Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-19T14:17:55.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermodynamic formalism for Haar systems in noncommutative integration: transverse functions and entropy of transverse measures

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  27 March 2020

ARTUR O. LOPES Open the ORCID record for ARTUR O. LOPES [Opens in a new window]  and
JAIRO K. MENGUE
ARTUR O. LOPES
Affiliation:
Universidade Federal do Rio Grande do Sul, Instituto de Matemática, Av. Bento Goncalves 9500, 90450-140Porto Alegre, RS, Brazil email arturoscar.lopes@gmail.com, jairokras@gmail.com
JAIRO K. MENGUE
Affiliation:
Universidade Federal do Rio Grande do Sul, Instituto de Matemática, Av. Bento Goncalves 9500, 90450-140Porto Alegre, RS, Brazil email arturoscar.lopes@gmail.com, jairokras@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider here a certain class of groupoids obtained via an equivalence relation (the so-called subgroupoids of pair groupoids). We generalize to Haar systems in these groupoids some results related to entropy and pressure which are well known in thermodynamic formalism. We introduce a transfer operator, where the equivalence relation (which defines the groupoid) plays the role of the dynamics and the corresponding transverse function plays the role of the a priori probability. We also introduce the concept of invariant transverse probability and of entropy for an invariant transverse probability, as well as of pressure for transverse functions. Moreover, we explore the relation between quasi-invariant probabilities and transverse measures. Some of the general results presented here are not for continuous modular functions but for the more general class of measurable modular functions.

Keywords

Haar systemstransverse functionstransverse measuresentropynoncommutative integration

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 6 , June 2021 , pp. 1835 - 1863
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aguiar, D., Cioletti, L. and Ruviaro, R.. A variational principle for the specific entropy for symbolic systems with uncountable alphabets. Math. Nachr. 291(17–18) (2018), 2506–2515.CrossRefGoogle Scholar
Anatharaman-Delaroche, C.. Ergodic theory and Von Neumann algebras: an introduction. Preprint, Univ. d’Orleans (France).Google Scholar
Anantharaman-Delaroche, C. and Renault, J.. Amenable groupoids. L’Enseignement Mathématique, Geneva, 2000, p. 196.Google Scholar
Baraviera, A. T., Cioletti, L. M., Lopes, A. O., Mohr, J. and Souza, R. R.. On the general one-dimensional XY model: positive and zero temperature, selection and non-selection. Rev. Math. Phys. 23(10) (2011), 1063–1113.CrossRefGoogle Scholar
Bissacot, R. and Kimura, B.. Gibbs measures on multidimensional subshifts, Preprint, 2016, Universidade Estadual de Sao Paulo.Google Scholar
de Castro, G. G.. $C^{\ast }$ -Algebras associadas a certas dinamicas e seus estados KMS. PhD Thesis, Prog. Pos. Mat., UFRGS, 2009.Google Scholar
de Castro, G. G. and Lopes, A. O.. KMS states, entropy, and a variational principle for pressure. Real Anal. Exchange 34(2) (2009), 333–346.CrossRefGoogle Scholar
de Castro, G. G., Lopes, A. O. and Mantovani, G.. Haar systems, KMS states on von Neumann algebras and C ∗ -algebras on dynamically defined groupoids and noncommutative integration. Modeling, Dynamics, Optimization and Bioeconomics IV (Proceedings in Mathematics and Statistics) . Eds. Pinto, A. and Zilberman, D.. Springer, New York, to appear.Google Scholar
Cioletti, L. and Lopes, A. O.. Interactions, specifications, DLR probabilities and the Ruelle operator in the one-dimensional lattice. Discrete Contin. Dyn. Syst. A 37(12) (2017), 6139–6152.CrossRefGoogle Scholar
Cioletti, L. and Melo, L. C.. On the extensions of generalized transfer operators. Preprint, 2020, Fundacao Universidade de Brasilia.Google Scholar
Connes, A.. Sur la Theorie non commutative de l’integration. Semminaire sur les Algèbres d’Operateurs (Lecture Notes in Mathematics, 725) . Ed. de la Harpe, P.. Springer, Berlin, 1979, pp. 19–143.CrossRefGoogle Scholar
Exel, R.. A new look at the crossed-product of a C ∗ -algebra by an endomorphism. Ergod. Th. & Dynam. Sys. 23(6) (2003), 1733–1750.CrossRefGoogle Scholar
Exel, R. and Lopes, A.. C ∗ -algebras, approximately proper equivalence relations and thermodynamic formalism. Ergod. Th. & Dynam. Sys. 24 (2004), 1051–1082.CrossRefGoogle Scholar
Exel, R. and Lopes, A.. C ∗ - algebras and thermodynamic formalism. São Paulo J. Math. Sci. 2(1) (2008), 285–307.CrossRefGoogle Scholar
Feldman, J. and Moore, C. C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc. 234 (1977), 289–324.CrossRefGoogle Scholar
Feldman, J. and Moore, C. C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. II. Trans. Amer. Math. Soc. 234 (1977), 325–359.CrossRefGoogle Scholar
Haydn, N. T. A. and Ruelle, D.. Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification. Comm. Math. Phys. 148 (1992), 155–167.CrossRefGoogle Scholar
Kastler, D.. On Connes’ noncommutative integration theory. Comm. Math. Phys. 85 (1982), 99–120.CrossRefGoogle Scholar
Krieger, W. A.. On non-singular transformations of a measure space. I. Z. Wahrsch. Verw. Geb. 11 (1969), 83–97.CrossRefGoogle Scholar
Krieger, W. A.. On non-singular transformations of a measure space. II. Z. Wahrsch. Verw. Geb. 11 (1969), 98–119.CrossRefGoogle Scholar
Kumjian, A. and Renault, J.. KMS states on C ∗ -Algebras associated to expansive maps. Proc. Amer. Math. Soc. 134(7) (2006), 2067–2078.CrossRefGoogle Scholar
Lopes, A. O. and Mantovani, G.. The KMS condition for the homoclinic equivalence relation and Gibbs probabilities. São Paulo J. Math. Sci. 13(1) (2019), 248–282.CrossRefGoogle Scholar
Lopes, A. O., Mengue, J. K., Mohr, J. and Souza, R. R.. Entropy and variational principle for one-dimensional lattice systems with a general a-priori probability: positive and zero temperature. Ergod. Th. & Dynam. Sys. 35(6) (2015), 1925–1961.CrossRefGoogle Scholar
Lopes, A. O. and Oliveira, E.. Continuous groupoids on the symbolic space, quasi-invariant probabilities for Haar systems and the Haar–Ruelle operator. Bull. Braz. Math. Soc. (N.S.) 50(3) (2019), 663–683.CrossRefGoogle Scholar
Miller, B.. The existence of measures of a given cocycle. I. Atomless, ergodic 𝜎-finite measures. Ergod. Th. & Dynam. Sys. 28(5) (2008), 1599–1613.CrossRefGoogle Scholar
Miller, B.. The existence of measures of a given cocycle. II. Probability measures. Ergod. Th. & Dynam. Sys. 28(5) (2008), 1615–1633.CrossRefGoogle Scholar
Nadkarni, M.. On the existence of an invariant measure. Nat. Acad. Sci. Lett. 13(4) (1990), 127–128.Google Scholar
Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990), 1–268.Google Scholar
Renault, J.. A Groupoid Approach to C ∗ -Algebras (Lecture Notes in Mathematics, 793) . Springer, Berlin, 1980.CrossRefGoogle Scholar
Renault, J.. The Radon-Nikodym problem for approximately proper equivalence relations. Ergod. Th. & Dynam. Sys. 25 (2005), 1643–1672.CrossRefGoogle Scholar
Renault, J.. C ∗ -Algebras and Dynamical Systems, XXVII Coloq. Bras. Mat. IMPA, Rio de Janeiro, 2009.Google Scholar
Ruelle, D.. Noncommutative algebras for hyperbolic diffeomorphisms. Invent. Math. 93 (1988), 1–13.CrossRefGoogle Scholar
Sakai, S.. C ∗ Algebras and W ∗ Algebras. Springer, New York, 1971.Google Scholar
Schmidt, K.. Klaus Unique ergodicity for quasi-invariant measures. Math. Z. 167(2) (1979), 169–172.CrossRefGoogle Scholar
Viana, M. and Oliveira, K.. Foundations of Ergodic Theory. Cambridge University Press, Cambridge, 2016.CrossRefGoogle Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 2000.Google Scholar
Weinstein, A.. Groupoids: unifying internal and external symmetry. A tour through some examples. Notices Amer. Math. Soc. 43(7) (1996), 744–752.Google Scholar
Weiss, B.. Measurable Dynamics: Conference in Modern Analysis and Probability. American Mathematical Society, Providence, RI, 1984, pp. 395–421.CrossRefGoogle Scholar