On large deviations for amenable group actions

Previous Article
Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials
DCDS Home
This Issue
Next Article
A powered Gronwall-type inequality and applications to stochastic differential equations

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
Related Papers

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.
[2]	M. Brin and A. Katok, On local entropy,, Lecture Notes in Mathematics, 1007 (1983), 30. 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. doi: 10.1017/S0143385712000429.
[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.
[5]	T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, to appear in J. Reine Angew. Math.,, , ().
[6]	A. Eizenberg, Y. Kifer and B. Weiss, Large deviations for $Z^d$-Actions,, Comm. Math. Physics, 164 (1994), 433. doi: 10.1007/BF02101485.
[7]	R. S. Ellis, Entropy, Large Deviations and Statistical Mechanics,, Springer-Verlag, (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. 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.
[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.
[11]	E. Lindenstrauss, Pointwise theorems for amenable groups,, Invent. Math., 146 (2001), 259. 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. doi: 10.1016/j.jfa.2011.09.020.
[13]	J. M. Ollagnier, Ergodic Theory and Statistical Mechanics,, Lecture Notes in Math. 1115, (1115). doi: 10.1007/BFb0101575.
[14]	J. M. Ollagnier and D. Pinchon, The variational principle,, Studia Math., 72 (1982), 151.
[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.
[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.
[17]	A. Shulman, Maximal ergodic theorems on groups,, Dep. Lit. NIINTI, (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.
[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.
[20]	B. Weiss, Actions of amenable groups,, in Topics in Dynamics and Ergodic Theory, 310 (2003), 226. doi: 10.1017/CBO9780511546716.012.
[21]	L.-S. Young, Some large deviation results for dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525. doi: 10.2307/2001318.
[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.

show all references

References:

[1]	R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X.
[2]	M. Brin and A. Katok, On local entropy,, Lecture Notes in Mathematics, 1007 (1983), 30. 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. doi: 10.1017/S0143385712000429.
[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.
[5]	T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, to appear in J. Reine Angew. Math.,, , ().
[6]	A. Eizenberg, Y. Kifer and B. Weiss, Large deviations for $Z^d$-Actions,, Comm. Math. Physics, 164 (1994), 433. doi: 10.1007/BF02101485.
[7]	R. S. Ellis, Entropy, Large Deviations and Statistical Mechanics,, Springer-Verlag, (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. 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.
[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.
[11]	E. Lindenstrauss, Pointwise theorems for amenable groups,, Invent. Math., 146 (2001), 259. 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. doi: 10.1016/j.jfa.2011.09.020.
[13]	J. M. Ollagnier, Ergodic Theory and Statistical Mechanics,, Lecture Notes in Math. 1115, (1115). doi: 10.1007/BFb0101575.
[14]	J. M. Ollagnier and D. Pinchon, The variational principle,, Studia Math., 72 (1982), 151.
[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.
[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.
[17]	A. Shulman, Maximal ergodic theorems on groups,, Dep. Lit. NIINTI, (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.
[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.
[20]	B. Weiss, Actions of amenable groups,, in Topics in Dynamics and Ergodic Theory, 310 (2003), 226. doi: 10.1017/CBO9780511546716.012.
[21]	L.-S. Young, Some large deviation results for dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525. doi: 10.2307/2001318.
[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.

[1]	Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195
[2]	Xiankun Ren. Periodic measures are dense in invariant measures for residually finite amenable group actions with specification. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1657-1667. doi: 10.3934/dcds.2018068
[3]	Jean-Paul Thouvenot. The work of Lewis Bowen on the entropy theory of non-amenable group actions. Journal of Modern Dynamics, 2019, 15: 133-141. doi: 10.3934/jmd.2019016
[4]	Masayuki Asaoka, Kenichiro Yamamoto. On the large deviation rates of non-entropy-approachable measures. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4401-4410. doi: 10.3934/dcds.2013.33.4401
[5]	Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469
[6]	Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221
[7]	Xiaojun Huang, Yuan Lian, Changrong Zhu. A Billingsley-type theorem for the pressure of an action of an amenable group. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 959-993. doi: 10.3934/dcds.2019040
[8]	Marcelo Sobottka. Topological quasi-group shifts. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 77-93. doi: 10.3934/dcds.2007.17.77
[9]	Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048
[10]	Yves Guivarc'h. On the spectrum of a large subgroup of a semisimple group. Journal of Modern Dynamics, 2008, 2 (1) : 15-42. doi: 10.3934/jmd.2008.2.15
[11]	Yoshikazu Katayama, Colin E. Sutherland and Masamichi Takesaki. The intrinsic invariant of an approximately finite dimensional factor and the cocycle conjugacy of discrete amenable group actions. Electronic Research Announcements, 1995, 1: 43-47.
[12]	Peng Zhang. Multiperiod mean semi-absolute deviation interval portfolio selection with entropy constraints. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1169-1187. doi: 10.3934/jimo.2016067
[13]	Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651
[14]	Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673
[15]	Jean-François Biasse. Subexponential time relations in the class group of large degree number fields. Advances in Mathematics of Communications, 2014, 8 (4) : 407-425. doi: 10.3934/amc.2014.8.407
[16]	Fritz Colonius. Invariance entropy, quasi-stationary measures and control sets. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2093-2123. doi: 10.3934/dcds.2018086
[17]	David Burguet. Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 873-899. doi: 10.3934/dcds.2010.26.873
[18]	Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701
[19]	Elon Lindenstrauss. Pointwise theorems for amenable groups. Electronic Research Announcements, 1999, 5: 82-90.
[20]	Welington Cordeiro, Manfred Denker, Xuan Zhang. On specification and measure expansiveness. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1941-1957. doi: 10.3934/dcds.2017082

2017 Impact Factor: 1.179

American Institute of Mathematical Sciences