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Communications in Mathematical Physics

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

Renormalization of random Jacobi operators

  • Oliver Knill
Article
  • 48 Downloads

Abstract

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±.

Keywords

Neural Network Complex System Hull Nonlinear Dynamics Function System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Oliver Knill
    • 1
  1. 1.MathematikdepartmentETH ZürichZürichSwitzerland

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