  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.
