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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
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