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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.

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