Skip to main content
SpringerLink
Log in

Limit Theorems for Random Non-uniformly Expanding or Hyperbolic Maps with Exponential Tails

  • Original Paper
  • Published:
Annales Henri Poincaré Aims and scope Submit manuscript

Abstract

We prove a Berry–Esseen theorem, a local central limit theorem and (local) large and (global) moderate deviations principles for i.i.d. (uniformly) random non-uniformly expanding or hyperbolic maps with exponential first return times. Using existing results the problem is reduced to certain random (Young) tower extensions, which is the main focus of this paper. On the random towers we will obtain our results using contraction properties of random complex equivariant cones with respect to the complex Hilbert projective metric.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. in the terminology of [7]  \(\mu _{\omega }\) are “sample stationary measures.”

  2. In terms of the maps \(\{f_{\omega }\}\), on the \(\ell \)th level of the tower \({\Delta }_{\omega }\) we have that \(s_{\omega }(x,y)+\ell \) is the time the random orbit of \(x_0\) and \(y_0\) stays together in the sense that all the returns to the random bases occur thorough the same atom, where \(x=(x_0,\ell )\) and \(y=(y_0,\ell )\).

  3. Here \(g\mathrm{d}m_{\omega }\) denotes the absolutely continuous measure w.r.t. \(m_{\omega }\) whose density is g.

  4. We refer to [41] for the definition of a canonical complexification. See also [24, Appendix A] for a summary of all the properties of real and complex cones which will be used in what follows.

References

  1. Alvés, J., Bahsoun, W., Ruziboev, M.: Almost Sure Rates of Mixing for Partially Hyperbolic Attractors, preprint arXiv:1904.12844

  2. Alvés, J., Dias, C., Luzzatto, S., Pinheiro, V.: SRB measures for partially hyperbolic systems whose central direction is weakly expanding. J. Eur. Math. Soc. 19, 2911–2946 (2017)

    Article  MathSciNet  Google Scholar 

  3. Aimino, R., Nicol, M., Vaienti, S.: Annealed and quenched limit theorems for random expanding dynamical systems. Probab. Theory Relat. Fields 162, 233–274 (2015)

    Article  MathSciNet  Google Scholar 

  4. Arnoux, P., Fisher, A.: Anosov families, renormalization and non-stationary subshifts. Ergod. Theory Dyn. Syst. 25, 661–709 (2005)

    Article  MathSciNet  Google Scholar 

  5. Baladi, V., Benedicks, M., Maume-Deschamps, V.: Almost sure rates of mixing for i.i.d. unimodal maps. Ann. Sci. École Norm. Sup. 35, 77–126 (2002)

    Article  MathSciNet  Google Scholar 

  6. Bahsoun, W., Bose, C.: Mixing rates and limit theorems for random intermittent maps. Nonlinearity 29(4), 1417–1433 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  7. Bahsoun, W., Bose, C., Duan, Y.: Decay of correlation for random intermittent maps. Nonlinearity 27(7), 1543–1554 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  8. Bahsoun, W., Bose, C., Ruziboev, M.: Quenched decay of correlations for slowly mixing systems. Trans. Am. Math. Soc. (2019)

  9. Coelho, Z., Parry, W.: Central limit asymptotics for shifts of finite type. Isr. J. Math. 69(2), 235 (1990)

    Article  MathSciNet  Google Scholar 

  10. Conze, J.-P., Raugi, A.: Limit Theorems for Sequential Expanding Dynamical Systems. AMS, Providence (2007)

    Book  Google Scholar 

  11. Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Probab. Theory Relat. Fields 100, 365–393 (1994)

    Article  MathSciNet  Google Scholar 

  12. Demers, M., Zhang, H.: A functional analytic approach to perturbations of the Lorentz gas. Commun. Math. Phys. 324, 767–830 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  13. Dragičević, D., Froyland, G., Gonzalez-Tokman, C., Vaienti, S.: Almost sure invariance principle for random piecewise expanding maps. Nonlinearity 31, 2252–2280 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  14. Dragičević, D., Froyland, G., Gonzalez-Tokman, C., Vaienti, S.: A spectral approach for quenched limit theorems for random expanding dynamical systems. Commun. Math. Phys. 360, 1121–1187 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  15. Dragičević, D., Froyland, G., Gonzalez-Tokman, C., Vaienti, S.: A spectral approach for quenched limit theorems for random hyperbolic dynamical systems. Trans. Am. Math. Soc. 373, 629–664 (2020)

    Article  MathSciNet  Google Scholar 

  16. Dragičević, D., Hafouta, Y.: Almost sure invariance principle for random dynamical systems via Gouëzel’s approach, accepted for publication in Nonlinearity, preprint avaiable on arxiv:1912.12332

  17. Dragičević, D., Hafouta, Y.: Limit theorems for random expanding or hyperbolic dynamical systems and vector-valued observables. Ann. Henri Poincaré 21, 3869–391 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  18. Dubois, L.: Projective metrics and contraction principles for complex cones. J. Lond. Math. Soc. 79, 719–737 (2009)

    Article  MathSciNet  Google Scholar 

  19. Dubois, L.: An explicit Berry–Esseen bound for uniformly expanding maps on the interval. Isr. J. Math. 186, 221–250 (2011)

    Article  MathSciNet  Google Scholar 

  20. Du, Z.: On Mixing Rates for Random Perturbations. Ph.D. thesis. National University of Singapore (2015)

  21. Guivarćh, Y., Hardy, J.: Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. H. Poincaré Probab. Statist. 24(1), 73–98 (1988)

  22. Gouëzel, S.: Berry–Esseen theorem and local limit theorem for non uniformly expanding maps. Ann. Inst. Henri Poincarë Prob. Stat. 41, 997–1024 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  23. Hafouta, Y., Kifer, Yu.: Berry–Esseen type estimates for nonconventional sums. Stoch. Proc. Appl. 126, 2430–2464 (2016)

    Article  MathSciNet  Google Scholar 

  24. Hafouta, Y., Kifer, Yu.: Nonconventional Limit Theorems and Random Dynamics. World Scientific, Singapore (2018)

    Book  Google Scholar 

  25. Hafouta, Y.: Limit theorems for some skew products with mixing base maps. Ergod. Theory Dyn. 41(1), 241–271 (2021)

    Article  MathSciNet  Google Scholar 

  26. Hafouta, Y.: Asymptotic moments and Edgeworth expansions for some processes in random dynamical environment. J. Stat. Phys 179, 945–971 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  27. Hafouta, Y.: Limit theorems for some time dependent expanding dynamical systems. Nonlinearity 33, 6421 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  28. Hafouta, Y.: A local limit theorem for number of multiple recurrences generated by some mixing processes with applications to Young towers. Accepted for publication in Journal d’Analyse Mathematique. arXiv:2003.08528v1

  29. Hafouta, Y.: On Eagleson’s theorem in the non-stationary setup, accepted for publication in Bull. Lond. Math. Soc. arXiv:2004.09333

  30. Hella, O., Stenlund, M.: Quenched normal approximation for random sequences of transformations. J. Stat. Phys. (2019). https://doi.org/10.1007/s10955-019-02390-5

    Article  MATH  Google Scholar 

  31. Haydn, N., Psiloyenis, Y.: Return times distribution for Markov towers with decay of correlations. Nonlinearity 27(6), 1323 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  32. Haydn, N., Nicol, M., Török, A., Vaienti, S.: Almost sure invariance principle for sequential and non-stationary dynamical systems. Trans. Am. Math. Soc. 369, 5293–5316 (2017)

    Article  MathSciNet  Google Scholar 

  33. Hennion, H., Hervé, L.: Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Mathematics, vol. 1766. Springer, Berlin (2001)

  34. Kifer, Y.: Perron–Frobenius theorem, large deviations, and random perturbations in random environments. Math. Z. 222, 677–698 (1996)

    Article  MathSciNet  Google Scholar 

  35. Kifer, Y.: Limit theorems for random transformations and processes in random environments. Trans. Am. Math. Soc. 350, 1481–1518 (1998)

    Article  MathSciNet  Google Scholar 

  36. Korepanov, A., Kosloff, Z., Melbourne, I.: Martingale-coboundary decomposition for families of dynamical systems. Ann. Inst. Henri Poincaré C Anal. Non linéaire 35(4), 859–885 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  37. Maume-Deschamps, V.: Projective metrics and mixing properties on towers. Trans. Am. Math. Soc. 353, 3371–3389 (2001)

    Article  MathSciNet  Google Scholar 

  38. Melbourne, I., Nicol, M.: Large deviations for nonuniformly hyperbolic systems. Trans. Am. Math. Soc. 360, 6661–6676 (2008)

    Article  MathSciNet  Google Scholar 

  39. Nándori, P., Szász, D., Varjú, T.: A central limit theorem for time-dependent dynamical systems. J. Stat. Phys. 146, 1213–1220 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  40. Nicol, M., Torok, A., Vaienti, S.: Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Theory Dyn. Syst. 38, 1127–1153 (2016)

    Article  MathSciNet  Google Scholar 

  41. Rugh, H.H.: Cones and gauges in complex spaces: spectral gaps and complex Perron–Frobenius theory. Ann. Math. 171, 1707–1752 (2010)

    Article  MathSciNet  Google Scholar 

  42. Su, Y.: Random Young Towers and Quenched Limit Laws, preprint. arXiv:1907.12199

  43. Young, L.S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 7, 585–650 (1998)

    Article  MathSciNet  Google Scholar 

  44. Young, L.S.: Recurrence time and rate of mixing. Isr. J. Math. 110, 153–188 (1999)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I would like to thank D. Dragičević for reading carefully a preliminary version of the paper and for some useful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, USA

    Yeor Hafouta

Authors
  1. Yeor Hafouta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yeor Hafouta.

Additional information

Communicated by Dmitry Dolgopyat.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hafouta, Y. Limit Theorems for Random Non-uniformly Expanding or Hyperbolic Maps with Exponential Tails. Ann. Henri Poincaré 23, 293–332 (2022). https://doi.org/10.1007/s00023-021-01094-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00023-021-01094-5

Search

Navigation