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A variational principle of scaled entropy for amenable group actions

Part of: Topological dynamics Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  09 January 2025

Yu Liu  and
Zhiming Li
Yu Liu
Affiliation:
School of Mathematics, Northwest University, Xi’an, P.R. China e-mail: liuyu_yzr@126.com
Zhiming Li*
Affiliation:
School of Mathematics, Northwest University, Xi’an, P.R. China e-mail: liuyu_yzr@126.com
*
e-mail: china-lizhiming@163.com
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Abstract

We study scaled topological entropy, scaled measure entropy, and scaled local entropy in the context of amenable group actions. In particular, a variational principle is established.

Keywords

Scaled topological entropyscaled measure entropyscaled local entropyvariational principle

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 37B40: Topological entropy

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 78 , Issue 2 , April 2026 , pp. 544 - 571
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Adler, R. L., Konheim, A. G., and McAndrew, M. H., Topological entropy . Trans. Amer. Math. Soc. 114(1965), 309–319.CrossRefGoogle Scholar
Brin, M. and Katok, A., On local entropy. Lecture Notes in Math., 1007, Springer-Verlag, Berlin, 1983, pp. 30–38.CrossRefGoogle Scholar
Bowen, L., Zero entropy is generic . Entropy 18(2016), no. 6, Paper No. 220, 20.CrossRefGoogle Scholar
Bowen, R., Topological entropy for non-compact sets . Trans. Amer. Math. Soc. 184(1973), 125–136.CrossRefGoogle Scholar
Choda, M., Entropy of automorphisms arising from dynamical systems through discrete groups with amenable actions . J. Funct. Anal. 217(2004), 181–191.CrossRefGoogle Scholar
Deninger, C., Fuglede-Kadison determinants and entropy for actions of discrete amenable groups . J. Amer. Math. Soc. 19(2006), 737–758.CrossRefGoogle Scholar
Dinaburg, E. I., The relation between topological entropy and metric entropy . Sovrem. Mat. 11(1970), 13–16.Google Scholar
Dou, D. and Zhang, R. F., A note on dimensional entropy for amenable group actions . Topol. Methods Nonlinear Anal. 51(2018): 599–608.Google Scholar
Dou, D., Zheng, D. M., and Zhou, X. M., Packing topological entropy for amenable group actions. Ergodic Theory Dynam. Systems 43(2023), no. 2, 480–514.Google Scholar
Federer, H., Geometric measure theory. Springer-Verlag, New York, 1969, xiv+676 pp.Google Scholar
Feng, D. J. and Huang, W., Variational principles for topological entropies of subsets . J. Funct. Anal. 263(2012), 2228–2254.CrossRefGoogle Scholar
Foreman, M. and Weiss, B., An anti-classification theorem for ergodic measure preserving transformations . J. Eur. Math. Soc. 6(2004), 277–292.CrossRefGoogle Scholar
Goodman, T. N. T., Relating topological entropy and measure entropy . Bull. Lond. Math. Soc. 3(1971), 176–180.CrossRefGoogle Scholar
Goodwyn, L. W., Comparing topological entropy with measure-theoretic entropy . Proc. Amer. Math. Soc. 94(1972), 366–388.Google Scholar
Huang, W., Ye, X. D. and Zhang, G. H., Local entropy theory for a countable discrete amenable group action . J. Funct. Anal. 261(2011), 1028–1082.CrossRefGoogle Scholar
Huang, X. J., Liu, J. S., and Zhu, C. R., Bowen topological entropy of subsets for amenable group actions . J. Math. Anal. Appl. 472(2019), 1678–1715.CrossRefGoogle Scholar
Huang, X. J., Li, Z. Q., and Zhou, Y. H., A variational principle of topological pressure on subsets for amenable group actions . Discrete Contin. Dyn. Syst. 40(2020), 2687–2703.CrossRefGoogle Scholar
Kong, D. and Chen, E., Slow entropy for noncompact sets and variational principle . J. Dynam. Differential Equations. 26(2014), 477–492.CrossRefGoogle Scholar
Kerr, D. and Li, H. F., Entropy and the variational principle for actions of sofic groups . Invent. Math. 186(2011), 501–558.CrossRefGoogle Scholar
Kerr, D. and Li, H. F., Soficity, amenability and dynamical entropy . Am. J. Math. 135(2013), 721–761.CrossRefGoogle Scholar
Kerr, D. and Li, H. F., Ergodic theory: Independence and dichotomies, Math. Springer, Cham, 2016, xxxiv+431 pp.CrossRefGoogle Scholar
Kieffer, J. C., A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space . Ann. Probab. 3(1975), no. 6, 1031–1037.CrossRefGoogle Scholar
Mattila, P., Geometry of sets and measures in Euclidean spaces. Cambridge University Press, Cambridge, 1995, xii+343 pp.CrossRefGoogle Scholar
Ornstein, D. and Weiss, B., Entropy and isomorphism theorems for actions of amenable groups . J. Anal. Math. 48(1987), 1–141.CrossRefGoogle Scholar
Pesin, Y. B., Dimension theory in dynamical systems: Contemporary views and applications. University of Chicago Press, Chicago 1997, xiv+304 pp.CrossRefGoogle Scholar
Pesin, Y. B. and Pitskel, B. S., Topological pressure and the variational principle for noncompact sets . Funct. Anal. Appl. 18(1984), 307–318.CrossRefGoogle Scholar
Ren, X. K. and Sun, W. X., Local entropy, metric entropy and topological entropy for countable discrete amenable group actions . Internat. J. Bifur. Chaos Appl Sci. Engrg. 26(2016), 165–175.CrossRefGoogle Scholar
Rudolph, D. J. and Weiss, B., Entropy and mixing for amenable group actions . Ann. Math. 151(2000), 1119–1150.CrossRefGoogle Scholar
Simpson, S. G., Symbolic dynamics: Entropy = dimension = complexity . Theory Comput. Syst. 56(2015), 527–543.CrossRefGoogle Scholar
Zhao, Y., Measure theoretic pressure for amenable group actionss . Colloq. Math. 148(2017), 87–106.CrossRefGoogle Scholar
Zheng, D. M. and Chen, E. C., Bowen entropy for actions of amenable groups . Israel J. Math. 212(2016), 895–911.CrossRefGoogle Scholar
Zheng, D. M. and Chen, E. C., Topological entropy of sets of generic points for actions of amenable groups . Sci. China Math. 61(2018), 869–880.CrossRefGoogle Scholar
Zhao, Y. and Pesin, Y., Scaled entropy for dynamical systems . J. Stat. Phys. 158(2015), 447–475.CrossRefGoogle Scholar