Abstract
By applying Grothendieck theory and Ruelle thermodynamic formalism, we prove that, for expansive dynamical systems and interaction potentials satisfying certain conditions of analyticity, the associated Gibbs states are unique. This allows us to draw an analogy between some quantities in classical thermodynamics and abstract dynamics in the spirit of the previous work of the authors [13].
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References
A. F. Beardon, “The geometry of discrete groups,” Grad. Texts in Math., (1983).
R. E. Bowen, “Periodic points and measures for Axiom-A diffeomorphisms, ” Trans. Amer. Math. Soc., 154, 377–397 (1971).
R. E. Bowen and C. Series, “Markov maps associated to Fuchsian groups,” Publ. IHES, 50, 153–170 (1979).
C. Earle and R. Hamilton, “A fixed point theorem for holomorphic mappings,” In: Global Analysis, Proc. Symp. Pure Math., vol. XIV, S. Chern and S. Smale (Eds.), AMS, Providence, RI (1970).
G. Gallavotti, Statistical Mechanics: A Short Treatise, Springer-Verlag, Berlin (1999).
A. Grothendieck, “Produits tensoriels topologiques et espaces nucléaires,” Mem. Amer. Math. Soc., 16 (1955).
A. Grothendieck, “La théorie de Fredholm,” Bull. Soc. Math. Fr., 84, 319–384 (1956).
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge (1995).
G. Keller, “Equilibrium states in ergodic theory,” London Math. Soc. Stud. Texts 42 (1998).
O. Jenkinson and M. Pollicott, “Calculating Hausdorff dimension of Julia sets and Kleinian limit sets,” Am. J. Math., 124, 495–545 (2002).
D. Mayer, “On the ς function related to the continued fraction transformation,” Bull. Soc. Math. Fr., 104, 195–203 (1976).
D. Mayer, “On composition operators on Banach spaces of holomorphic functions,” J. Funct. Anal., 35, 191–206 (1980).
A. M. Mesón and F. Vericat, “Relations between some quantities in classical thermodynamics and abstract dynamics. Beyond hyperbolicity,” J. Dynam. Control Syst., 9, 437–448 (2003).
A. M. Mesón and F. Vericat, “Geometric constructions and multifractal analysis for boundary hyperbolic maps,” Dynam. Syst., 17, 203–213 (2002).
A. M. Mesón and F. Vericat, “Dimension theory and Fuchsian groups,” Acta Appl. Math., 89, 95–121 (2004).
P. J. Nicholls, Ergodic Theory of Discrete Groups, Cambridge Univ. Press, Cambridge (1989).
W. Parry and M. Pollicott, “An analogue of the prime number theorem for closed orbits of Axiom-A flows,” Ann. Math., 118, 573–591 (1983).
S. J. Patterson, “The limit set of a Fuschian group,” Acta Math., 136, 241–273 (1976).
D. Ruelle, “A measure associated with Axiom-A attractors,” Am. J. Math., 98, 619–654 (1976).
D. Ruelle, Thermodynamic Formalism, Encyclopedia Math., Addison-Wesley (1978).
C. Series, Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, T. Bedford, M. Keane and C. Series (Eds.), Oxford University Press, Oxford (1991).
C. Series, “Geometrical Markov coding on surfaces of constant negative curvature,” Ergodic Theory Dynam. Syst., 6 (1986).
Ya. G. Sinai, “Gibbs neasures in ergodic theory,” Russ. Math. Surv., 27, No. 21 (1972).
F. Takens and E. Verbitski, “Multifractal analysis of local entropies for expansive homeomorphisms with specifications,” Commun. Math. Phys, 203, 593–612 (1999).
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, Berlin (1982).
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.
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Mesón, A.M., Vericat, F. On the uniqueness of Gibbs states in some dynamical systems. J Math Sci 161, 250–260 (2009). https://doi.org/10.1007/s10958-009-9550-8
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DOI: https://doi.org/10.1007/s10958-009-9550-8