Martingale–coboundary decomposition for families of dynamical systems

Korepanov, A.; Kosloff, Z.; Melbourne, I.

doi:10.1016/j.anihpc.2017.08.005

Korepanov, A. ; Kosloff, Z. ; Melbourne, I.

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 4, pp. 859-885.

Résumé

We prove statistical limit laws for sequences of Birkhoff sums of the type $\sum_{j = 0}^{n - 1} v_{n} \circ T_{n}^{j}$ where $T_{n}$ is a family of nonuniformly hyperbolic transformations.

The key ingredient is a new martingale–coboundary decomposition for nonuniformly hyperbolic transformations which is useful already in the case when the family $T_{n}$ is replaced by a fixed transformation T, and which is particularly effective in the case when $T_{n}$ varies with n.

In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family $T_{n}$ consists of intermittent maps, unimodal maps (along the Collet–Eckmann parameters), Viana maps, and externally forced dispersing billiards.

As an application, we prove a homogenisation result for discrete fast–slow systems where the fast dynamics is generated by a family of nonuniformly hyperbolic transformations.

MR Zbl | 1 citation dans Numdam

DOI : 10.1016/j.anihpc.2017.08.005

Mots-clés : Martingale–coboundary decomposition, Nonuniform hyperbolicity, Statistical limit laws, Homogenisation, Fast–slow systems

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     author = {Korepanov, A. and Kosloff, Z. and Melbourne, I.},
     title = {Martingale{\textendash}coboundary decomposition for families of dynamical systems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {859--885},
     publisher = {Elsevier},
     volume = {35},
     number = {4},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.08.005},
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     zbl = {1406.37027},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2017.08.005/}
}

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Korepanov, A.; Kosloff, Z.; Melbourne, I. Martingale–coboundary decomposition for families of dynamical systems. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 4, pp. 859-885. doi : 10.1016/j.anihpc.2017.08.005. https://www.numdam.org/articles/10.1016/j.anihpc.2017.08.005/

Bibliographie
Cité par

[1] Alves, J.F. SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. Éc. Norm. Supér., Volume 33 (2000), pp. 1–32 | Numdam | MR | Zbl

[2] Alves, J.F. Strong statistical stability of non-uniformly expanding maps, Nonlinearity, Volume 17 (2004), pp. 1193–1215 | DOI | MR | Zbl

[3] Alves, J.F.; Azevedo, D. Statistical properties of diffeomorphisms with weak invariant manifolds, Discrete Contin. Dyn. Syst., Volume 36 (2016), pp. 1–41 | MR

[4] Alves, J.F.; Freitas, J.M.; Luzzatto, S.; Vaienti, S. From rates of mixing to recurrence times via large deviations, Adv. Math., Volume 228 (2011), pp. 1203–1236 | DOI | MR | Zbl

[5] Alves, J.F.; Luzzatto, S.; Pinheiro, V. Markov structures and decay of correlations for non-uniformly expanding dynamical systems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 22 (2005), pp. 817–839 | DOI | Numdam | MR | Zbl

[6] Alves, J.F.; Pinheiro, V. Slow rates of mixing for dynamical systems with hyperbolic structures, J. Stat. Phys., Volume 131 (2008), pp. 505–534 | DOI | MR | Zbl

[7] Alves, J.F.; Pinheiro, V. Gibbs–Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction, Adv. Math., Volume 223 (2010), pp. 1706–1730 | DOI | MR | Zbl

[8] Alves, J.F.; Viana, M. Statistical stability for robust classes of maps with non-uniform expansion, Ergod. Theory Dyn. Syst., Volume 22 (2002), pp. 1–32 | DOI | MR | Zbl

[9] M. Antoniou, I. Melbourne, Rate of convergence in the weak invariance principle for deterministic systems, in preparation.

[10] Araujo, V.; Pacifico, M.J.; Pujals, E.R.; Viana, M. Singular-hyperbolic attractors are chaotic, Trans. Am. Math. Soc., Volume 361 (2009), pp. 2431–2485 | MR | Zbl

[11] Baladi, V.; Todd, M. Linear response for intermittent maps, Commun. Math. Phys., Volume 347 (2016), pp. 857–874 | DOI | MR | Zbl

[12] Benedicks, M.; Carleson, L. On iterations of $1 - a x^{2}$ on $(- 1, 1)$ , Ann. Math., Volume 122 (1985), pp. 1–25 | DOI | MR | Zbl

[13] Billingsley, P. Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999 | MR

[14] Bunimovič, L.A. The ergodic properties of billiards that are nearly scattering, Dokl. Akad. Nauk SSSR, Volume 211 (1973), pp. 1024–1026 | MR

[15] Burkholder, D.L. Distribution function inequalities for martingales, Ann. Probab., Volume 1 (1973), pp. 19–42 | DOI | MR | Zbl

[16] Chernov, N.; Korepanov, A. Spatial structure of Sinai–Ruelle–Bowen measures, Physica D, Volume 285 (2014), pp. 1–7 | DOI | MR | Zbl

[17] Chernov, N.; Zhang, H.-K. A family of chaotic billiards with variable mixing rates, Stoch. Dyn., Volume 5 (2005), pp. 535–553 | DOI | MR | Zbl

[18] Chernov, N.I. Sinai billiards under small external forces, Ann. Henri Poincaré, Volume 2 (2001), pp. 197–236 | DOI | MR | Zbl

[19] Chernov, N.I.; Zhang, H.-K. Billiards with polynomial mixing rates, Nonlinearity, Volume 18 (2005), pp. 1527–1553 | DOI | MR | Zbl

[20] Collet, P.; Eckmann, J.-P. Positive Liapunov exponents and absolute continuity for maps of the interval, Ergod. Theory Dyn. Syst., Volume 3 (1983), pp. 13–46 | DOI | MR | Zbl

[21] Cuny, C.; Merlevède, F. Strong invariance principles with rate for “reverse” martingales and applications, J. Theor. Probab. (2015), pp. 137–183 | MR | Zbl

[22] Dedecker, J.; Merlevède, F. Moment bounds for dependent sequences in smooth Banach spaces, Stoch. Process. Appl., Volume 125 (2015), pp. 3401–3429 | DOI | MR | Zbl

[23] Demers, M.F.; Zhang, H.-K. A functional analytic approach to perturbations of the Lorentz gas, Commun. Math. Phys., Volume 324 (2013), pp. 767–830 | DOI | MR

[24] Dolgopyat, D. Limit theorems for partially hyperbolic systems, Trans. Am. Math. Soc., Volume 356 (2004), pp. 1637–1689 | MR | Zbl

[25] Freitas, J.M.; Todd, M. The statistical stability of equilibrium states for interval maps, Nonlinearity, Volume 22 (2009), pp. 259–281 | DOI | MR | Zbl

[26] Gordin, M.I. The central limit theorem for stationary processes, Sov. Math. Dokl., Volume 10 (1969), pp. 1174–1176 | MR | Zbl

[27] Gottwald, G.A.; Melbourne, I. Homogenization for deterministic maps and multiplicative noise, Proc. R. Soc. Lond. A (2013) | MR | Zbl

[28] Gouëzel, S. Sharp polynomial estimates for the decay of correlations, Isr. J. Math., Volume 139 (2004), pp. 29–65 | DOI | MR | Zbl

[29] Gouëzel, S. Statistical properties of a skew product with a curve of neutral points, Ergod. Theory Dyn. Syst., Volume 27 (2007), pp. 123–151 | DOI | MR | Zbl

[30] Gouëzel, S.; Melbourne, I. Moment bounds and concentration inequalities for slowly mixing dynamical systems, Electron. J. Probab., Volume 19 (2014), pp. 1–30 | DOI | MR | Zbl

[31] Hall, P.; Heyde, C.C. Martingale Limit Theory and Its Application, Probability and Mathematical Statistics, Academic Press, New York, 1980 | MR | Zbl

[32] Jakobson, M.V. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Commun. Math. Phys., Volume 81 (1981), pp. 39–88 | DOI | MR | Zbl

[33] Kelly, D.; Melbourne, I. Smooth approximation of stochastic differential equations, Ann. Probab., Volume 44 (2016), pp. 479–520 | DOI | MR | Zbl

[34] Kelly, D.; Melbourne, I. Homogenization for deterministic fast–slow systems with multidimensional multiplicative noise, J. Funct. Anal., Volume 272 (2017), pp. 4063–4102 | DOI | MR | Zbl

[35] Kipnis, C.; Varadhan, S.R.S. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Commun. Math. Phys., Volume 104 (1986), pp. 1–19 | DOI | MR | Zbl

[36] Korepanov, A. Linear response for intermittent maps with summable and nonsummable decay of correlations, Nonlinearity, Volume 29 (2016), pp. 1735–1754 | DOI | MR | Zbl

[37] Korepanov, A.; Kosloff, Z.; Melbourne, I. Averaging and rates of averaging for uniform families of deterministic fast–slow skew product systems, Stud. Math., Volume 238 (2017), pp. 59–89 | DOI | MR | Zbl

[38] Korepanov, A.; Kosloff, Z.; Melbourne, I. Explicit coupling argument for nonuniformly expanding maps, Proc. R. Soc. Edinb. (2017) (in press)

[39] Liverani, C.; Ledrappier, F.; Lewowicz, J.; Newhouse, S. Central limit theorem for deterministic systems, International Conference on Dynamical Systems, Pitman Research Notes in Math., vol. 362, Longman Group Ltd, Harlow, 1996, pp. 56–75 | MR | Zbl

[40] Liverani, C.; Saussol, B.; Vaienti, S. A probabilistic approach to intermittency, Ergod. Theory Dyn. Syst., Volume 19 (1999), pp. 671–685 | DOI | MR | Zbl

[41] Maxwell, M.; Woodroofe, M. Central limit theorems for additive functionals of Markov chains, Ann. Probab., Volume 28 (2000), pp. 713–724 | DOI | MR | Zbl

[42] McLeish, D.L. Dependent central limit theorems and invariance principles, Ann. Probab., Volume 2 (1974), pp. 620–628 | DOI | MR | Zbl

[43] Melbourne, I.; Nicol, M. Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., Volume 260 (2005), pp. 131–146 | DOI | MR | Zbl

[44] Melbourne, I.; Nicol, M. Large deviations for nonuniformly hyperbolic systems, Trans. Am. Math. Soc., Volume 360 (2008), pp. 6661–6676 | DOI | MR | Zbl

[45] Melbourne, I.; Stuart, A. A note on diffusion limits of chaotic skew product flows, Nonlinearity (2011), pp. 1361–1367 | DOI | MR | Zbl

[46] Melbourne, I.; Török, A. Statistical limit theorems for suspension flows, Isr. J. Math., Volume 144 (2004), pp. 191–209 | DOI | MR | Zbl

[47] Melbourne, I.; Török, A. Convergence of moments for Axiom A and nonuniformly hyperbolic flows, Ergod. Theory Dyn. Syst., Volume 32 (2012), pp. 1091–1100 | DOI | MR | Zbl

[48] Melbourne, I.; Varandas, P. A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion, Stoch. Dyn., Volume 16 (2016) (13 pages) | DOI | MR | Zbl

[49] Melbourne, I.; Zweimüller, R. Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems, Ann. Inst. Henri Poincaré B, Probab. Stat., Volume 51 (2015), pp. 545–556 | DOI | Numdam | MR | Zbl

[50] Merlevède, F.; Peligrad, M.; Utev, S. Recent advances in invariance principles for stationary sequences, Probab. Surv., Volume 3 (2006), pp. 1–36 | DOI | MR | Zbl

[51] Philipp, W.; Stout, W.F. Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables, Memoirs of the Amer. Math. Soc., vol. 161, Amer. Math. Soc., Providence, RI, 1975 | MR | Zbl

[52] Pomeau, Y.; Manneville, P. Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys., Volume 74 (1980), pp. 189–197 | DOI | MR

[53] Rio, E. Théorie asymptotique des processus aléatoires faiblement dépendants, Mathématiques & Applications (Berlin), Mathematics & Applications, vol. 31, Springer-Verlag, Berlin, 2000 | MR | Zbl

[54] Tyran-Kamińska, M. An invariance principle for maps with polynomial decay of correlations, Commun. Math. Phys., Volume 260 (2005), pp. 1–15 | DOI | MR | Zbl

[55] Viana, M. Multidimensional nonhyperbolic attractors, Publ. Math. IHÉS (1997), pp. 63–96 | Numdam | MR | Zbl

[56] Young, L.-S. Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., Volume 147 (1998), pp. 585–650 | MR | Zbl

[57] Young, L.-S. Recurrence times and rates of mixing, Isr. J. Math., Volume 110 (1999), pp. 153–188 | MR | Zbl

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