American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies
  • DCDS Home
  • This Issue
  • Next Article
    Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation
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

[1]

Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146
[2]

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
[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]

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
[5]

Michel Coornaert, Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 779-793. doi: 10.3934/dcds.2005.13.779
[6]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[7]

Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363
[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]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[10]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[11]

Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201
[12]

Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827
[13]

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
[14]

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.

[15]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147
[16]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
[17]

Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086
[18]

Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75
[19]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[20]

Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (169)
  • HTML views (120)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences