Local linear operators and multicontour solutions of homogeneous coupled map lattices

Theorems are proved establishing a relationship between the spectra of the linear operators of the formA+Ωg _iB_ig_i ⁻¹ andA+B _i, whereg _i∈G, andG is a group acting by linear isometric operators. It is assumed that the closed operatorsA andB _i possess the following property: ‖B _iA⁻¹gB_jA⁻¹‖→0 asd(e,g)→∞. Hered is a left-invariant metric onG ande is the unit ofG. Moreover, the operatorA is invariant with respect to the action of the groupG. These theorems are applied to the proof of the existence of multicontour solutions of dynamical systems on lattices. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 37–47, January, 1999.