Inventiones mathematicae

, Volume 213, Issue 3, pp 811–1016 | Cite as

Limit theorems for fast–slow partially hyperbolic systems

  • Jacopo De Simoi
  • Carlangelo Liverani


We prove several limit theorems for a simple class of partially hyperbolic fast–slow systems. We start with some well know results on averaging, then we give a substantial refinement of known large (and moderate) deviation results and conclude with a completely new result (a local limit theorem) on the distribution of the process determined by the fluctuations around the average. The method of proof is based on a mixture of standard pairs and transfer operators that we expect to be applicable in a much wider generality.

Mathematics Subject Classification

37A25 37C30 37D30 37A50 60F17 


  1. 1.
    Anosov, D.V.: Averaging in systems of ordinary differential equations with rapidly oscillating solutions. Izv. Akad. Nauk SSSR Ser. Mat. 24, 721–742 (1960)MathSciNetGoogle Scholar
  2. 2.
    Avila, A., Gouëzel, S., Yoccoz, J.-C.: Exponential mixing for the Teichmüller flow. Publ. Math. Inst. Hautes Études Sci. 104, 143–211 (2006)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bakhtin, V.I.: Cramér asymptotics in the averaging method for systems with fast hyperbolic motions. Tr. Mat. Inst. Steklova 244(Din. Sist. i Smezhnye Vopr. Geom.), 65–86 (2004)Google Scholar
  4. 4.
    Bakhtin, V.I.: Cramér’s asymptotics in systems with fast and slow motions. Stoch. Stoch. Rep. 75(5), 319–341 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baladi, V.: Positive Transfer Operators and Decay of Correlations, Volume 16 of Advanced Series in Nonlinear Dynamics. World Scientific Publishing Co., Inc., River Edge (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Baladi, V., Tsujii, M.: Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier 57(1), 127–154 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Baladi, V., Vallée, B.: Euclidean algorithms are Gaussian. J. Number Theory 110(2), 331–386 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Baladi, V., Vallée, B.: Exponential decay of correlations for surface semi-flows without finite Markov partitions. Proc. Am. Math. Soc. 133(3), 865–874 (2005). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Blank, M., Keller, G., Liverani, C.: Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15(6), 1905–1973 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chernov, N.I.: Markov approximations and decay of correlations for Anosov flows. Ann. Math. (2) 147(2), 269–324 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Collier, D., Morris, I.D.: Approximating the maximum ergodic average via periodic orbits. Ergod. Theory Dyn. Syst. 28(4), 1081–1090 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    De Simoi, J., Liverani, C.: The martingale approach after Varadhan and Dolgopyat. In: Hyperbolic Dynamics, Fluctuations and Large Deviations, Volume 89 of Proceedings of Symposium in Pure Mathematics, pp. 311–339. Amer. Math. Soc., Providence (2015)Google Scholar
  13. 13.
    De Simoi, J., Liverani, C.: Statistical properties of mostly contracting fast–slow partially hyperbolic systems. Invent. Math. 206(1), 147–227 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    De Simoi, J., Liverani, C., Poquet, C., Volk, D.: Fast–slow partially hyperbolic systems versus Freidlin–Wentzell random systems. J. Stat. Phys. 166(3–4), 650–679 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, Volume 38 of Stochastic Modelling and Applied Probability. Springer, Berlin (2010). Corrected reprint of the second (1998) editionGoogle Scholar
  16. 16.
    Demers, M.F., Liverani, C.: Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Am. Math. Soc. 360(9), 4777–4814 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Demers, M.F., Zhang, H.-K.: Spectral analysis of the transfer operator for the Lorentz gas. J. Mod. Dyn. 5(4), 665–709 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dolgopyat, D.: On decay of correlations in Anosov flows. Ann. Math. (2) 147(2), 357–390 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Dolgopyat, D.: Prevalence of rapid mixing. II. Topological prevalence. Ergod. Theory Dyn. Syst. 20(4), 1045–1059 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dolgopyat, D.: Averaging and invariant measures. Mosc. Math. J. 5(3), 537–576 (2005). 742MathSciNetzbMATHGoogle Scholar
  21. 21.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
  22. 22.
    Faure, F.: Semiclassical origin of the spectral gap for transfer operators of a partially expanding map. Nonlinearity 24(5), 1473–1498 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Faure, F., Sjöstrand, J.: Upper bound on the density of Ruelle resonances for Anosov flows. Commun. Math. Phys. 308(2), 325–364 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Faure, F., Tsujii, M.: Band structure of the Ruelle spectrum of contact Anosov flows. C. R. Math. Acad. Sci. Paris 351(9–10), 385–391 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, 2nd edn. Wiley, New York (1971)zbMATHGoogle Scholar
  26. 26.
    Giulietti, P., Liverani, C., Pollicott, M.: Anosov flows and dynamical zeta functions. Ann. Math. (2) 178(2), 687–773 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gottwald, G.A., Melbourne, I.: Homogenization for deterministic maps and multiplicative noise. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469(2156), 20130201 (2013). 16MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gouëzel, S.: Local limit theorem for nonuniformly partially hyperbolic skew-products and Farey sequences. Duke Math. J. 147(2), 193–284 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Gouëzel, S., Liverani, C.: Banach spaces adapted to Anosov systems. Ergod. Theory Dyn. Syst. 26(1), 189–217 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Gouëzel, S., Liverani, C.: Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. J. Differ. Geom. 79(3), 433–477 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Guivarc’h, Y., Hardy, J.: Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. H. Poincaré Probab. Stat. 24(1), 73–98 (1988)zbMATHGoogle Scholar
  32. 32.
    Hennion, H.: Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Am. Math. Soc. 118(2), 627–634 (1993)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 editionCrossRefGoogle Scholar
  34. 34.
    Keller, G.: Equilibrium States in Ergodic Theory, Volume 42 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  35. 35.
    Keller, G., Liverani, C.: Stability of the spectrum for transfer operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28(1), 141–152 (1999)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Kifer, Y.: Averaging principle for fully coupled dynamical systems and large deviations. Ergod. Theory Dyn. Syst. 24(3), 847–871 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kifer, Y.: Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging. Mem. Am. Math. Soc. 201(944), viii+129 (2009)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Komiya, H.: Elementary proof for Sion’s minimax theorem. Kodai Math. J. 11(1), 5–7 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Lieb, E.H., Loss, M.: Analysis, Volume 14 of Graduate Studies in Mathematics. American Mathematical Society, Providence (1997)Google Scholar
  40. 40.
    Liverani, C.: Decay of correlations. Ann. Math. (2) 142(2), 239–301 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Liverani, C.: Decay of correlations for piecewise expanding maps. J. Stat. Phys. 78(3–4), 1111–1129 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Liverani, C.: On contact Anosov flows. Ann. Math. (2) 159(3), 1275–1312 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Liverani, C.: Multidimensional expanding maps with singularities: a pedestrian approach. Ergod. Theory Dyn. Syst. 33(1), 168–182 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Melbourne, I., Stuart, A.M.: A note on diffusion limits of chaotic skew-product flows. Nonlinearity 24(4), 1361–1367 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Parthasarathy, K.R.: On the category of ergodic measures. Ill. J. Math. 5, 648–656 (1961)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Pollicott, M.: On the mixing of Axiom A attracting flows and a conjecture of Ruelle. Ergod. Theory Dyn. Syst. 19(2), 535–548 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Rockafellar, R.T.: Convex analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997). Reprint of the 1970 original, Princeton PaperbacksGoogle Scholar
  48. 48.
    Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Tsujii, M.: Exponential mixing for generic volume-preserving anosov flows in dimension three. Preprint arXiv:1601.00063
  50. 50.
    Tsujii, M.: Quasi-compactness of transfer operators for contact Anosov flows. Nonlinearity 23(7), 1495–1545 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Young, L.-S.: Large deviations in dynamical systems. Trans. Am. Math. Soc. 318(2), 525–543 (1990)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Dipartimento di MatematicaII Università di Roma (Tor Vergata)RomeItaly

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