Abstract
We introduce the notion of localized topological pressure for continuous maps on compact metric spaces. The localized pressure of a continuous potential \(\varphi \) is computed by considering only those \((n,\epsilon )\)-separated sets whose statistical sums with respect to an m-dimensional potential \(\Phi \) are “close” to a given value \(w\in {\mathbb R}^m\). We then establish for several classes of systems and potentials \(\varphi \) and \(\Phi \) a local version of the variational principle. We also construct examples showing that the assumptions in the localized variational principle are fairly sharp. Next, we study localized equilibrium states and show that even in the case of subshifts of finite type and Hölder continuous potentials, there are several new phenomena that do not occur in the theory of classical equilibrium states. In particular, we construct an example with infinitely many ergodic localized equilibrium states. We also show that for systems with strong thermodynamic properties and w in the interior of the rotation set of \(\Phi \) there is at least one and at most finitely many localized equilibrium states.
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Barreira, L., Gelfert, K.: Dimension estimates in smooth dynamics: a survey of recent results. Ergod. Theory Dyn. Syst. 31, 641–671 (2011)
Barreira, L., Saussol, B., Schmeling, J.: Higher-dimensional multifractal analysis. J. Math. Pures Appl. 9, 67–91 (2002)
Barreira, L., Saussol, B.: Variational principles and mixed multifractal spectra. Trans. Am. Math. Soc. 353, 3919–3944 (2001)
Bowen, R.: Lecture Notes in Mathematics. Equilibrium states and the ergodic theory of Anosov diffeomorphisms, vol. 470. Springer, Berlin (1975)
Bowen, R.: Hausdorff dimension of quasicircles. Inst. Hautes Études Sci. Publ. Math. 50, 11–25 (1979)
Bowen, R.: Some systems with unique equilibrium states. Math. Syst. Theory 8, 193–202 (1974/75)
Climenhaga, V.: Topological pressure of simultaneous level sets. Nonlinearity 26, 241–268 (2013)
Climenhaga, V., Thompson, D.J.: Equilibrium states beyond specification and the Bowen property. J. Lond. Math. Soc. 87(2), 401–427 (2013)
Haydn, N., Ruelle, D.: Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification. Commun. Math. Phys. 148, 155–167 (1992)
Feng, D.J., Lau, K.S., Wu, J.: Ergodic limits on the conformal repeller. Adv. Math. 169, 58–91 (2002)
Fisher, T., Diaz, L., Pacifico, M., Vieitez, J.: Entropy-expansiveness for partially hyperbolic diffeomorphisms. Discret. Contin. Dyn. Syst. 32, 4195–4207 (2012)
He, L., Lv, J., Zhou, L.: Definition of measure-theoretic pressure using spanning sets. Acta Math. Sinica (English Series) 20, 709–718 (2004)
Jenkinson, O.: Rotation, entropy, and equilibrium states. Trans. Am. Math. Soc. 353, 3713–3739 (2001)
Katok, A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math., Inst. Hautes Étud. Sci. 51, 137–173 (1980)
Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge (1995)
Keller, G.: London Mathematical Society Student Texts. Equilibrium states in ergodic theory, vol. 42, p. x+178. Cambridge University Press, Cambridge (1998)
Kucherenko, T., Wolf, C.: Geometry and entropy of generalized rotation sets. Isr. J. Math. 2014, 791–829 (1999)
Manning, A., McCluskey, H.: Hausdorff dimension for horseshoes. Ergod. Theory Dyn. Syst. 3, 251–260 (1983)
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge University Press, Cambridge (1995)
Misiurewicz, M.: Rotation Theory, Misiurewicz’s webpage
Newhouse, S.: Continuity properties of entropy. Ann. Math. 129(2), 215–235 (1989)
Pesin, Y.: Chicago Lectures in Mathematics. Dimension theory in dynamical systems: contemporary views and applications. Chicago University Press, Chicago (1997)
Pesin, Ya., Pitskel, B.: Topological pressure and the variational principle for noncompact sets. Funktsional. Anal. i Prilozhen. 18, 50–63 (1984). (Russian)
Pfister, C.-E., Sullivan, W.G.: On the topological entropy of saturated sets. Ergod. Theory Dyn. Syst. 2, 929–956 (2007)
Poincaré, H.: Sur les cousbes définies par les équations différentielles, Euvres Complètes, tome 1. Gauthier-Villars, Paris (1952)
Przytycki, F., Urbanski, M.: London Mathematical Society Lecture Note Series. Conformal fractals: ergodic theory methods, vol. 371 (x+354 pp). Cambridge University Press, Cambridge (2010)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Ruelle, D.: Repellers for real analytic maps. Ergod. Theory Dyn. Syst. 2, 99–107 (1982)
Ruelle, D.: Thermodynamic Formalism. Cambridge University Press, Cambridge (2004)
Takens, F., Verbitskiy, E.: On the variational principle for the topological entropy of certain non-compact sets. Ergod. Theory Dyn. Syst. 23, 317–348 (2003)
Thompson, D.: A variational principle for topological pressure for certain non-compact sets. J. Lond. Math. Soc. 81, 585–602 (2009)
Thompson, D.: A thermodynamic definition of topological pressure for non-compact sets. Ergod. Theory Dyn. Syst. 31, 527–547 (2011)
Walters, P.: Graduate Texts in Mathematics. An introduction to ergodic theory, vol. 79. Springer, New York (1981)
Ziemian, K.: Rotation sets for subshifts of finite type. Fund. Math. 146, 189–201 (1995)
Acknowledgments
This work was partially supported by grants from the PSC-CUNY (TRADA-45-278 to Tamara Kucherenko), (TRADA-45-356 to Christian Wolf) and by a grant from the Simons Foundation (#209846 to Christian Wolf).
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Kucherenko, T., Wolf, C. Localized Pressure and Equilibrium States. J Stat Phys 160, 1529–1544 (2015). https://doi.org/10.1007/s10955-015-1289-7
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DOI: https://doi.org/10.1007/s10955-015-1289-7