We consider Jacobi matrices built on equilibrium measures of hyperbolic polynomials. We show their property, which, on one side, is related to almost periodicity of such matrices, and, on the other side, is a sort of noncommutative
Perron-Frobenius-Ruelle theorem. While proving these key property one is naturally brought to consider a two-weight Hilbert transform. Its boundedness can be proved in our situation, while the general two-weight Hilbert transform
boundedness criterion is not yet available.
We will mention other problems in spectral theory of Jacobi matrices, where this paradigm of nonhomogeneous harmonic analysis---two weight Hilbert transform---appears in the natural way.