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Transactions of the American Mathematical Society

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Statistical mechanics on a compact set with $ Z^{v}$ action satisfying expansiveness and specification


Author: David Ruelie
Journal: Trans. Amer. Math. Soc. 185 (1973), 237-251
MSC: Primary 28A65; Secondary 82.28, 58F99, 54H20
DOI: https://doi.org/10.1090/S0002-9947-1973-0417391-6
MathSciNet review: 0417391
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Abstract: We consider a compact set $ \Omega $ with a homeomorphism (or more generally a $ {{\mathbf{Z}}^\nu }$ action) such that expansiveness and Bowen's specification condition hold. The entropy is a function on invariant probability measures. The pressure (a concept borrowed from statistical mechanics) is defined as function on $ \mathcal{C}(\Omega )$--the real continuous functions on $ \Omega $. The entropy and pressure are shown to be dual in a certain sense, and this duality is investigated.


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  • [1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. MR 30 #5291. MR 0175106 (30:5291)
  • [2] R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377-397. MR 43 #8084. MR 0282372 (43:8084)
  • [3] -, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 725-747. MR 43 #2740. MR 0277003 (43:2740)
  • [4] G. Choquet and P. A. Meyer, Existence et unicité des représentations intégrales dans les convexes compacts quelconques, Ann. Inst. Fourier (Grenoble) 13 (1963), fasc. 1, 139-154. MR 26 #6748; MR 30 #1203. MR 0149258 (26:6748)
  • [5] E. I. Dinaburg, A correlation between topological entropy and metric entropy, Dokl. Akad. Nauk SSSR 190 (1970), 19-22 = Soviet Math. Dokl. 11 (1970), 13-16. MR 41 #425. MR 0255765 (41:425)
  • [6] N. Dunford and J. T. Schwartz, Linear operators. I. General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [7] G. Gallavotti and S. Miracle-Sole, Statistical mechanics of lattice systems, Comm. Math. Phys. 5 (1967), 317-323. MR 36 #1173. MR 0218084 (36:1173)
  • [8] L. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc. 23 (1969), 679-688. MR 40 #299. MR 0247030 (40:299)
  • [9] W. H. Gottschalk and G. A. Hedlund, Topological dynamics, Amer. Math. Soc. Colloq. Publ., vol 36, Amer. Math. Soc., Providence, R.I., 1955. MR 17, 650. MR 0074810 (17:650e)
  • [10] K. Jacobs, Lecture notes on ergodic theory. I, II, Mat. Inst., Aarhus Univ., Aarhus, 1963, pp. 1-207, 208-505. MR 28 #1247; #3138. MR 0159922 (28:3138)
  • [11] O. E. Lanford and D. W. Robinson, Statistical mechanics of quantum spin systems. III, Comm. Math. Phys. 9 (1968), 327-338. MR 38 #3012. MR 0234696 (38:3012)
  • [12] D. W. Robinson and D. Ruelle, Mean entropy of states in classical statistical mechanics, Comm. Math. Phys. 5 (1967), 288-300. MR 37 #1146. MR 0225553 (37:1146)
  • [13] D. Ruelle, A variational formulation of equilibrium statistical mechanics and the Gibbs phase rule, Comm. Math. Phys. 5 (1967), 324-329. MR 36 #699. MR 0217610 (36:699)
  • [14] -, Statistical mechanics. Rigorous results, Benjamin, New York, 1969. MR 44 #6279. MR 0289084 (44:6279)
  • [15] K. Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math. 11 (1970), 99-109. MR 44 #3349. MR 0286135 (44:3349)
  • [16] Ja. G. Sinaĭ, Markov partitions and Y-diffeomorphisms, Funkcional Anal. i Priložen. 2 (1968), no.l, 64-89 = Functional Anal. Appl. 2 (1968), 61-82. MR 38 #1361. MR 0233038 (38:1361)
  • [17] -, Construction of Markov partitionings, Funkcional. Anal. i Priložen. 2 (1968), no. 3, 70-80. (Russian) MR 40 #3591. MR 0250352 (40:3591)
  • [18] -, Invariant measures for Anosov's dynamical systems, Proc. Internat. Congress Math. (Nice, 1970), vol. 2, Gauthier-Villars, Paris, 1971, pp. 929-940.
  • [19] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 37 #3598. MR 0228014 (37:3598)
  • [20] M. Smorodinsky, Ergodic theory, entropy, Lecture Notes in Math., vol. 214, Springer-Verlag, Berlin, 1971. MR 0422582 (54:10568)

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DOI: https://doi.org/10.1090/S0002-9947-1973-0417391-6
Article copyright: © Copyright 1973 American Mathematical Society

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