  Our main result asserts that a certain natural non-linear operator on Jacobi
matrices built by a hyperbolic polynomial with real Julia set is a contraction
in operator norm if the polynomial is sufficiently hyperbolic. This allows us
to get for such polynomials the solution of a problem of Bellissard, in other
words, to prove the limit periodicity of the limit Jacobi matrix. This fact
does not require the iteration of the same fixed polynomial, and therefore it
gives a wide class of limit periodic Jacobi matrices with singular continuous
spectrum.
