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December  2016, 36(12): 7191-7206. doi: 10.3934/dcds.2016113

## On large deviations for amenable group actions

 1 School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210023, China, China 2 School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, Jiangsu

Received  August 2015 Revised  July 2016 Published  October 2016

By proving an amenable version of Katok's entropy formula and handling the quasi tiling techniques, we establish large deviations bounds for countable discrete amenable group actions. This generalizes the classical results of Lai-Sang Young [21].
Citation: Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
##### References:
 [1] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. [2] M. Brin and A. Katok, On local entropy, Lecture Notes in Mathematics, Springer, Berlin, 1007 (1983), 30-38. doi: 10.1007/BFb0061408. [3] N. Chung, Topological pressure and the variational principle for actions of sofic groups, Ergod. Th. Dynam. Sys., 33 (2013), 1363-1390. doi: 10.1017/S0143385712000429. [4] N. Chung and H. Li, Homoclinic group, IE group, and expansive algebraic actions, Invent. Math., 199 (2015), 805-858. doi: 10.1007/s00222-014-0524-1. [5] T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, to appear in J. Reine Angew. Math., arXiv:1502.02413v1. [6] A. Eizenberg, Y. Kifer and B. Weiss, Large deviations for $Z^d$-Actions, Comm. Math. Physics, 164 (1994), 433-454. doi: 10.1007/BF02101485. [7] R. S. Ellis, Entropy, Large Deviations and Statistical Mechanics, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4613-8533-2. [8] W. Huang, X. Ye and G. Zhang, Local entropy theory for a countable discrete amenable group action, J. Funct. Anal., 261 (2011), 1028-1082. doi: 10.1016/j.jfa.2011.04.014. [9] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. I.H.E.S., 51 (1980), 137-173. [10] Y. Kifer, Multidimensional random subshifts of finite type and their large deviations, Probability Theory and Related Fields, 103 (1995), 223-248. doi: 10.1007/BF01204216. [11] E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295. doi: 10.1007/s002220100162. [12] B. Liang and K. Yan, Topological pressure for sub-additive potentials of amenable group actions, J. Funct. Anal., 262 (2012), 584-601. doi: 10.1016/j.jfa.2011.09.020. [13] J. M. Ollagnier, Ergodic Theory and Statistical Mechanics, Lecture Notes in Math. 1115, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0101575. [14] J. M. Ollagnier and D. Pinchon, The variational principle, Studia Math., 72 (1982), 151-159. [15] D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math., 48 (1987), 1-141. doi: 10.1007/BF02790325. [16] L. Rey-Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems, Ergod. Th. Dynam. Sys., 28 (2008), 587-612. doi: 10.1017/S0143385707000478. [17] A. Shulman, Maximal ergodic theorems on groups, Dep. Lit. NIINTI, No.2184, 1988. [18] A. M. Stepin and A. T. Tagi-Zade, Variational characterization of topological pressure of the amenable groups of transformations, Dokl. Akad. Nauk SSSR, 254 (1980), 545-549 (in Russian). [19] T. Ward and Q. Zhang, The Abramov-Rokhlin entropy addition formula for amenable group action, Monatshefte für Mathematik, 114 (1992), 317-329. doi: 10.1007/BF01299386. [20] B. Weiss, Actions of amenable groups, in Topics in Dynamics and Ergodic Theory, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 310 (2003), 226-262. doi: 10.1017/CBO9780511546716.012. [21] L.-S. Young, Some large deviation results for dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543. doi: 10.2307/2001318. [22] D. Zheng and E. Chen, Bowen entropy for actions of amenable groups, Israel J. Math., 212 (2016), 895-911. doi: 10.1007/s11856-016-1312-y.

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##### References:
 [1] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. [2] M. Brin and A. Katok, On local entropy, Lecture Notes in Mathematics, Springer, Berlin, 1007 (1983), 30-38. doi: 10.1007/BFb0061408. [3] N. Chung, Topological pressure and the variational principle for actions of sofic groups, Ergod. Th. Dynam. Sys., 33 (2013), 1363-1390. doi: 10.1017/S0143385712000429. [4] N. Chung and H. Li, Homoclinic group, IE group, and expansive algebraic actions, Invent. Math., 199 (2015), 805-858. doi: 10.1007/s00222-014-0524-1. [5] T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, to appear in J. Reine Angew. Math., arXiv:1502.02413v1. [6] A. Eizenberg, Y. Kifer and B. Weiss, Large deviations for $Z^d$-Actions, Comm. Math. Physics, 164 (1994), 433-454. doi: 10.1007/BF02101485. [7] R. S. Ellis, Entropy, Large Deviations and Statistical Mechanics, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4613-8533-2. [8] W. Huang, X. Ye and G. Zhang, Local entropy theory for a countable discrete amenable group action, J. Funct. Anal., 261 (2011), 1028-1082. doi: 10.1016/j.jfa.2011.04.014. [9] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. I.H.E.S., 51 (1980), 137-173. [10] Y. Kifer, Multidimensional random subshifts of finite type and their large deviations, Probability Theory and Related Fields, 103 (1995), 223-248. doi: 10.1007/BF01204216. [11] E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295. doi: 10.1007/s002220100162. [12] B. Liang and K. Yan, Topological pressure for sub-additive potentials of amenable group actions, J. Funct. Anal., 262 (2012), 584-601. doi: 10.1016/j.jfa.2011.09.020. [13] J. M. Ollagnier, Ergodic Theory and Statistical Mechanics, Lecture Notes in Math. 1115, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0101575. [14] J. M. Ollagnier and D. Pinchon, The variational principle, Studia Math., 72 (1982), 151-159. [15] D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math., 48 (1987), 1-141. doi: 10.1007/BF02790325. [16] L. Rey-Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems, Ergod. Th. Dynam. Sys., 28 (2008), 587-612. doi: 10.1017/S0143385707000478. [17] A. Shulman, Maximal ergodic theorems on groups, Dep. Lit. NIINTI, No.2184, 1988. [18] A. M. Stepin and A. T. Tagi-Zade, Variational characterization of topological pressure of the amenable groups of transformations, Dokl. Akad. Nauk SSSR, 254 (1980), 545-549 (in Russian). [19] T. Ward and Q. Zhang, The Abramov-Rokhlin entropy addition formula for amenable group action, Monatshefte für Mathematik, 114 (1992), 317-329. doi: 10.1007/BF01299386. [20] B. Weiss, Actions of amenable groups, in Topics in Dynamics and Ergodic Theory, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 310 (2003), 226-262. doi: 10.1017/CBO9780511546716.012. [21] L.-S. Young, Some large deviation results for dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543. doi: 10.2307/2001318. [22] D. Zheng and E. Chen, Bowen entropy for actions of amenable groups, Israel J. Math., 212 (2016), 895-911. doi: 10.1007/s11856-016-1312-y.
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