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Noncommutative algebras for hyperbolic diffeomorphisms. (English) Zbl 0657.58026
The author presents a general definition of Gibbs states (cf. D. Capocaccia [Commun. Math. Phys. 48, 85-88 (1976)]). Each Gibbs state $\alpha$ is related with a state $\stackrel{^}{\alpha }$ on a suitable ${C}^{*}$- algebra such that $\stackrel{^}{\alpha }$ satisfies the Kubo-Martin-Schwinger boundary condition with respect to an explicitly given modular group of automorphisms. The case of a Smale space [cf. the author, Thermodynamic formalism. The mathematical structures of classical equilibriums. Statistical mechanics (1978; Zbl 0401.28016)]) is discussed. In this case according to a result of N. T. A. Haydn [Ergodic Theory Dyn. Syst. 7, 119-132 (1987; Zbl 0616.58025)] the corresponding definition of Gibbs states is equivalent to that of Sinai [cf. Ya. G. Sinai, Usp. Mat. Nauk 27, No.4(166), 21-64 (1972; Zbl 0246.28008)] with the use of Markov partitions. Finally, the author defines Gibbs states $\rho$ in the case of a ${C}^{2}$-diffeomorphism of a compact manifold having an ergodic invariant measure with no zero characteristic exponent. Each state $\rho$ is related with a state $\stackrel{^}{\rho }$ on a von Neumann algebra such that $\stackrel{^}{\rho }$ satisfies the Kubo-Martin-Schwinger boundary condition with respect to a modular group of automorphisms.
Reviewer: L.N.Stoyanov

##### MSC:
 37D99 Dynamical systems with hyperbolic behavior
##### References:
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