Abstract
In this paper, a definition of entropy for ℤ k+ (k ≥ 2)-actions due to Friedland is studied. Unlike the traditional definition, it may take a nonzero value for actions whose generators have finite (even zero) entropy as single transformations. Some basic properties are investigated and its value for the ℤ k+ -actions on circles generated by expanding endomorphisms is given. Moreover, an upper bound of this entropy for the ℤ k+ -actions on tori generated by expanding endomorphisms is obtained via the preimage entropies, which are entropy-like invariants depending on the “inverse orbits” structure of the system.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bowen, R.: Entropy for group endomorphisms and homogenuous spaces. Trans. Amer. Math. Soc., 153, 401–414 (1971)
Cheng, W.-C., Newhouse, S.: Pre-image entropy. Ergodic Theory Dynam. Systems, 25, 1091–1113 (2005)
Friedland, S.: Entropy of graphs, semi-groups and groups. In: Ergodic Theory of ℤd-actions (M. Pollicott and K. Schmidt eds.), London Math. Soc. Lecture Note Ser. 228, Cambridge Univ. Press, Cambridge, 1996, 319–343
Geller, W., Pollicott, M.: An entropy for ℤ2-actions with finite entropy generators. Fund. Math., 157, 209–220 (1998)
Huang, W., Ye, X., Zhang, G.: Local entropy theory for a countable discrete amenable group action. J. Funct. Anal., 261, 1028–1082 (2011)
Hurley, M.: On topological entropy of maps. Ergodic Theory Dynam. Systems, 15, 557–568 (1995)
Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995
Nitecki, Z., Przytycki, F.: Preimage entropy for mappings. Int. J. Bifur. and Chaos, 9, 1815–1843 (1999)
Nitecki, Z.: Topological entropy and the preimage structure of maps. Real Anal. Exchange, 29, 7–39 (2003/2004)
Ornstein, D., Weiss, B.: Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math., 48, 1–141 (1987)
Parry, W.: Intrinsic Markov chains. Trans. Amer. Math. Soc., 112, 55–66 (1964)
Ruelle, D.: Statistical mechanics on a compact set with ℤν-action satisfying expansiveness and specification. Trans. Amer. Math. Soc., 185, 237–251 (1973)
Schmidt, K.: Dynamical Systems of Algebraic Origin, Berlin Birkhauser-Verlag, New York, 1995
Walters, P.: An Introduction to Ergodic Theory, Springer, New York, 1982
Zeng, F., Yan, K., Zhang, G.: Pre-image pressure and invariant measures. Ergodic Theory Dynam. Systems, 27, 1037–1052 (2007)
Zhang, J., Zhu, Y., He, L.: Preimage entropy for nonautonomous dynamical systems. Acta Math. Sin., Chin. Ser., 48, 693–702 (2005)
Zhu, Y.: Preimage entropy for random dynamical systems. Discrete Contin. Dyn. Syst., 18, 829–851 (2007)
Zhu, Y., Li, Z., Li, X.: Preimage pressure for random transformations. Ergodic Theory Dynam. Systems, 29, 1669–1687 (2009)
Zhu, Y., Liu, Z., Xu, X., et al.: Entropy of nonautonomous dynamical systems. J. Korean Math. Soc., 49, 165–185 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by National Natural Science Foundation of China (Grant No. 11071054), the Key Project of Chinese Ministry of Education (Grant No. 211020), the Program for New Century Excellent Talents in University (Grant No. 11-0935) and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (Grant No. 11126011)
Rights and permissions
About this article
Cite this article
Zhu, Y.J., Zhang, W.D. On an entropy of ℤ k+ -actions. Acta. Math. Sin.-English Ser. 30, 467–480 (2014). https://doi.org/10.1007/s10114-014-2357-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-014-2357-7