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On Gibbs and equilibrium states

Published online by Cambridge University Press:  19 September 2008

Nicolai T. A. Haydn
Nicolai T. A. Haydn
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV47AL, England
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Abstract

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Ruelle and Capocaccia gave a new definition of Gibbs states on Smale spaces. Equilibrium states of suitable functions thereon are known to be Gibbs states. The converse is discussed in this paper, where the problem is reduced to shift spaces and solved there by constructing suitable conjugating homeomorphisms in order to verify the conditions for Gibbs states which Bowen gave for shift spaces, where the equivalence to equilibrium states is known.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 1 , March 1987 , pp. 119 - 132
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[1]Bowen, R.. Equilibrium Slates and the Ergodic Theory of Anosov Diffeomorphisms. Springer Lecture Notes in Maths. # 470, 1975.Google Scholar
[2]Capocaccia, D.. A definition of Gibbs states for a compact set with Zν-action. Comm. Math. Phys. 48, (1976), 85–88.CrossRefGoogle Scholar
[3]Manning, A.. Axiom A diffeomorphisms have rational Zeta functions. Bull London Math. Soc. 3 (1971), 215–220.CrossRefGoogle Scholar
[4]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley, 1978.Google Scholar