Abstract
D. Ruelle considered a general setting where he is able to characterize equilibrium states for Hölder potentials based on properties of conjugating homeomorphism in the so called Smale spaces. On this setting he also shows a relation of KMS states of \(C^*\)-algebras with equilibrium probabilities of Thermodynamic Formalism. A later paper by N. Haydn and D. Ruelle presents a shorter proof of this equivalence. Here we consider similar problems but now on the symbolic space \(\Omega = \{1,2,\ldots ,d\}^{{\mathbb {Z}} - \{ 0 \} }\) and the dynamics will be given by the shift \(\tau \). In the case of potentials depending on a finite coordinates we will present a simplified proof of the equivalence mentioned above which is the main issue of the papers by D. Ruelle and N. Haydn. The class of conjugating homeomorphism is explicit and reduced to a minimal set of conditions. We also present with details (following D. Ruelle) the relation of these probabilities with the KMS dynamical \(C^*\)-state on the \(C^*\)-algebra associated to the groupoid defined by the homoclinic equivalence relation. The topics presented here are not new but we believe the main ideas of the proof of the results by Ruelle and Haydn will be quite transparent in our exposition.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baravieira, A., Leplaideur, R., Lopes, A.O.: Ergodic Optimization, Zero Temperature Limits and the Max-Plus Algebra. XXIX Coloq. Bras. Mat - IMPA, Rio de Janeiro (2013)
Bissacot, R., Kimura, B.: Gibbs Measures on Multidimensional Sub-shifts. Preprint (2016)
Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diemorphisms. Springer, Berlin (1975)
Bratteli, O., Robinson, D.: Operator Algebras and Quantum Statistical Mechanics I. Springer, Berlin (2003)
Castro, G., Lopes, A.O., Mantovani, G.: Haar systems, KMS states on von Neumann algebras and \(C^*\)-algebras on dynamically defined groupoids and Noncommutative Integration. Preprint (2017)
Cioletti, L., Lopes, A.O.: Interactions, specifications, DLR probabilities and the Ruelle operator in the one-dimensional lattice. Discrete Contin. Dyn. Syst. Ser. A 37(12), 6139–6152 (2017)
DellAntonio, G.: Lectures on the Mathematics of Quantum Mechanics II. Atlantis Press, Paris (2016)
Exel, R., Lopes, A.O.: \(C^*\)-algebras, approximately proper equivalence relations and Thermodynamic Formalism. Ergod. Theory Dyn. Syst. 24, 1051–1082 (2004)
Exel, R., Lopes, A.O.: C*-algebras and Thermodynamic Formalism. Sao Paulo J. Math. Sci. 2(1), 285–307 (2008)
Haydn, N.T.A.: On Gibbs and equilibrium states. Ergod. Theory Dyn. Syst. 7, 119–132 (1987)
Haydn, N.T.A., Ruelle, D.: Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification. Commun. Math. Phys. 148, 155–167 (1992)
Kumjian, A., Renault, J.: KMS states on \( C^*\)-algebras associated to expansive maps. Proc. AMS 134(7), 2067–2078 (2006)
Lopes, A.O., Oliveira, E.: Continuous groupoids on the symbolic space, quasi-invariant probabilities for Haar systems and the Haar–Ruelle operator, to appear in Bull Braz. Math. Soc. (on line 2018)
Putnam, I.: \(C^*\)-algebras from Smale spaces. Can. J. Math. 48(1), 175–195 (1996)
Putnam, I., Spielberg, J.: The structure of \(C^*\)-algebras associated with hyperbolic dynamical systems. J. Funct. Anal. 163(2), 279–299 (1999)
Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Asterisque, 187–188 (1990). http://www.numdam.org/item/AST_1990__187-188__1_0/
Pedersen, G.K.: \(C^{*}\) Algebras and Their Automorphism Groups. Academic Press, New York (1979)
Renault, J.: A Groupoid Approach to \(C^*\)-Algebras. Lecture Notes in Mathematics, vol. 793. Springer, Berlin (1980)
Renault, J.: \(C^*\)-Algebras and Dynamical Systems. XXVII Coloquio Bras. de Matematica - IMPA, Rio de Janeiro (2009)
Ruelle, D.: Noncommutative algebras for hyperbolic diffeomorphisms. Invent. Math. 93, 1–13 (1988)
Ruelle, D.: Thermodynamic Formalism, 2nd edn. Cambridge University Press, Cambridge (2004)
Thomsen, C.K.: \(C^*\)-Algebras of Homoclinic and Heteroclinic Structure in Expansive Dynamics, vol. 206, no. 970. Memoirs of the American Mathematical Society, Providence (2010)
Author information
Authors and Affiliations
Corresponding author
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
Lopes, A.O., Mantovani, G. The KMS condition for the homoclinic equivalence relation and Gibbs probabilities. São Paulo J. Math. Sci. 13, 248–282 (2019). https://doi.org/10.1007/s40863-019-00118-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40863-019-00118-7