We study the correspondence between almost periodic difference operators and algebraic curves (spectral surfaces). An especial role plays the parametrization of the spectral curves in terms of, so called, branching divisors. The multiplication operator by the covering map with respect to the natural basis in the Hardy space on the surface is the $2d+1$--diagonal matrix; the $d$--root of the product of the Green functions (counting their multiplicities) with respect to all infinite points on the surface is the symbol of the shift operator. We demonstrate an application of our general construction to a particular covering, which generate widely discussed almost periodic CMV matrices. We discuss an important theme: covering of one spectral surface by another one and related to this operation transformations on the set of multidiagonal operators (so called Renormalization Equations). We proof several new results dealing with Renormalization Equations for periodic Jacobi matrices (polynomial coverings) and the case of a rational double covering.
Band structure of the Ruelle spectrum of contact Anosov flows
若X是光滑紧致流形M上的切触Anosov向量场，V是M上的光滑函数，则已知微分算子A=-X+V在特定的索伯列夫空间中具有称为Ruelle-Pollicott共振的离散谱.我们证明，|Im（z）|大的A的特征值被限制到垂直带，并且在带之间的间隙中，A的预解式关于|Im（z）|.在每一个孤立的频带中，本征值的密度由外尔定律给出。在第一波段，大部分特征值集中在< D >函数D（x）=V（x）-1/2div（X）/E_u的空间平均值的垂直线Re（z）=上，其中Eu为不稳定分布。该谱给出了动力学相关函数的渐近展开式。
