On large deviations for amenable group actions

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

On large deviations for amenable group actions

Dongmei Zheng ^1, , Ercai Chen ^2, and Jiahong Yang ^1,

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

Abstract
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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].

Keywords: Large deviation, specification, amenable group, quasi tiling., entropy.

Mathematics Subject Classification: Primary: 37A15, 37A60; Secondary: 60F1.

Citation: Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar
[2]	M. Brin and A. Katok, On local entropy,, Lecture Notes in Mathematics, 1007 (1983), 30. doi: 10.1007/BFb0061408. Google Scholar
[3]	N. Chung, Topological pressure and the variational principle for actions of sofic groups,, Ergod. Th. Dynam. Sys., 33 (2013), 1363. doi: 10.1017/S0143385712000429. Google Scholar
[4]	N. Chung and H. Li, Homoclinic group, IE group, and expansive algebraic actions,, Invent. Math., 199 (2015), 805. doi: 10.1007/s00222-014-0524-1. Google Scholar
[5]	T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, to appear in J. Reine Angew. Math.,, , (). Google Scholar
[6]	A. Eizenberg, Y. Kifer and B. Weiss, Large deviations for $Z^d$-Actions,, Comm. Math. Physics, 164 (1994), 433. doi: 10.1007/BF02101485. Google Scholar
[7]	R. S. Ellis, Entropy, Large Deviations and Statistical Mechanics,, Springer-Verlag, (1985). doi: 10.1007/978-1-4613-8533-2. Google Scholar
[8]	W. Huang, X. Ye and G. Zhang, Local entropy theory for a countable discrete amenable group action,, J. Funct. Anal., 261 (2011), 1028. doi: 10.1016/j.jfa.2011.04.014. Google Scholar
[9]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Publ. Math. I.H.E.S., 51 (1980), 137. Google Scholar
[10]	Y. Kifer, Multidimensional random subshifts of finite type and their large deviations,, Probability Theory and Related Fields, 103 (1995), 223. doi: 10.1007/BF01204216. Google Scholar
[11]	E. Lindenstrauss, Pointwise theorems for amenable groups,, Invent. Math., 146 (2001), 259. doi: 10.1007/s002220100162. Google Scholar
[12]	B. Liang and K. Yan, Topological pressure for sub-additive potentials of amenable group actions,, J. Funct. Anal., 262 (2012), 584. doi: 10.1016/j.jfa.2011.09.020. Google Scholar
[13]	J. M. Ollagnier, Ergodic Theory and Statistical Mechanics,, Lecture Notes in Math. 1115, (1115). doi: 10.1007/BFb0101575. Google Scholar
[14]	J. M. Ollagnier and D. Pinchon, The variational principle,, Studia Math., 72 (1982), 151. Google Scholar
[15]	D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups,, J. Anal. Math., 48 (1987), 1. doi: 10.1007/BF02790325. Google Scholar
[16]	L. Rey-Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems,, Ergod. Th. Dynam. Sys., 28 (2008), 587. doi: 10.1017/S0143385707000478. Google Scholar
[17]	A. Shulman, Maximal ergodic theorems on groups,, Dep. Lit. NIINTI, (1988). Google Scholar
[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. Google Scholar
[19]	T. Ward and Q. Zhang, The Abramov-Rokhlin entropy addition formula for amenable group action,, Monatshefte für Mathematik, 114 (1992), 317. doi: 10.1007/BF01299386. Google Scholar
[20]	B. Weiss, Actions of amenable groups,, in Topics in Dynamics and Ergodic Theory, 310 (2003), 226. doi: 10.1017/CBO9780511546716.012. Google Scholar
[21]	L.-S. Young, Some large deviation results for dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525. doi: 10.2307/2001318. Google Scholar
[22]	D. Zheng and E. Chen, Bowen entropy for actions of amenable groups,, Israel J. Math., 212 (2016), 895. doi: 10.1007/s11856-016-1312-y. Google Scholar

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References:

[1]	R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar
[2]	M. Brin and A. Katok, On local entropy,, Lecture Notes in Mathematics, 1007 (1983), 30. doi: 10.1007/BFb0061408. Google Scholar
[3]	N. Chung, Topological pressure and the variational principle for actions of sofic groups,, Ergod. Th. Dynam. Sys., 33 (2013), 1363. doi: 10.1017/S0143385712000429. Google Scholar
[4]	N. Chung and H. Li, Homoclinic group, IE group, and expansive algebraic actions,, Invent. Math., 199 (2015), 805. doi: 10.1007/s00222-014-0524-1. Google Scholar
[5]	T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, to appear in J. Reine Angew. Math.,, , (). Google Scholar
[6]	A. Eizenberg, Y. Kifer and B. Weiss, Large deviations for $Z^d$-Actions,, Comm. Math. Physics, 164 (1994), 433. doi: 10.1007/BF02101485. Google Scholar
[7]	R. S. Ellis, Entropy, Large Deviations and Statistical Mechanics,, Springer-Verlag, (1985). doi: 10.1007/978-1-4613-8533-2. Google Scholar
[8]	W. Huang, X. Ye and G. Zhang, Local entropy theory for a countable discrete amenable group action,, J. Funct. Anal., 261 (2011), 1028. doi: 10.1016/j.jfa.2011.04.014. Google Scholar
[9]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Publ. Math. I.H.E.S., 51 (1980), 137. Google Scholar
[10]	Y. Kifer, Multidimensional random subshifts of finite type and their large deviations,, Probability Theory and Related Fields, 103 (1995), 223. doi: 10.1007/BF01204216. Google Scholar
[11]	E. Lindenstrauss, Pointwise theorems for amenable groups,, Invent. Math., 146 (2001), 259. doi: 10.1007/s002220100162. Google Scholar
[12]	B. Liang and K. Yan, Topological pressure for sub-additive potentials of amenable group actions,, J. Funct. Anal., 262 (2012), 584. doi: 10.1016/j.jfa.2011.09.020. Google Scholar
[13]	J. M. Ollagnier, Ergodic Theory and Statistical Mechanics,, Lecture Notes in Math. 1115, (1115). doi: 10.1007/BFb0101575. Google Scholar
[14]	J. M. Ollagnier and D. Pinchon, The variational principle,, Studia Math., 72 (1982), 151. Google Scholar
[15]	D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups,, J. Anal. Math., 48 (1987), 1. doi: 10.1007/BF02790325. Google Scholar
[16]	L. Rey-Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems,, Ergod. Th. Dynam. Sys., 28 (2008), 587. doi: 10.1017/S0143385707000478. Google Scholar
[17]	A. Shulman, Maximal ergodic theorems on groups,, Dep. Lit. NIINTI, (1988). Google Scholar
[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. Google Scholar
[19]	T. Ward and Q. Zhang, The Abramov-Rokhlin entropy addition formula for amenable group action,, Monatshefte für Mathematik, 114 (1992), 317. doi: 10.1007/BF01299386. Google Scholar
[20]	B. Weiss, Actions of amenable groups,, in Topics in Dynamics and Ergodic Theory, 310 (2003), 226. doi: 10.1017/CBO9780511546716.012. Google Scholar
[21]	L.-S. Young, Some large deviation results for dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525. doi: 10.2307/2001318. Google Scholar
[22]	D. Zheng and E. Chen, Bowen entropy for actions of amenable groups,, Israel J. Math., 212 (2016), 895. doi: 10.1007/s11856-016-1312-y. Google Scholar

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