Renormalization of Random Jacobi Operators

We construct a Cantor set ̂ of limit-periodic Jacobi operators having the spectrum on the Julia set J of the quadratic map z ι-> z + E for large negative real numbers E. The density of states of each of these operators is equal to the unique equilibrium measure μ on J. The Jacobi operators in $ are defined over the von Neumann-Kakutani system, a group translation on the compact topological group of dyadic integers. The Cantor set $ is an attractor of the iterated function system built up by the two renormalisation maps Φ± : L = ψ(D2L + E) ι-> D±. To prove the contraction property, we use an explicit interpolation of the Backlund transformations by Toda flows. We show that the attractor $ is identical to the hull of the fixed point