Abstract
In this paper, we give some definitions of the topological tail pressures for sub-additive potentials and prove that they are equivalent if the potentials are continuous. Under some assumptions, we get a variational principle which exhibits the relationship between topological tail pressure and measure-theoretic tail entropy. Finally, we define a new measure-theoretic tail pressure for sub-additive potentials and some interesting properties of it are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barreira, L.: Measures of maximal dimension for hyperbolic diffeomorphisms. Commum. Math. Phys., 239, 93–113 (2003)
Burguet, D.: A direct proof of the tail variational principle and its extension to maps. Ergodic Theory Dynam. Systems, 29, 357–369 (2009)
Buzzi, J.: Intrinsic ergodicity for smooth interval maps. Israel J. Math., 100, 125–161 (1997)
Cao, Y., Feng, D., Huang, W.: The thermodynamic formalism for subadditive potentials. Discrete Contin. Dynam. Syst. Ser. A, 20, 639–657 (2008)
Cao, Y., Hu, H., Zhao, Y.: Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure. Ergodic Theory Dynam. Systems, 33(3), 831–850 (2013)
Chen, B., Ding, B., Cao, Y.: The local variational principle of topological pressure for subadditive potentials. Nonlinear Anal., 73, 3525–3536 (2010)
Ding, D., Zhang, Z., Ma, X., et al.: Variational principle for topological tail pressures with sub-additive upper semi-contiunous potentials. Dyn. Syst., 31(2), 198–220 (2016)
Downarowicz, T.: Entropy structure. J. Anal. Math., 96, 57–116 (2005)
Falconer, K.: A subadditive thermodynamic formalism for mixing repellers. J. Phys. A, 21, 737–742 (1998)
He, L., Lv, J., Zhou, L.: Definition of measure-theoretic pressure using spanning sets. Acta Math. Sin., English Series, 20, 709–718 (2004)
Li, Y., Chen, E., Cheng, W.: Tail pressure and the tail entropy function. Ergodic Theory Dynam. Systems, 32, 1400–1417 (2012)
Lindenstrauss, E.: Mean dimension, small entropy factors and an embedding theorem. Inst. Hautes Etudes Sci. Publ. Math., 89, 227–262 (1999)
Misiurewicz, M.: Topological conditional entropy. Studia Math., 55, 175–200 (1976)
Munroe, M.: Introduction to Measure and Integration, Inc. Addison-Wesley Publishing Company, Cambridge 42, MA, 1953
Ruelle, D.: Statistical mechanics on a compact set with Zv action satisfying expansivencess and specification. Trans. Amer. Math. Soc., 187, 237–251 (1973)
Walters, P.: A variational principle for the pressure of continuous transformations. Amer. J. Math., 97(4), 937–971 (1975)
Walters, P.: An Introduction on Ergodic Theory, Springer, New York, 1982
Zhao, Y.: A note on the measure-theoretic pressure in subadditive case. Chinese Annals of Math., Series A, 3, 325–332 (2008)
Zhao, Y., Cao, Y.: Measure-theoretic pressure for subadditive potentials. Nonlinear Analysis, 70, 2237–2247 (2009)
Zhou, X., Zhang, Y., Chen, E.: Topological conditional entropy for amenable group actions. Proc. Amer. Math. Soc., 143, 141–150 (2015)
Zhou, Y.: Tail variational principle for a countable discrete amenable group action. J. Math. Anal. Appl., 433, 1513–1530 (2016)
Zhou, Y.: On the tail pressure. Topol. Methods Nonlinear Anal., 47, 681–692 (2016)
Acknowledgements
We are very grateful to the referees for their carefully reading and suggestions which improves the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSFC (Grant No. 11471056) and Foundation and Frontier Research Program of Chongqing (Grant No. cstc2016jcyjA0312)
Rights and permissions
About this article
Cite this article
Peng, J.L., Zhou, Y.H. Sub-additive topological and measure-theoretic tail pressures. Acta. Math. Sin.-English Ser. 33, 1617–1631 (2017). https://doi.org/10.1007/s10114-017-6594-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-017-6594-4