Communications in Mathematical Physics

, Volume 164, Issue 1, pp 195–215 | Cite as

Renormalization of random Jacobi operators

  • Oliver Knill


We construct a Cantor set
of limit-periodic Jacobi operators having the spectrum on the Julia setJ of the quadratic mapzz2+E for large negative real numbersE. The density of states of each of these operators is equal to the unique equilibrium measure μ onJ. 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(D ± 2 +E) ↦D±. To prove the contraction property, we use an explicit interpolation of the Bäcklund transformations by Toda flows. We show that the attractor
is identical to the hull of the fixed pointL+ of Φ±.


Neural Network Complex System Hull Nonlinear Dynamics Function System 
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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Oliver Knill
    • 1
  1. 1.MathematikdepartmentETH ZürichZürichSwitzerland

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