zbMATH — the first resource for mathematics

The spectrum of weakly coupled map lattices. (English) Zbl 1023.37009
Summary: We consider weakly coupled analytic expanding circle maps on the lattice $$\mathbb{Z}^d$$ (for $$d\geq 1$$), with small coupling strength $$\varepsilon$$ and coupling between two sites decaying exponentially with the distance. We study the spectrum of the associated (Perron-Frobenius) transfer operators. We give a Fréchet space on which the operator associated to the full system has a simple eigenvalue at 1 (corresponding to the SRB measure $$\mu_\varepsilon$$ previously obtained by J. Bricmont and A. Kupiainen [Nonlinearity 8, 379-396 (1995; Zbl 0836.58027)]) and the rest of the spectrum, except maybe for continuous spectrum, is inside a disc of radius smaller than one. For $$d=1$$ we also construct Banach spaces of densities with respect to $$\mu_\varepsilon$$ on which perturbation theory, applied to the difference of fixed high iterates of the normalized coupled and uncoupled transfer operators, yields localization of the full spectrum of the coupled operator (i.e., the first spectral gap and beyond). As a side-effect, we show that the whole spectra of the truncated coupled transfer operators (on bounded analytic functions) are $${\mathcal O}(\varepsilon)$$-close to the truncated uncoupled spectra, uniformly in the spatial size. Our method uses polymer expansions and also gives the exponential decay of time-correlations for a larger class of observables than those considered in [loc. cit.].

MSC:
 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
Full Text:
References:
 [1] Baladi, V., Periodic orbits and dynamical spectra, Ergodic theory dynamical systems, (1997), Preprint to appear [2] Bricmont, J.; Kupiainen, A., Coupled analytic maps, Nonlinearity, 8, 379-396, (1995) · Zbl 0836.58027 [3] Bricmont, J.; Kupiainen, A., High temperature expansions and dynamical systems, Comm. math. phys., 178, 703-732, (1996) · Zbl 0859.58037 [4] Bunimovich, L.A., Coupled map lattices: one step forward and two two steps back, Phys. D (chaos, order and patterns: aspects of nonlinearity—the “gran finale,” Como 1993), 86, 248-255, (1995) · Zbl 0890.58029 [5] Bunimovich, L.A.; Sinai, Ya.G., Spacetime chaos in coupled map lattices, Nonlinearity, 1, 491-516, (1988) · Zbl 0679.58028 [6] Dunford, N.; Schwartz, L., Linear operators, (1957), John Wiley & Sons New York, Part One [7] Fischer, T.; Rugh, H.H., Transfer operators for coupled analytic maps, (1997), Preprint [8] Järvenpää, E., A SRB-measure for globally coupled circle maps, Nonlinearity, 10, 1435-1469, (1997) · Zbl 0908.58031 [9] Jiang, M., Equilibrium states for lattice models of hyperbolic type, Nonlinearity, 8, 631-659, (1995) · Zbl 0836.58032 [10] Jiang, M.; Pesin, Ya.B., Equilibrium measures for coupled map lattices: existence, uniqueness and finite-dimensional approximations, Comm. math. phys., (1997), Preprint, to appear [11] () [12] Kato, T., Perturbation theory for linear operators, (1976), Springer-Verlag Berlin [13] Keller, G.; Künzle, M., Transfer operators for coupled map lattices, Ergodic theory dynamical systems, 12, 297-318, (1992) · Zbl 0737.58032 [14] Krzyzewski, K.; Szlenk, W., On invariant measures for expanding differentiable mappings, Studia math., 33, 83-92, (1969) · Zbl 0176.00901 [15] Maes, C.; Van Moffaert, A., Stochastic stability of weakly coupled lattice maps, Nonlinearity, 10, 715-730, (1997) · Zbl 0962.82010 [16] Mayer, D., Continued fractions and related transformations, (), 83-92 [17] Pesin, Ya.G.; Sinai, Ya.G., Space-time chaos in chains of weakly interacting hyperbolic mappings, dynamical systems and statistical mechanincs (Moscow, 1991), Amer. math. soc., (1991), Providence, RI · Zbl 0850.70250 [18] Rudin, W., Real and complex analysis, (1985), Tata Mc Graw-Hill New Delhi, India · Zbl 0613.26001 [19] Ruelle, D., Zeta functions for expanding maps and Anosov flows, Invent. math., 34, 231-242, (1976) · Zbl 0329.58014 [20] Ruelle, D., An extension of the theory of Fredholm determinants, Inst. hautes études sci. publ. math., 72, 175-193, (1990) · Zbl 0732.47003 [21] Simon, B., () [22] Trèves, F., Topological vector spaces, distributions and kernels, (1967), Academic Press New York · Zbl 0171.10402 [23] Volevich, V.L., Construction of an analogue of the Bowen-Ruelle-Sinai measure for a multidimensional lattice of interacting hyperbolic mappings, Mat. sb., 17-36, (1993) · Zbl 0823.58025 [24] Wilson, R., Introduction to graph theory, (1985), Longman Inc New York [25] Yosida, K., Functional analysis, (1980), Springer Verlag Berlin Heidelberg New York · Zbl 0217.16001 [26] Young, L.-S., Ergodic theory of chaotic dynamical systems, Notices amer. math. soc., (1997), to appear
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.