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A thermodynamic definition of topological pressure for non-compact sets

Published online by Cambridge University Press:  26 March 2010

DANIEL J. THOMPSON
DANIEL J. THOMPSON*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, USA (email: thompson@math.psu.edu)
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Abstract

We give a new definition of topological pressure for arbitrary (non-compact, non-invariant) Borel subsets of metric spaces. This new quantity is defined via a suitable variational principle, leading to an alternative definition of an equilibrium state. We study the properties of this new quantity and compare it with existing notions of topological pressure. We are particularly interested in the situation when the ambient metric space is assumed to be compact. We motivate our definition by applying it to some interesting examples, including the level sets of the pointwise Lyapunov exponent for the Manneville–Pomeau family of maps.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 2 , April 2011 , pp. 527 - 547
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Adler, R. L., Konheim, A. G. and McAndrew, M. H.. Topological entropy. Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
[2]Barreira, L.. Dimension and Recurrence in Hyperbolic Dynamics (Progress in Mathematics, 272). Birkhäuser, Basel, 2008.Google Scholar
[3]Barreira, L. and Schmeling, J.. Sets of non-typical points have full topological entropy and full Hausdorff dimension. Israel J. Math. 116 (2000), 29–70.CrossRefGoogle Scholar
[4]Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–510.CrossRefGoogle Scholar
[5]Bowen, R.. Topological entropy for non-compact sets. Trans. Amer. Math. Soc. 184 (1973), 125–136.CrossRefGoogle Scholar
[6]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
[7]Bruin, H. and Todd, M.. Equilibrium states for interval maps: potentials with sup ϕ−inf ϕ<h top(f). Comm. Math. Phys. 283(3) (2008), 579–611.CrossRefGoogle Scholar
[8]Buzzi, J.. Specification on the interval. Trans. Amer. Math. Soc. 349(7) (1997), 2737–2754.CrossRefGoogle Scholar
[9]Dai, X. and Jiang, Y.. Distance entropy of dynamical systems on non-compact phase spaces. Discrete Contin. Dyn. Syst. 20 (2008), 313–333.CrossRefGoogle Scholar
[10]Denker, M., Grillenberger, C. and Sigmund, K.. Ergodic Theory on Compact Spaces (Lecture Notes in Mathematics, 527). Springer, Berlin, 1976.CrossRefGoogle Scholar
[11]Ercai, C., Küpper, T. and Lin, S.. Topological entropy for divergence points. Ergod. Th. & Dynam. Sys. 25(4) (2005), 1173–1208.CrossRefGoogle Scholar
[12]Gurevic, B. M. and Savchenko, V.. Thermodynamic formalism for countable symbolic Markov chains. Soviet Math. Dokl. 53 (1998), 245–344.Google Scholar
[13]Handel, M., Kitchens, B. and Rudolph, D.. Metrics and entropy for non-compact spaces. Israel J. Math. 91 (1995), 253–271.CrossRefGoogle Scholar
[14]Hasselblatt, B., Nitecki, Z. and Propp, J.. Topological entropy for nonuniformly continuous maps. Discrete Contin. Dynam. Syst. 22(1–2) (2008), 201–213.Google Scholar
[15]Ledrappier, F. and Walters, P.. A relativised variational principle for continuous transformations. J. Lond. Math. Soc. 16 (1977), 568–576.CrossRefGoogle Scholar
[16]Pesin, Y. and Senti, S.. Equilibrium measures for maps with inducing schemes. J. Mod. Dyn. 2(3) (2008), 397–427.CrossRefGoogle Scholar
[17]Pesin, Y. B.. Dimension Theory in Dimensional Systems: Contemporary Views and Applications. University of Chicago Press, Chicago, 1997.CrossRefGoogle Scholar
[18]Pesin, Y. B. and Pitskel, B. S.. Topological pressure and the variational principle for non-compact sets (English translation). Funct. Anal. Appl. 18 (1984), 307–318.CrossRefGoogle Scholar
[19]Pfister, C.-E. and Sullivan, W. G.. On the topological entropy of saturated sets. Ergod. Th. & Dynam. Sys. 27 (2007), 929–956.CrossRefGoogle Scholar
[20]Pollicott, M., Sharp, R. and Yuri, M.. Large deviations for maps with indifferent fixed points. Nonlinearity 11 (1998), 1173–1184.CrossRefGoogle Scholar
[21]Prellberg, T. and Slawny, J.. Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Stat. Phys. 66(1–2) (1992), 503–514.CrossRefGoogle Scholar
[22]Sarig, O.. Thermodynamic formalism for countable Markov shifts. Ergod. Th. & Dynam. Sys. 19 (1999), 1565–1593.CrossRefGoogle Scholar
[23]Takens, F. and Verbitskiy, E.. On the variational principle for the topological entropy of certain non-compact sets. Ergod. Th. & Dynam. Sys. 23(1) (2003), 317–348.CrossRefGoogle Scholar
[24]Thompson, D. J.. The irregular set for maps with the specification property has full topological pressure. Dyn. Syst. 25(1) (2010), 25–51.CrossRefGoogle Scholar
[25]Thompson, D. J.. A variational principle for topological pressure for certain non-compact sets. J. Lond. Math. Soc. 80(3) (2009), 585–602.CrossRefGoogle Scholar
[26]Urbanski, M.. Parabolic Cantor sets. Fund. Math. 151(3) (1996), 241–277.Google Scholar
[27]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.CrossRefGoogle Scholar