Skip to main content Accessibility help
Thermodynamic Formalism
  • Cited by 265
  • 2nd edition
  • David Ruelle, Institut des Hautes Études Scientifiques, France
  • Export citation
  • Recommend to librarian
  • Buy the print book

Book description

Reissued in the Cambridge Mathematical Library this classic book outlines the theory of thermodynamic formalism which was developed to describe the properties of certain physical systems consisting of a large number of subunits. It is aimed at mathematicians interested in ergodic theory, topological dynamics, constructive quantum field theory, the study of certain differentiable dynamical systems, notably Anosov diffeomorphisms and flows. It is also of interest to theoretical physicists concerned with the conceptual basis of equilibrium statistical mechanics. The level of the presentation is generally advanced, the objective being to provide an efficient research tool and a text for use in graduate teaching. Background material on mathematics has been collected in appendices to help the reader. Extra material is given in the form of updates of problems that were open at the original time of writing and as a new preface specially written for this new edition by the author.


‘This is the second edition of the already classical book on the theory of thermodynamic formalism by David Ruelle.‘

Source: Monatshefte für Mathematik

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.


Abramov, L. M., “On the entropy of a flow,” Dokl. Akad. Nauk SSSR 128, 873–875 (1959). English translation, Amer. Math. Soc. Transl., Ser. 2, 49, 167–170 (1966)
Adler, R. L., Konheim, A. G., and McAndrew, M. H., “Topological entropy,” Trans. Amer. Math. Soc. 114, 309–319 (1965)
Anosov, D. V., “Geodesic flows on a compact Riemann manifold of negative curvature,” Trudy Mat. Inst. Steklov 90 (1967). English translation, Proc. Steklov Math. Inst. 90 (1967)
Araki, H., “Gibbs states of a one-dimensional quantum lattice,” Commun. Math. Phys. 14, 120–157 (1969)
Berger, R., “The undecidability of the domino problem,” Mem. Amer. Math. Soc., No. 66, 1966
P. Billingsley, Ergodic Theory and Information. John Wiley, New York, 1965
N. Bourbaki, Eléments de mathématique. Intégration. Chapitres 1, 2, 3, et 4, 2e éd. Hermann, Paris, 1965
N. Bourbaki, Eléments de mathématique. Intégration. Chapitre 5, 2e éd. Hermann, Paris, 1967
Bowen, R., “Markov partitions for axiom A diffeomorphisms,” Amer. J. Math. 92, 725–747 (1970)
Bowen, R., “Markov partitions and minimal sets for axiom A diffeomorphisms,” Amer. J. Math. 92, 907–918 (1970)
Bowen, R., “Entropy for group endomorphisms and homogeneous spaces,” Trans. Amer. Math. Soc. 153, 401–414 (1971)
Bowen, R., “Symbolic dynamics for hyperbolic flows,” Amer. J. Math. 95, 429–459 (1973)
Bowen, R., “Some systems with unique equilibrium states,” Math. Systems Theory 8, 193–202 (1974)
R. Bowen Equilibrium States and the Ergodic Theory of Anosov diffeomorphisms. Lecture Notes in Math. No. 470. Springer, Berlin, 1975
Bowen, R. and Ruelle, D., “The ergodic theory of axiom A flows,” Inventiones Math. 29, 181–202 (1975)
Bunimovič, L. A., “Imbedding of Bernoulli shifts in certain special flows,” Uspehi Mat. Nauk 28, No. 3, 171–172 (1973)
Capocaccia, D., “A definition of Gibbs state for a compact set with Zν action,” Commun. Math. Phys. 48, 85–88 (1976)
Choquet, G. and Meyer, P.-A., “Existence et unicité des représentations intégrales dans les convexes compacts quelconques,” Ann. Inst. Fourier 13, 139–154 (1963)
K. L. Chung, Markov Chains with Stationary Transition Probabilities. Springer, Berlin, 1967
Conze, J. P., “Entropie d'un groupe abélien de transformations,” Zeitschr. Wahrschein-lichkeitstheorie Verw. Gebiete 25, 11–30 (1972)
Denker, M., “Remarques sur la pression pour les transformations continues,” C. R. Acad. Sci. Paris 279, A967–A970 (1974)
M. Denker, C. Grillenberger, and K. Sigmund, Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics No. 527. Springer, Berlin, 1976
Dinaburg, E. I., “The relation between topological entropy and metric entropy,” Dokl. Akad. Nauk SSSR 190, No. 1, 19–22 (1970). English translation, Soviet Math. Dok.11, 13–16 (1970)
Dobrushin, R. L., “The description of a random field by means of conditional probabilities and conditions of its regularity,” Teorija Verojatn. i ee Prim. 13, 201–229 (1968). English translation, Theory Prob. Applications 13, 197–224 (1968)
Dobrushin, R. L., “Gibbsian random fields for lattice systems with pairwise interactions,” Funkts. Analiz i ego Pril. 2, No. 4, 31–43 (1968). English translation, Functional Anal. Appl.2, 292–301 (1968)
Dobrushin, R. L., “The problem of uniqueness of a Gibbsian random field and the problem of phase transitions,” Funkts. Analiz i ego Pril. 2, No. 4, 44–57 (1968). English translation, Functional Anal. Appl.2, 302–312 (1968)
Dobrushin, R. L., “Analyticity of correlation functions in one-dimensional classical systems with slowly decreasing potentials,” Commun. Math. Phys. 32, 269–289 (1973)
Dyson, F. J., “Existence of a phase-transition in a one-dimensional Ising ferromagnet,” Commun. Math. Phys. 12, 91–107 (1969)
Elsanousi, S. A., “A variational principle for the pressure of a continuous Z2-action on a compact metric space,” Amer. J. Math. 99, 77–106 (1977)
Fisher, M. E., “The theory of condensation and the critical point,” Physics 3, 255–283 (1967)
Franks, J. M., “A reduced zeta function for diffeomorphisms,” Amer. J. Math. 100, No. 2 (1978)
Furstenberg, H. and Kesten, H., “Products of random matrices,” Ann. Math. Statist. 31, 457–469 (1960)
Gallavotti, G., “Ising model and Bernoulli schemes in one dimension,” Commun. Math. Phys. 32, 183–190 (1973)
Gallavotti, G., “Funzioni zeta ed insiemi basilari,” Accad. Lincei. Rend. Sc. fis. mat. e nat. 61, 309–317 (1976)
Gallavotti, G. and Miracle-Sole, S., “Statistical mechanics of lattice systems,” Commun. Math. Phys. 5, 317–323 (1967)
F. R. Gantmaher, The Theory of Matrices. Nauka, Moscow, 1967. English translation, Chelsea, New York, 1964
H.-O. Georgii, Phasenübergang 1. Art bei Gittergasmodellen. Lecture Notes in Physics No. 16. Springer, Berlin, 1972
Georgii, H.-O., “Two remarks on extremal equilibrium states,” Commun. Math. Phys. 32, 107–118 (1973)
Goodman, T. N. T., “Relating topological entropy and measure entropy,” Bull. London Math. Soc. 3, 176–180 (1971)
Goodwyn, L. W., “Topological entropy bounds measure-theoretic entropy,” Proc. Amer. Math. Soc. 23, 679–688 (1969)
Griffiths, R. B. and Ruelle, D., “Strict convexity (‘continuity’) of the pressure in lattice systems,” Commun. Math. Phys. 23, 169–175 (1971)
Gurevič, B. M., “Topological entropy of enumerable Markov chains,” Dokl. Akad. Nauk SSSR 187, No. 4, 754–757 (1969). English translation, Soviet Math. Dokl. 10, 911–915 (1969)
Gurevič, B. M. and Oseledec, V. I., “Gibbs distributions and dissipativeness of U-diffeomorphisms,” Dokl. Akad. Nauk SSSR 209, No. 5, 1021–1023 (1973). English translation, Soviet Math. Dokl.14, 570–573 (1973)
H. Halmos, Measure Theory. D. Van Nostrand, Princeton, 1950
Hirsch, M. W., “Expanding maps and transformation groups,” in Global Analysis Proc. Symp. Pure Math. 14, 1970, pp. 125–131
Israel, R. B., “Existence of phase transitions for long-range interactions,” Commun. Math. Phys. 43, 59–68 (1975)
R. B. Israel Tangents to the Pressure as Invariant Equilibrium States in Statistical Mechanics of Lattice Systems, Princeton University Press, Princeton, 1978
Keane, M., “Sur les mesures invariantes d'un recouvrement régulier,” C. R. Acad. Sci. Paris 272, A585–A587 (1971)
G. Köthe, Topologische lineare Räume I. Springer, Berlin, 1960
O. E. Lanford, “Selected topics in functional analysis,” in Mécanique statistique et théorie quantique des champs. Les Houches 1970. (C. De Witt, and R. Stora, eds.), pp. 109–214. Gordon and Breach, New York, 1971
O. E. Lanford “Entropy and equilibrium states in classical statistical mechanics,” in Statistical mechanics and mathematical problems, Lecture Notes in Physics No. 20, pp. 1–113. Springer, Berlin, 1973
Lanford, O. E. and Robinson, D. W., “Statistical mechanics of quantum spin systems III,” Commun. Math. Phys. 9, 327–338 (1968)
Lanford, O. E. and Ruelle, D., “Observables at infinity and states with short range correlations in statistical mechanics,” Commun. Math. Phys. 13, 194–215 (1969)
Lasota, A. and Yorke, J. A., “On the existence of invariant measures for piecewise monotonic transformations,” Trans. Amer. Math. Soc. 186, 481–488 (1973)
Ledrappier, F.Mesures d'équilibre sur un réseau,” Commun. Math. Phys. 33, 119–128 (1973)
Ledrappier, F., “Principe variationnel et systémes dynamiques symboliques,” Z. Wahrschein-lichkeitstheorie Verw. Gebiete 30, 185–202 (1974)
Ledrappier, F. and Walters, P., “A relativised variational principle for continuous transformations,” J. London Math. Soc. 16, 568–576 (1977)
Livšic, A. N., “Homology properties of Y-systems,” Mat. Zametki 10, No. 5, 555–564 (1971). English translation, Math. Notes10, 758–763 (1971)
Livšic, A. N., “Cohomology of dynamical systems,” Izv. Akad. Nauk SSSR. Ser. Mat. 36, No. 6, 1296–1320 (1972). English translation, Math. USSR Izvestija6, 1276–1301 (1972)
Manning, A., “Axiom A diffeomorphisms have rational zeta functions,” Bull. London Math. Soc. 3, 215–220 (1971)
A. Manning “Topological entropy and the first homology group,” in Dynamical Systems. Warwick 1974, Lecture Notes in Mathematics No. 468, pp. 185–190. Spinger, Berlin, 1975
Mazur, S., “Öber konvexe Mengen in linearen normierten Räumen,” Studia Math. 4, 70–84 (1933)
Misiurewicz, M., “A short proof of the variational Principle for a Z+N action on a compact space,” Astérisque 40, 147–157 (1976)
D. S. Ornstein, Ergodic Theory, Randomness, and Dynamical Systems. Yale Mathematical Monographs 5. Yale University Press, New Haven, 1974
Oseledec, V. I., “A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems,” Trudy Moscov. Mat. Obšč. 19, 179–210 (1968). English translation, Trans. Moscow Math. Soc.19, 197–231 (1968)
Parry, W., “Intrinsic Markov chains,” Trans. Amer. Math. Soc. 112, 55–66 (1964)
W. Parry “Topological Markov chains and suspensions,” Warwick preprint, 1974
R. Phelps, Lectures on Choquet's Theorem. Van Nostrand Mathematical Studies No. 7. D. Van Nostrand, Princeton, 1966
C. J. Preston, Gibbs States on Countable Sets. Cambridge Tracts in Mathematics No. 68. Cambridge University Press, Cambridge, 1974
C. J. Preston, Random Fields. Lecture Notes in Mathematics No. 534. Springer, Berlin, 1976
Ratner, M., “The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature,” Israel J. Math. 16, 181–197 (1973)
Ratner, M., “Anosov flows with Gibbs measures are also Bernoullian,” Israel J. Math. 17, 380–391 (1974)
Robinson, R. M., “Undecidability and nonperiodicity for tilings of the plane,” Inventiones Math. 12, 177–209 (1971)
Robinson, D. W. and Ruelle, D., “Mean entropy of states in classical statistical mechanics.” Commun. Math. Phys. 5, 288–300 (1967)
Rohlin, V. A., “On the fundamental ideas of measure theory,” Mat. Sbornik (N. S.) 25, 107–150 (1949). English translation, Amer. Math. Soc. Transl., Ser. 1, 10, 1–54 (1952)
Ruelle, D., “A variational formulation of equilibrium statistical mechanics and the Gibbs phase rule,” Commun. Math. Phys. 5, 324–329 (1967)
Ruelle, D., “Statistical mechanics of a one-dimensional lattice gas,” Commun. Math. Phys. 9, 267–278 (1968)
D. Ruelle Statistical Mechanics. Rigorous Results. Benjamin, New York, 1969
Ruelle, D., “Statistical mechanics on a compact set with Zθ-action satisfying expansiveness and specification,” Bull. Amer. Math. Soc. 78, 988–991 (1972); Trans. Amer. Math. Soc. 185, 237–251 (1973)
Ruelle, D., “A measure associated with axiom A attractors,” Amer. J. Math. 98, 619–654 (1976)
Ruelle, D., “Generalized zeta-functions for axiom A basic sets,” Bull. Amer. Math. Soc. 82, 153–156 (1976)
Ruelle, D., “Zeta-functions for expanding maps and Anosov flows,” Inventiones Math. 34, 231–242 (1976)
Ruelle, D., “A heuristic theory of phase transitions,” Commun. Math. Phys., 53, 195–208 (1977)
Ruelle, D. and Sullivan, D., “Currents, flows and diffeomorphisms,” Topology 14, 319–327 (1975)
P. Shields, The Theory of Bernoulli Shifts. University of Chicago Press, Chicago, 1973
Shub, M., “Endomorphisms of compact differentiable manifolds,” Amer. J. Math. 91, 175–199 (1969)
B. Simon, The Pφ2Euclidean (Quantum) Field Theory. Princeton University Press, Princeton, 1974
Sinai, Ia. G., “Markov partitions and C-diffeomorphisms,” Funkts. Analiz i Ego Pril. 2, No. 1, 64–89 (1968). English translation, Functional Anal. Appl. 2, 61–82 (1968)
Sinai, Ia. G., “Construction of Markov partition,” Funkts. Analiz i Ego Pril. 2, No. 3, 70–80 (1968). English translation, Functional Anal. Appl. 2, 245–253 (1968)
Ia. G. Sinai “Mesures invariantes des Y-systémes,” in Actes, Congrés intern. Math., Nice, 1970, Vol. 2 pp. 929–940. Gauthier-Villars, Paris, 1971
Sinai, Ia. G., “Gibbsian measures in ergodic theory,” Uspehi Mat. Nauk 27, No. 4, 21–64 (1972). English translation, Russian Math. Surveys 27, No. 4, 21–69 (1972)
Smale, S., “Differentiable dynamical systems,” Bull. Amer. Math. Soc. 73, 747–817 (1967)
M. Smorodinsky, Ergodic Theory, Entropy. Lecture Notes in Mathematics No. 214. Springer, Berlin, 1971
Sullivan, W. G., “Potentials for almost Markovian random fields,” Commun. Math. Phys. 33, 61–74 (1973)
G. Velo and A. S. Wightman (eds.), Constructive Quantum Field Theory. Lecture Notes in Physics No. 25. Springer, Berlin, 1973
Walters, P., “A variational principle for the pressure of continuous transformations,” Amer. J. Math. 97, 937–971 (1976)
P. Walters Ergodic Theory. Introductory Lectures. Lecture Notes in Mathematics No. 458. Springer, Berlin, 1975
Walters, P., “A generalized Ruelle Perron—Frobenius theorem and some applications,” Astérisque 40, 183–192 (1976)
Walters, P., “Invariant measures and equilibrium states for some mappings which expand distances,” Trans. Amer. Math. Soc., to appear
Williams, R. F., “Classification of subshifts of finite type,” Ann. of Math. 98, 120–153 (1973). Errata, Ann. of Math.99, 380–381 (1974)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.