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Almost Sure Rates of Mixing for Random Intermittent Maps

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Differential Equations and Dynamical Systems (USUZCAMP 2017)

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Abstract

We consider a family \(\mathcal F\) of maps with two branches and a common neutral fixed point 0 such that the order of tangency at 0 belongs to some interval \([\alpha _0, \alpha _1]\subset (0, 1)\). Maps in \(\mathcal F\) do not necessarily share common Markov partition. At each step a member of \(\mathcal F\) is chosen independently with respect to the uniform distribution on \([\alpha _0, \alpha _1]\). We show that the construction of the random tower in Bahsoun et al. (Quenched Decay of Correlations for Slowly Mixing Systems, 2018, [5]) with general return time can be carried out for random compositions of such maps. Thus their general results are applicable and gives upper bounds for the quenched decay of correlations of form the \(n^{1-1/\alpha _0+\delta }\) for the any \(\delta >0\).

Dedicated to Abdulla Azamov and Leonid Bunimovich on the occasion of their 70th birthday.

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Notes

  1. 1.

    A subclass of the so called Pomeau–Manneville maps introduced in [18], and popularised by Liverani, Saussol and Vaienti in [16]. Such systems have attracted the attention of both mathematicians and physicists (see [15] for a recent work in this area).

  2. 2.

    Such extensions can be constructed easily. For example, for \(f\in \mathcal {F}\) it is sufficient to take \(\tilde{f}(x)=a(x-x_\alpha )^4+b(x-x_\alpha )^3+c(x-x_\alpha )^2+d(x-x_\alpha )+1\) with \(a<bc/d\), where a, b, c are the Taylor coefficients of f at \(x=x_\alpha \).

References

  1. Aimino, R., Hu, H., Nicol, M., Török, A., Vaienti, S.: Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete Contin. Dyn. Syst. 35(3), 793–806 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Ayyer, A., Liverani, C., Stenlund, M.: Quenched CLT for random toral automorphisms. Discrete Contin. Dyn. Syst. 24(2), 331–348 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Bahsoun, W., Bose, C., Ruziboev, M.: Quenched Decay of Correlations for Slowly Mixing Systems. arXiv:1706.04158. Cited 30 Jan 2018

  6. Baladi, V., Benedicks, M., Maume-Deschamps, V.: Almost sure rates of mixing for i.i.d. unimodal maps. (English, French summary) Ann. Sci. École Norm. Sup. (4) 35(1), 77–126 (2002)

    Article  MathSciNet  Google Scholar 

  7. Buzzi, J.: Exponential decay of correlations for random Lasota-Yorke maps. Comm. Math. Phys. 208(1), 25–54 (1999)

    Article  MathSciNet  Google Scholar 

  8. de Melo, W., van Strien, S.: One-Dimensional Dynamics. Springer, Berlin (1993)

    Book  Google Scholar 

  9. Gouëzel, S.: Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139, 29–65 (2004)

    Article  MathSciNet  Google Scholar 

  10. Haydn, N., Nicol, M., Török, A., Vaienti, S.: Almost sure invariance principle for sequential and non-stationary dynamical systems. Trans. Amer. Math. Soc. 369(8), 5293–5316 (2017)

    Article  MathSciNet  Google Scholar 

  11. Haydn, N., Rousseau, J., Yang, F.: Exponential Law for Random Maps on Compact Manifolds. arXiv:1705.05869. Cited 16 May 2017

  12. Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Amer. Stat. Soc. 58(30), 13–30 (1963)

    Article  MathSciNet  Google Scholar 

  13. Keller, G.: Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahr. Verw. Geb. 69, 461–478 (1985)

    Article  MathSciNet  Google Scholar 

  14. Kifer, Y.: Limit theorems for random transformations and processes in random environments. Trans. Amer. Math. Soc. 350(4), 1481–1518 (1998)

    Article  MathSciNet  Google Scholar 

  15. Leppänen, J., Stenlund, M.: Quasistatic dynamics with intermittency. Math. Phys. Anal. Geom. 19(2), Art. 8, 23 pp (2016)

    Google Scholar 

  16. Liverani, C., Saussol, B., Vaienti, S.: A probabilistic approach to intermittency. Ergod. Theory Dyn. Syst. 19, 671–685 (1999)

    Article  MathSciNet  Google Scholar 

  17. Nicol, M., Török, A., Vaienti, S.: Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Theory Dyn. Syst. https://doi.org/10.1017/etds.2016.69

    Article  MathSciNet  Google Scholar 

  18. Pomeau, Y., Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74, 189–197 (1980)

    Article  MathSciNet  Google Scholar 

  19. Saussol, B.: Absolutely continuous invariant measures for multidimensional expanding maps. Israel J. Math. 116, 223–248 (2000)

    Article  MathSciNet  Google Scholar 

  20. Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research was supported by The Leverhulme Trust through the research grant RPG-2015-346. The author would like to thank Wael Bahsoun for useful discussions during the preparation of the paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK

    Marks Ruziboev

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  1. Marks Ruziboev
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Correspondence to Marks Ruziboev .

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Editors and Affiliations

  1. Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan

    Abdulla Azamov

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA

    Leonid Bunimovich

  3. Turin Polytechnic University, Tashkent, Uzbekistan

    Akhtam Dzhalilov

  4. Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA, USA

    Hong-Kun Zhang

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Ruziboev, M. (2018). Almost Sure Rates of Mixing for Random Intermittent Maps. In: Azamov, A., Bunimovich, L., Dzhalilov, A., Zhang, HK. (eds) Differential Equations and Dynamical Systems. USUZCAMP 2017. Springer Proceedings in Mathematics & Statistics, vol 268. Springer, Cham. https://doi.org/10.1007/978-3-030-01476-6_11

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