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September  2018, 38(9): 4467-4482. doi: 10.3934/dcds.2018195

The Katok's entropy formula for amenable group actions

Xiaojun Huang 1,, , Jinsong Liu 2,3, and  Changrong Zhu 1,

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2. 

HLM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
3. 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: hxj@cqu.edu.cn

Received  September 2017 Revised  April 2018 Published  June 2018

Fund Project: The first and second authors are supported by NSF of China No.11471318 and No.11671057; The second author is also supported by NSF of China No.11688101; the third author is supported by NSF of China No.11671058

In this paper we generalize Katok's entropy formula to a large class of infinite countably amenable group actions.

Keywords: Amenable group, topological entropy, Shannon-McMillan-Breiman theorem.
Mathematics Subject Classification: Primary: 37A35; Secondary: 37B40.
Citation: 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
References:
[1]

R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.  Google Scholar

[2]

L. Bowen and A. Nevo, Pointwise ergodic theorems beyond amenable groups, Ergodic Theory Dynam. Systems, 33 (2013), 777-820.  doi: 10.1017/S0143385712000041.  Google Scholar

[3]

R. Bowen, Entropy for group automorphisms and homogenous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[4]

M. Choda, Entropy of automorphisms arising from dynamical systems through discrete groups with amenable actions, J. Funct. Anal., 217 (2004), 181-191.  doi: 10.1016/j.jfa.2004.03.016.  Google Scholar

[5]

C. Deninger, Fuglede-Kadison determinants and entropy for actions of discrete amenable groups, J. Amer. Math. Soc., 19 (2006), 737-758.  doi: 10.1090/S0894-0347-06-00519-4.  Google Scholar

[6]

E. Dinaburg, On the relations among various entropy characteristics of dynamical system, Math. of the USSR-Izvestija, 5 (1971), 337-378.  doi: 10.1070/IM1971v005n02ABEH001050.  Google Scholar

[7]

A. Dooley and V. Golodets, The spectrum of completely positive entropy actions of countable amenable groups, J. Funct. Anal., 196 (2002), 1-18.  doi: 10.1006/jfan.2002.3966.  Google Scholar

[8]

M. Hochman, Return times, recurrence densities and entropy for actions of some discrete amenable groups, J. Anal. Math., 100 (2006), 1-51.  doi: 10.1007/BF02916754.  Google Scholar

[9]

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.  Google Scholar

[10]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.   Google Scholar

[11]

D. Kerr and H. Li, Ergodic Theory: Independence and Dichotomies, Springer, 2016. doi: 10.1007/978-3-319-49847-8.  Google Scholar

[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.  Google Scholar

[13]

E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.  doi: 10.1007/s002220100162.  Google Scholar

[14]

D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.  Google Scholar

[15]

F. Pogorzelski, Almost-additive ergodic theorems for amenable groups, J. Funct. Anal., 265 (2013), 1615-1666.  doi: 10.1016/j.jfa.2013.06.009.  Google Scholar

[16]

D. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Ann. of Math., 151 (2000), 1119-1150.  doi: 10.2307/121130.  Google Scholar

show all references

References:
[1]

R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.  Google Scholar

[2]

L. Bowen and A. Nevo, Pointwise ergodic theorems beyond amenable groups, Ergodic Theory Dynam. Systems, 33 (2013), 777-820.  doi: 10.1017/S0143385712000041.  Google Scholar

[3]

R. Bowen, Entropy for group automorphisms and homogenous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[4]

M. Choda, Entropy of automorphisms arising from dynamical systems through discrete groups with amenable actions, J. Funct. Anal., 217 (2004), 181-191.  doi: 10.1016/j.jfa.2004.03.016.  Google Scholar

[5]

C. Deninger, Fuglede-Kadison determinants and entropy for actions of discrete amenable groups, J. Amer. Math. Soc., 19 (2006), 737-758.  doi: 10.1090/S0894-0347-06-00519-4.  Google Scholar

[6]

E. Dinaburg, On the relations among various entropy characteristics of dynamical system, Math. of the USSR-Izvestija, 5 (1971), 337-378.  doi: 10.1070/IM1971v005n02ABEH001050.  Google Scholar

[7]

A. Dooley and V. Golodets, The spectrum of completely positive entropy actions of countable amenable groups, J. Funct. Anal., 196 (2002), 1-18.  doi: 10.1006/jfan.2002.3966.  Google Scholar

[8]

M. Hochman, Return times, recurrence densities and entropy for actions of some discrete amenable groups, J. Anal. Math., 100 (2006), 1-51.  doi: 10.1007/BF02916754.  Google Scholar

[9]

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.  Google Scholar

[10]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.   Google Scholar

[11]

D. Kerr and H. Li, Ergodic Theory: Independence and Dichotomies, Springer, 2016. doi: 10.1007/978-3-319-49847-8.  Google Scholar

[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.  Google Scholar

[13]

E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.  doi: 10.1007/s002220100162.  Google Scholar

[14]

D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.  Google Scholar

[15]

F. Pogorzelski, Almost-additive ergodic theorems for amenable groups, J. Funct. Anal., 265 (2013), 1615-1666.  doi: 10.1016/j.jfa.2013.06.009.  Google Scholar

[16]

D. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Ann. of Math., 151 (2000), 1119-1150.  doi: 10.2307/121130.  Google Scholar

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