Large and moderate deviations for slowly mixing dynamical systems

Melbourne, Ian

doi:10.1090/S0002-9939-08-09751-7

Large and moderate deviations for slowly mixing dynamical systems
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by Ian Melbourne PDF

Proc. Amer. Math. Soc. 137 (2009), 1735-1741 Request permission

Abstract:

We obtain results on large deviations for a large class of nonuniformly hyperbolic dynamical systems with polynomial decay of correlations $1/n^\beta$, $\beta >0$. This includes systems modelled by Young towers with polynomial tails, extending recent work of M. Nicol and the author which assumed $\beta >1$. As a byproduct of the proof, we obtain slightly stronger results even when $\beta >1$. The results are sharp in the sense that there exist examples (such as Pomeau-Manneville intermittency maps) for which the obtained rates are best possible. In addition, we obtain results on moderate deviations.

References

Vítor Araújo, Large deviations bound for semiflows over a non-uniformly expanding base, Bull. Braz. Math. Soc. (N.S.) 38 (2007), no. 3, 335–376. MR 2344203, DOI 10.1007/s00574-007-0049-y
V. Araújo and M. J. Pacifico, Large deviations for non-uniformly expanding maps, J. Stat. Phys. 125 (2006), no. 2, 415–457. MR 2270016, DOI 10.1007/s10955-006-9183-y
N. Chernov and R. Markarian, Dispersing billiards with cusps: slow decay of correlations, Comm. Math. Phys. 270 (2007), no. 3, 727–758. MR 2276463, DOI 10.1007/s00220-006-0169-z
N. Chernov and H.-K. Zhang, Billiards with polynomial mixing rates, Nonlinearity 18 (2005), no. 4, 1527–1553. MR 2150341, DOI 10.1088/0951-7715/18/4/006
N. Chernov and H.-K. Zhang, Improved estimates for correlations in billiards, Comm. Math. Phys. 277 (2008), no. 2, 305–321. MR 2358286, DOI 10.1007/s00220-007-0360-x
Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, 2nd ed., Applications of Mathematics (New York), vol. 38, Springer-Verlag, New York, 1998. MR 1619036, DOI 10.1007/978-1-4612-5320-4
Richard S. Ellis, Entropy, large deviations, and statistical mechanics, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 271, Springer-Verlag, New York, 1985. MR 793553, DOI 10.1007/978-1-4613-8533-2
Hubert Hennion and Loïc Hervé, Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness, Lecture Notes in Mathematics, vol. 1766, Springer-Verlag, Berlin, 2001. MR 1862393, DOI 10.1007/b87874
Huyi Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 495–524. MR 2054191, DOI 10.1017/S0143385703000671
Gerhard Keller and Tomasz Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Comm. Math. Phys. 149 (1992), no. 1, 31–69. MR 1182410, DOI 10.1007/BF02096623
Yuri Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc. 321 (1990), no. 2, 505–524. MR 1025756, DOI 10.1090/S0002-9947-1990-1025756-7
Carlangelo Liverani, Benoît Saussol, and Sandro Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems 19 (1999), no. 3, 671–685. MR 1695915, DOI 10.1017/S0143385799133856
Artur O. Lopes, Entropy and large deviation, Nonlinearity 3 (1990), no. 2, 527–546. MR 1054587, DOI 10.1088/0951-7715/3/2/013
Roberto Markarian, Billiards with polynomial decay of correlations, Ergodic Theory Dynam. Systems 24 (2004), no. 1, 177–197. MR 2041267, DOI 10.1017/S0143385703000270
I. Melbourne and M. Nicol. Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360 (2008) 6661–6676.
Florence Merlevède, Magda Peligrad, and Sergey Utev, Recent advances in invariance principles for stationary sequences, Probab. Surv. 3 (2006), 1–36. MR 2206313, DOI 10.1214/154957806100000202
Steven Orey and Stephan Pelikan, Large deviation principles for stationary processes, Ann. Probab. 16 (1988), no. 4, 1481–1495. MR 958198
M. Pollicott and R. Sharp. Large deviations for intermittent maps. Preprint, 2008.
Mark Pollicott, Richard Sharp, and Michiko Yuri, Large deviations for maps with indifferent fixed points, Nonlinearity 11 (1998), no. 4, 1173–1184. MR 1632614, DOI 10.1088/0951-7715/11/4/023
Yves Pomeau and Paul Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys. 74 (1980), no. 2, 189–197. MR 576270, DOI 10.1007/BF01197757
Luc Rey-Bellet and Lai-Sang Young, Large deviations in non-uniformly hyperbolic dynamical systems, Ergodic Theory Dynam. Systems 28 (2008), no. 2, 587–612. MR 2408394, DOI 10.1017/S0143385707000478
Emmanuel Rio, Théorie asymptotique des processus aléatoires faiblement dépendants, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 31, Springer-Verlag, Berlin, 2000 (French). MR 2117923
Simon Waddington, Large deviation asymptotics for Anosov flows, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 4, 445–484. MR 1404318, DOI 10.1016/S0294-1449(16)30110-X
Lai-Sang Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc. 318 (1990), no. 2, 525–543. MR 975689, DOI 10.1090/S0002-9947-1990-0975689-7
Lai-Sang Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), no. 3, 585–650. MR 1637655, DOI 10.2307/120960
Lai-Sang Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153–188. MR 1750438, DOI 10.1007/BF02808180