Large deviations for systems with non-uniform structure

Climenhaga, Vaughn; Thompson, Daniel; Yamamoto, Kenichiro

doi:10.1090/tran/6786

Large deviations for systems with non-uniform structure
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by Vaughn Climenhaga, Daniel J. Thompson and Kenichiro Yamamoto PDF

Trans. Amer. Math. Soc. 369 (2017), 4167-4192 Request permission

Abstract:

We use a weak Gibbs property and a weak form of specification to derive level-2 large deviations principles for symbolic systems equipped with a large class of reference measures. This has applications to a broad class of symbolic systems, including $\beta$-shifts, $S$-gap shifts, and their factors. A crucial step in our approach is to prove a ‘horseshoe theorem’ for these systems.

References

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
Simon Baker and Andrei E. Ghenciu, Dynamical properties of $S$-gap shifts and other shift spaces, J. Math. Anal. Appl. 430 (2015), no. 2, 633–647. MR 3351972, DOI 10.1016/j.jmaa.2015.04.092
Anne Bertrand, Specification, synchronisation, average length, Coding theory and applications (Cachan, 1986) Lecture Notes in Comput. Sci., vol. 311, Springer, Berlin, 1988, pp. 86–95 (English, with French summary). MR 960710, DOI 10.1007/3-540-19368-5_{9}
Anne Bertrand-Mathis, Développement en base $\theta$; répartition modulo un de la suite $(x\theta ^n)_{n\geq 0}$; langages codés et $\theta$-shift, Bull. Soc. Math. France 114 (1986), no. 3, 271–323 (French, with English summary). MR 878240, DOI 10.24033/bsmf.2058
Yong Moo Chung, Large deviations on Markov towers, Nonlinearity 24 (2011), no. 4, 1229–1252. MR 2776118, DOI 10.1088/0951-7715/24/4/011
Yong Moo Chung and Hiroki Takahasi, Large deviation principle for Benedicks-Carleson quadratic maps, Comm. Math. Phys. 315 (2012), no. 3, 803–826. MR 2981814, DOI 10.1007/s00220-012-1540-x
V. Climenhaga, Specification and towers in shift spaces, preprint (2015), arXiv:1502.00931v1.
Vaughn Climenhaga and Daniel J. Thompson, Intrinsic ergodicity beyond specification: $\beta$-shifts, $S$-gap shifts, and their factors, Israel J. Math. 192 (2012), no. 2, 785–817. MR 3009742, DOI 10.1007/s11856-012-0052-x
Vaughn Climenhaga and Daniel J. Thompson, Equilibrium states beyond specification and the Bowen property, J. Lond. Math. Soc. (2) 87 (2013), no. 2, 401–427. MR 3046278, DOI 10.1112/jlms/jds054
Vaughn Climenhaga and Daniel J. Thompson, Intrinsic ergodicity via obstruction entropies, Ergodic Theory Dynam. Systems 34 (2014), no. 6, 1816–1831. MR 3272773, DOI 10.1017/etds.2013.16
Vaughn Climenhaga and Daniel J. Thompson, Unique equilibrium states for flows and homeomorphisms with non-uniform structure, Adv. Math. 303 (2016), 745–799. MR 3552538, DOI 10.1016/j.aim.2016.07.029
Henri Comman and Juan Rivera-Letelier, Large deviation principles for non-uniformly hyperbolic rational maps, Ergodic Theory Dynam. Systems 31 (2011), no. 2, 321–349. MR 2776378, DOI 10.1017/S0143385709001163
Dawoud Ahmadi Dastjerdi and Somaye Jangjoo, Dynamics and topology of $S$-gap shifts, Topology Appl. 159 (2012), no. 10-11, 2654–2661. MR 2923435, DOI 10.1016/j.topol.2012.04.002
M. Denker, Large deviation and the pressure function, Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions, Prague, 1990, Academia Publ. House of the Czechoslovak Acad. of Science, 1992, pp. 21–33.
A. Eizenberg, Y. Kifer, and B. Weiss, Large deviations for $\textbf {Z}^d$-actions, Comm. Math. Phys. 164 (1994), no. 3, 433–454. MR 1291239, DOI 10.1007/BF02101485
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
Richard S. Ellis, The theory of large deviations and applications to statistical mechanics, Long-range interacting systems, Oxford Univ. Press, Oxford, 2010, pp. 13, 228–277. MR 2759544
Hans Föllmer and Steven Orey, Large deviations for the empirical field of a Gibbs measure, Ann. Probab. 16 (1988), no. 3, 961–977. MR 942749
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173. MR 573822, DOI 10.1007/BF02684777
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
Dominik Kwietniak, Piotr Oprocha, and MichałRams, On entropy of dynamical systems with almost specification, Israel J. Math. 213 (2016), no. 1, 475–503. MR 3509480, DOI 10.1007/s11856-016-1339-0
Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. MR 1369092, DOI 10.1017/CBO9780511626302
Artur O. Lopes, Entropy and large deviation, Nonlinearity 3 (1990), no. 2, 527–546. MR 1054587, DOI 10.1088/0951-7715/3/2/013
Ian Melbourne, Large and moderate deviations for slowly mixing dynamical systems, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1735–1741. MR 2470832, DOI 10.1090/S0002-9939-08-09751-7
Ian Melbourne and Matthew Nicol, Large deviations for nonuniformly hyperbolic systems, Trans. Amer. Math. Soc. 360 (2008), no. 12, 6661–6676. MR 2434305, DOI 10.1090/S0002-9947-08-04520-0
Steven Orey and Stephan Pelikan, Large deviation principles for stationary processes, Ann. Probab. 16 (1988), no. 4, 1481–1495. MR 958198
Steven Orey and Stephan Pelikan, Deviations of trajectory averages and the defect in Pesin’s formula for Anosov diffeomorphisms, Trans. Amer. Math. Soc. 315 (1989), no. 2, 741–753. MR 935534, DOI 10.1090/S0002-9947-1989-0935534-4
C.-E. Pfister and W. G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Applications to the $\beta$-shifts, Nonlinearity 18 (2005), no. 1, 237–261. MR 2109476, DOI 10.1088/0951-7715/18/1/013
C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems 27 (2007), no. 3, 929–956. MR 2322186, DOI 10.1017/S0143385706000824
Mark Pollicott and Richard Sharp, Large deviations for intermittent maps, Nonlinearity 22 (2009), no. 9, 2079–2092. MR 2534293, DOI 10.1088/0951-7715/22/9/001
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
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
Y\B{o}ichir\B{o} Takahashi, Entropy functional (free energy) for dynamical systems and their random perturbations, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 437–467. MR 780769, DOI 10.1016/S0924-6509(08)70404-5
Y\B{o}ichir\B{o} Takahashi, Two aspects of large deviation theory for large time, Probabilistic methods in mathematical physics (Katata/Kyoto, 1985) Academic Press, Boston, MA, 1987, pp. 363–384. MR 933831
Paulo Varandas, Non-uniform specification and large deviations for weak Gibbs measures, J. Stat. Phys. 146 (2012), no. 2, 330–358. MR 2873016, DOI 10.1007/s10955-011-0392-7
Peter Walters, Equilibrium states for $\beta$-transformations and related transformations, Math. Z. 159 (1978), no. 1, 65–88. MR 466492, DOI 10.1007/BF01174569
Kenichiro Yamamoto, On the weaker forms of the specification property and their applications, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3807–3814. MR 2529890, DOI 10.1090/S0002-9939-09-09937-7
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