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Renormalization of random Jacobi operators

Abstract

We construct a Cantor set

of limit-periodic Jacobi operators having the spectrum on the Julia setJ of the quadratic mapzz 2+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±.

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Communicated by B. Simon

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Knill, O. Renormalization of random Jacobi operators. Commun.Math. Phys. 164, 195–215 (1994). https://doi.org/10.1007/BF02108812

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Keywords

  • Neural Network
  • Complex System
  • Hull
  • Nonlinear Dynamics
  • Function System