The paper describes relations between Liouville type theorems for solutionsof a periodic elliptic equation (or a system) on an abelian cover of a compactRiemannian manifold and the structure of the dispersion relation for thisequation at the edges of the spectrum. Here one says that the Liouville theoremholds if the space of solutions of any given polynomial growth is finitedimensional. The necessary and sufficient condition for a Liouville typetheorem to hold is that the real Fermi surface of the elliptic operatorconsists of finitely many points (modulo the reciprocal lattice). Thus, such atheorem generically is expected to hold at the edges of the spectrum. Theprecise description of the spaces of polynomially growing solutions dependsupon a `homogenized' constant coefficient operator determined by the analyticstructure of the dispersion relation. In most cases, simple explicit formulasare found for the dimensions of the spaces of polynomially growing solutions interms of the dispersion curves. The role of the base of the covering (inparticular its dimension) is rather limited, while the deck group is of themost importance. The results are also established for overdetermined elliptic systems, whichin particular leads to Liouville theorems for polynomially growing holomorphicfunctions on abelian coverings of compact analytic manifolds. Analogous theorems hold for abelian coverings of compact combinatorial orquantum graphs.
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