Topological conditional entropy for amenable group actions
HTML articles powered by AMS MathViewer
- by Xiaoyao Zhou, Yaqing Zhang and Ercai Chen
- Proc. Amer. Math. Soc. 143 (2015), 141-150
- DOI: https://doi.org/10.1090/S0002-9939-2014-12175-7
- Published electronically: August 22, 2014
- PDF | Request permission
Abstract:
We introduce the topological conditional entropy for countable discrete amenable group actions and establish a variational principle for it.References
- Lewis Bowen, Sofic entropy and amenable groups, Ergodic Theory Dynam. Systems 32 (2012), no. 2, 427–466. MR 2901354, DOI 10.1017/S0143385711000253
- Rufus Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc. 184 (1973), 125–136. MR 338317, DOI 10.1090/S0002-9947-1973-0338317-X
- Mike Boyle and Tomasz Downarowicz, The entropy theory of symbolic extensions, Invent. Math. 156 (2004), no. 1, 119–161. MR 2047659, DOI 10.1007/s00222-003-0335-2
- David Burguet, A direct proof of the tail variational principle and its extension to maps, Ergodic Theory Dynam. Systems 29 (2009), no. 2, 357–369. MR 2486774, DOI 10.1017/S0143385708080425
- L. Wayne Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc. 23 (1969), 679–688. MR 247030, DOI 10.1090/S0002-9939-1969-0247030-3
- T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc. 3 (1971), 176–180. MR 289746, DOI 10.1112/blms/3.2.176
- Wen Huang, Xiangdong Ye, and Guohua Zhang, Local entropy theory for a countable discrete amenable group action, J. Funct. Anal. 261 (2011), no. 4, 1028–1082. MR 2803841, DOI 10.1016/j.jfa.2011.04.014
- F. Ledrappier, A variational principle for the topological conditional entropy, Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978) Lecture Notes in Math., vol. 729, Springer, Berlin, 1979, pp. 78–88. MR 550412
- Yuan Li, Ercai Chen, and Wen-Chiao Cheng, Tail pressure and the tail entropy function, Ergodic Theory Dynam. Systems 32 (2012), no. 4, 1400–1417. MR 2955319, DOI 10.1017/S0143385711000204
- Bingbing Liang and Kesong Yan, Topological pressure for sub-additive potentials of amenable group actions, J. Funct. Anal. 262 (2012), no. 2, 584–601. MR 2854714, DOI 10.1016/j.jfa.2011.09.020
- M. Misiurewicz, A short proof of the variational principle for a $Z^{n}_{+}$ action on a compact space, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), no. 12, 1069–1075 (English, with Russian summary). MR 430213
- MichałMisiurewicz, Topological conditional entropy, Studia Math. 55 (1976), no. 2, 175–200. MR 415587, DOI 10.4064/sm-55-2-175-200
- Jean Moulin Ollagnier, Ergodic theory and statistical mechanics, Lecture Notes in Mathematics, vol. 1115, Springer-Verlag, Berlin, 1985. MR 781932, DOI 10.1007/BFb0101575
- Donald S. Ornstein and Benjamin Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987), 1–141. MR 910005, DOI 10.1007/BF02790325
- Ya. B. Pesin and B. S. Pitskel′, Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen. 18 (1984), no. 4, 50–63, 96 (Russian, with English summary). MR 775933
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
Bibliographic Information
- Xiaoyao Zhou
- Affiliation: School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, Jiangsu, People’s Republic of China
- Email: zhouxiaoyaodeyouxian@126.com
- Yaqing Zhang
- Affiliation: School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, Jiangsu, People’s Republic of China
- Email: zhangyaqing45@126.com
- Ercai Chen
- Affiliation: School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, Jiangsu, People’s Republic of China – and – Center of Nonlinear Science, Nanjing University, Nanjing 210093, Jiangsu, People’s Republic of China
- Email: ecchen@njnu.edu.cn
- Received by editor(s): December 4, 2012
- Received by editor(s) in revised form: February 19, 2013
- Published electronically: August 22, 2014
- Communicated by: Yingfei Yi
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 141-150
- MSC (2010): Primary 37D35, 37A35
- DOI: https://doi.org/10.1090/S0002-9939-2014-12175-7
- MathSciNet review: 3272739