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Floquet spectrum of weakly coupled map lattices. (English) Zbl 0990.37019
This paper is devoted to weakly coupled analytic expanding circle maps on the lattice \(\mathbb{Z}^d\) (for \(d\geq 1\)), with small coupling strength \(\varepsilon\) and summable decay of the two sites coupling. The authors study the spectrum of the associated transfer operators \(L_\varepsilon\). They manage localization of the full spectrum \(L_\varepsilon^{n_0}\), where \(n_0\) is a high iterate. They show that the time-transfer operator \(L_\varepsilon^{n_0}\) commutes with the spatial translations and yield to a description of part of their joint eigenvalues. As a result the authors exhibit smooth curves of eigenvalues and eigenspaces of \(L_\varepsilon^{n_0}\) as functions of the eigenvalues \(e^{i\alpha}\) of the spatial translation.

37C60 Nonautonomous smooth dynamical systems
37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
47A10 Spectrum, resolvent
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