zbMATH — the first resource for mathematics

Infinite-dimensional SRB measures. (English) Zbl 1194.82061
Summary: We review the basic steps leading to the construction of a Sinai-Ruelle-Bowen (SRB) measure for an infinite lattice of weakly coupled expanding circle maps, and we show that this measure has exponential decay of space-time correlations. First, using the Perron-Frobenius operator, one connects the dynamical system of coupled maps on a \(d\)-dimensional lattice to an equilibrium statistical mechanical model on a lattice of dimension \(d + 1\). This lattice model is, for weakly coupled maps, in a high-temperature phase, and we use a general, but very elementary, method to prove exponential decay of correlations at high temperatures.

82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
Full Text: DOI
[1] Blank, M., Singular effects in chaotic dynamical systems, Russian acad. sci. dokl. math., 47, 1-5, (1993) · Zbl 0820.58039
[2] Bowen, R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms, () · Zbl 0308.28010
[3] Bricmont, J.; Kupiainen, A., Coupled analytic maps, Nonlinearity, 8, 379-396, (1995) · Zbl 0836.58027
[4] Bricmont, J.; Kupiainen, A., High temperature expansions and dynamical systems, Commun. math. phys., 178, 703-732, (1996) · Zbl 0859.58037
[5] Brydges, D.C., A short course on cluster expansions, (), 139-183
[6] Bunimovich, L.A.; Sinai, Y.G., Space-time chaos in coupled map lattices, Nonlinearity, 1, 491-516, (1988) · Zbl 0679.58028
[7] P.Collet, Some ergodic properties of maps of the interval, lectures at the CIMPA Summer School “Dynamical Systems and Frustrated Systems”, to appear.
[8] Dobrushin, R.L., Gibbsian random fields for lattice systems with pairwise interactions, Funct. anal. appl., 2, 292-301, (1968) · Zbl 1183.82023
[9] Dobrushin, R.L., The description of a random field by means of conditional probabilities and conditions on its regularity, Theory probab. appl., 13, 197-224, (1968) · Zbl 0184.40403
[10] von Dreifus, H.; Klein, A.; Perez, J.F., Taming Griffiths’ singularities: infinite differentiability of quenched correlation functions, Comm. math. phys., 170, 21-39, (1995) · Zbl 0820.60086
[11] Eckmann, J.P.; Ruelle, D., Ergodic theory of chaos and strange atrractors, Rev. mod. phys., 57, 617-656, (1985) · Zbl 0989.37516
[12] van Enter, A.C.D.; Fernandez, R.; Sokal, A., Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory, J. statist. phys., 72, 879-1167, (1993) · Zbl 1101.82314
[13] Fisher, M., Critical temperatures for anisotropic Ising lattices. II. general upper bounds, Phys. rev., 162, 480-485, (1967)
[14] Gross, L., Decay of correlations in classical lattice models at high temperature, Comm. math. phys., 68, 9-27, (1979) · Zbl 0442.60097
[15] Israel, R.B., High-temperature analyticity in classical lattice systems, Comm. math. phys., 50, 245-257, (1976)
[16] Jiang, M.; Jiang, M., Equilibrium states for lattice models of hyperbolic type, (), 8, 631-659, (1995), see also · Zbl 0836.58032
[17] M. Jiang and A. Mazel, Uniqueness of Gibbs states and exponential decay of correlations for some lattice models, preprint. · Zbl 1042.82520
[18] ()
[19] ()
[20] Katok, A.; Hasselblatt, B., Introduction to the modern theory of dynamical systems, (1995), Cambridge University Press Cambridge · Zbl 0878.58020
[21] Keller, G.; Künzle, M.; Künzle, M., Transfer operators for coupled map lattices, (), 12, 297-318, (1992), see also · Zbl 0737.58032
[22] Miller, J.; Huse, D.A., Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled-map lattice, Phys. rev. E, 48, 2528-2535, (1993)
[23] Pesin, Y.G.; Sinai, Y.G., Space-time chaos in chains of weakly coupled hyperbolic maps, () · Zbl 1187.37051
[24] Pomeau, Y., Periodic behaviour of cellular automata, J. statist. phys., 70, 1379-1382, (1993) · Zbl 0936.37002
[25] Ruelle, D., Thermodynamic formalism, (1978), Addison-Wesley Reading, MA
[26] Simon, B., ()
[27] Sinai, Y.G., Gibbs measures in ergodic theory, Russian math. surveys, 27, 21-64, (1972) · Zbl 0246.28008
[28] Sinai, Y.G., Topics in ergodic theory, (1994), Princeton University Press Princeton, NJ · Zbl 0805.58005
[29] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, (1988), Springer New York · Zbl 0662.35001
[30] Volevich, D.L., Kinetics of coupled map lattices, Nonlinearity, 4, 37-45, (1991) · Zbl 0778.58043
[31] Volevich, D.L., The Sinai-Bowen-Ruelle measure for a multidimensional lattice of interacting hyperbolic mappings, Russian acad. dokl. math., 47, 117-121, (1993) · Zbl 0823.58025
[32] Volevich, D.L., Construction of an analogue of Bowen-Ruelle-Sinai measure for a multidimensional lattice of interacting hyperbolic mappings, Russian acad. math. sbornik, 79, 347-363, (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.