eslami, P.: Inducing schemes for multi-dimensional piecewise expanding maps. E-prints arXiv:2002.06679 (2020)

  31. Field, M., Melbourne, I., Törok, A.: Stability of mixing and rapid mixing for hyperbolic flows. Ann. Math. 166, 269–291 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gouëzel, S.: Decay of correlations for nonuniformly expanding systems. Bull. Soc. Math. France 134(1), 1–31 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180(1), 119–140 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  34. Keller, G.: Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrsch. Verw. Gebiete 69(3), 461–478 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  35. Leplaideur, R., Yang, D.: SRB measure for higher dimensional singular partially hyperbolic attractors. Annales de l‘Institut Fourier 67(2), 2703–2717 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Melbourne, I.: Superpolynomial and polynomial mixing for semiflows and flows. Nonlinearity 31(10), R268–R316 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Melbourne, I., Török, A.: Statistical limit theorems for suspension flows. Israel J. Math. 144, 191–209 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  38. Metzger, R., Morales, C.: Sectional-hyperbolic systems. Ergod. Theory Dyn. Syst. 28, 1587–1597 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Morales, C.A.: Examples of singular-hyperbolic attracting sets. Dyn. Sys. Int. J. 22(3), 339–349 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. Morales, C.A., Pacifico, M.J., Pujals, E.R.: Singular hyperbolic systems. Proc. Am. Math. Soc. 127(11), 3393–3401 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. Morales, C.A., Pacifico, M.J., Pujals, E.R.: Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers. Ann. Math. 160(2), 375–432 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Newhouse, S.: On a differentiable linearization theorem of Philip Hartman. In: Pesin, K.A.Y., Hertz, F. (eds.) Modern Theory of Dynamical Systems: A Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, vol. 692, pp. 209–262. Americam Mathematical Society, New York (2017)

    Chapter  Google Scholar 

  43. Palis, J., de Melo, W.: Geometric Theory of Dynamical Systems. Springer (1982)

  44. Pesin, Y., Sinai, Y.: Gibbs measures for partially hyperbolic attractors. Ergod. Theory Dyn. Syst. 2, 417–438 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  45. Philipp, W., Stout, W.: Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables. Memoirs of the American Mathematical Society. American Mathematical Society, American Mathematical Society (1975)

  46. Pollicott, M.: On the rate of mixing of Axiom A flows. Invent. Math. 81(3), 413–426 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ruelle, D.: A measure associated with Axiom A attractors. Am. J. Math. 98, 619–654 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  48. Sinai, Y.: Gibbs measures in ergodic theory. Russ. Math. Surv. 27, 21–69 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  49. Tsujii, M., Zhang, Z.: Smooth mixing anosov flows in dimension three are exponential mixing. E-prints arXiv:2006.04293 (2020)

  50. Tucker, W.: The Lorenz attractor exists. C. R. Acad. Sci. Paris 328, 1197–1202 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This is based on the PhD thesis of E. Trindade at the Instituto de Matematica e Estatistica-Universidade Federal da Bahia (UFBA) under a CAPES scholarship. E.T. thanks the Mathematics and Statistics Institute at UFBA for the use of its facilities and the financial support from CAPES during his M.Sc. and Ph.D. studies. We thank A. Castro; Y. Lima; D. Smania and P. Varandas for many comments and suggestions which greatly improved the text. We also thank the anonymous referee for the useful suggestions that improved the text.

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  1. Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, Salvador, 40170-110, Brazil

    Vitor Araújo

  2. Instituto Federal de Educação Ciência e Tecnologia da Bahia (IFBA), Campus Porto Seguro, Rod. Br 367 Km 57,5 - Fontana I, Porto Seguro, BA, 45810-000, Brazil

    Edvan Trindade

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Correspondence to Vitor Araújo.

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V.A. was partially supported by CNPq-Brazil (Grant No. 300985/2019-3) and E.T. was partially supported by CAPES-Brazil (Grant No. 0001).

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Araújo, V., Trindade, E. Robust Exponential Mixing and Convergence to Equilibrium for Singular-Hyperbolic Attracting Sets. J Dyn Diff Equat 35, 2487–2536 (2023). https://doi.org/10.1007/s10884-021-10100-7

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