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The Katok's entropy formula for amenable group actions
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 |
In this paper we generalize Katok's entropy formula to a large class of infinite countably amenable group actions.
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. |
[2] |
L. Bowen and A. Nevo,
Pointwise ergodic theorems beyond amenable groups, Ergodic Theory Dynam. Systems, 33 (2013), 777-820.
doi: 10.1017/S0143385712000041. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
A. Katok,
Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.
|
[11] |
D. Kerr and H. Li,
Ergodic Theory: Independence and Dichotomies, Springer, 2016.
doi: 10.1007/978-3-319-49847-8. |
[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] |
E. Lindenstrauss,
Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.
doi: 10.1007/s002220100162. |
[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. |
[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. |
[16] |
D. Rudolph and B. Weiss,
Entropy and mixing for amenable group actions, Ann. of Math., 151 (2000), 1119-1150.
doi: 10.2307/121130. |
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. |
[2] |
L. Bowen and A. Nevo,
Pointwise ergodic theorems beyond amenable groups, Ergodic Theory Dynam. Systems, 33 (2013), 777-820.
doi: 10.1017/S0143385712000041. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
A. Katok,
Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.
|
[11] |
D. Kerr and H. Li,
Ergodic Theory: Independence and Dichotomies, Springer, 2016.
doi: 10.1007/978-3-319-49847-8. |
[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] |
E. Lindenstrauss,
Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.
doi: 10.1007/s002220100162. |
[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. |
[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. |
[16] |
D. Rudolph and B. Weiss,
Entropy and mixing for amenable group actions, Ann. of Math., 151 (2000), 1119-1150.
doi: 10.2307/121130. |
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