RENORMALIZATION OF RANDOM JACOBI OPERATORS

We construct a Cantor set J of limit-periodic Jacobi operators having the spectrum on the Julia set J of the quadratic map z bar arrow point ing right z2 + E for large negative real numbers E. The density of sta tes of each of these operators is equal to the unique equilibrium meas ure mu on J. The Jacobi operators in J are defined over the von Neuman n-Kakutani system, a group translation on the compact topological grou p of dyadic integers. The Cantor set J is an attractor of the iterated function system built up by the two renormalisation maps PHI+/- : L = psi(D+/-2 + E) bar arrow pointing right D+/-. To prove the contractio n property, we use an explicit interpolation of the Backlund transform ations by Toda flows. We show that the attractor J is identical to the hull of the fixed point L+ of PHI+.