## Large and moderate deviations for slowly mixing dynamical systems

HTML articles powered by AMS MathViewer

- 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

## Additional Information

**Ian Melbourne**- Affiliation: Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom
- MR Author ID: 123300
- Email: ism@math.uh.edu
- Received by editor(s): June 9, 2008
- Published electronically: November 26, 2008
- Additional Notes: This research was supported in part by EPSRC Grant EP/D055520/1.
- Communicated by: Bryna Kra
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**137**(2009), 1735-1741 - MSC (2000): Primary 37D25, 37A50, 60F10
- DOI: https://doi.org/10.1090/S0002-9939-08-09751-7
- MathSciNet review: 2470832