Abstract
The SRB measures of a hyperbolic system are widely accepted as the measures that are physically relevant. It has been shown by Ruelle that they depend smoothly on the system. Furthermore, Ruelle showed by a separate argument that the first derivative, i.e., the linear response function, admits a geometric interpretation.
In this paper, we consider thermodynamic limits of SRB measures in lattices of coupled hyperbolic attractors. In a previous paper, using Markov partitions and thermodynamic formalism, we had established the smooth dependence of thermodynamic limits of SRB measures. Here, we establish that the linear response function admits a geometric interpretation. The formula is analogous to the one found by Ruelle for finite dimensional systems if one term is reinterpreted appropriately. We show that the limiting derivative is the thermodynamic limit of the derivatives in finite volume. We also obtain similar results for the derivatives of the entropy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bonetto, F., Falco, P., Giuliani, A.: Analyticity of the SRB measure of a lattice of coupled Anosov diffeomorphisms of the torus. J. Math. Phys. 45(8), 3282–3309 (2004)
Bricmont, J., Kupiainen, A.: Coupled analytic maps. Nonlinearity 8, 379–396 (1995)
Bricmont, J., Kupiainen, A.: High temperature expansions and dynamical systems. Commun. Math. Phys. 178, 703-732 (1996)
Bricmont, J., Kupiainen, A.: Infinite-dimensional SRB measures. Phys. D 103(1–4), 18–33 (1997)
Contreras, G.: Regularity of topological and metric entropy of hyperbolic flows. Math. Z. 210(1), 97–111 (1992)
Contreras, G.: The derivatives of equilibrium states. Bol. Soc. Brasil. Mat. (N.S.) 26(2), 211–228 (1995)
Dolgopyat, D.: On differentiability of SRB states for partially hyperbolic systems. Invent. Math. 155(2), 389–449 (2004)
Jiang, M.: Equilibrium states for lattice models of hyperbolic type. Nonlinearity 8(5), 631-659 (1995)
Jiang, M., de la Llave, R.: Smooth dependence of thermodynamic limits of SRB measures. Commun. Math. Phys. 211, 303–333 (2000)
Jiang, M.: The entropy formula for SRB-measures of lattice dynamical systems. J. Stat. Phys. 95(3–4), 791–803 (1999)
Jiang, M.: Sinai-Ruelle-Bowen measures for lattice dynamical systems. J. Stat. Phys. 111 3/4, 863–902 (2003)
Jiang, M., Pesin, Y.B.: Equilibrium measures for coupled map lattices: existence, uniqueness, and finite-dimensional approximation. Commun. Math. Phys. 193, 675–711 (1998)
Mañé, R.: Ergodic Theory and Differential Dynamics. New York: Springer-Verlag, 1987
Pollicott, M.: Stability of mixing rates for Axiom A attractors. Nonlinearity. 16(2), 567–578 (2003)
Ruelle, D.: Thermodynamic Formalism. In: Encyclopedia of Mathematics and Its Applications, No.5, New York: Addison Wesley, 1978
Ruelle, D.: Differentiation of SRB states. Commun. Math. Phys. 187, 227–241 (1997)
Ruelle, D.: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 95(1-2), 393–468 (1999)
Ruelle, D.: Differentiation of SRB states: correction and complements. Commun. Math. Phys. 234, 185–190 (2003)
Ruelle, D.: Application of hyperbolic dynamics to physics: some problems and conjectures. Bull. Amer. Math. Soc. (N.S.) 41(3), 275–278 (2004) (electronic)
Ruelle, D.: Differentiation of SRB states for hyperbolic flows. http://www.ma.utexas.edu/mp-arc/04-246, 2004
Ruelle, D.: Statistical Mechanics, Reprint of Third Edition, River Edge, NJ: World Scientific, 1999
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J.L. Lebowitz
Supported in part by NSF grants.
Rights and permissions
About this article
Cite this article
Jiang, M., Llave, R. Linear Response Function for Coupled Hyperbolic Attractors. Commun. Math. Phys. 261, 379–404 (2006). https://doi.org/10.1007/s00220-005-1446-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-005-1446-y