Abstract
The localized pressure of a continuous potential \(\varphi \) is computed by considering only those \((F_n, \epsilon )\)-separated sets whose statistical sums with respect to an m-dimensional potential are “close” to a given value in \({\mathbb {R}}^m\). This article is devoted to establishing Katok’s pressure formula and a variational principle of localized pressure for amenable group actions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adler, R., Konheim, A., McAndrew, M.: Topological entropy. Trans. Am. Math. Soc. 114, 309–319 (1965)
Huang, X., Liu, J., Zhu, C.: The Katok’s entropy formula for amenable group actions. Discrete Contin. Dyn. Syst. 38, 4467–4482 (2018)
Huang, X., Li, Z., Zhou, Y.: A variational principle of topological pressure on subsets for amenable group actions. Discrete Contin. Dyn. Syst. 40, 2687–2703 (2020)
Katok, A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes Études Sci. Publ. Math. 51, 137–173 (1980)
Kerr, D., Li, H.: Entropy and variational principle for actions of sofic groups. Invent. Math. 186, 501–558 (2011)
Kerr, D., Li, H.: Ergodic Theory: Independence and Dichotomies. Springer Monographs in Mathematics. Springer, Cham (2016)
Kucherenko, T., Wolf, C.: Geometry and entropy of generalized rotation sets. Israel J. Math. 199, 791–829 (2014)
Kucherenko, T., Wolf, C.: Localized pressure and equilibrium states. J. Stat. Phys. 160, 1529–1544 (2015)
Lindenstrauss, E.: Pointwise theorem for amenable groups. Invent. Math. 146, 259–295 (2001)
Mendoza, L.: Ergodic attractors for diffeomorphisms of surfaces. J. Lond. Math. Soc. 37, 362–374 (1988)
Misiurewicz, M.: 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, 1069–1075 (1976)
Ornstein, D., Weiss, B.: Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48, 1–141 (1987)
Ruelle, D.: Statistical mechanics on compact set with \(Z^v\) action satisfying expansiveness and specification. Trans. Am. Math. Soc. 187, 237–251 (1973)
Walters, P.: An Introduction to Ergodic Theory. Springer, New York (1982)
Zhang, R.: Topological pressure of generic points for amenable group actions. J. Dyn. Differ. Equ. 30, 1583–1606 (2018)
Zhu, C.: Conditional variational principles of conditional entropies for amenable group actions. Nonlinearity 34, 5163–5185 (2021)
Acknowledgements
We would like to thank the anonymous referee for useful comments which improve the paper. Yunping Wang was supported by the Natural Science Foundation of Zhejiang Province(Q22A011940) and NNSF of China (12071222, 12101446 and 12101340). Cao Zhao was supported by NNSF of China (11901419).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declared that they have no conflicts of interest to this work.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Wang, Y., Zhao, C. Localized topological pressure for amenable group actions. Anal.Math.Phys. 12, 74 (2022). https://doi.org/10.1007/s13324-022-00684-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-022-00684-8