Spectrum of Lebesgue Measure Zero for Jacobi Matrices of Quasicrystals

Abstract

We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. We characterize the spectrum of these operators via non-uniformity of the transfer matrices and vanishing of the Lyapunov exponent. For aperiodic, minimal subshifts satisfying the so-called Boshernitzan condition this gives that the spectrum is supported on a Cantor set with Lebesgue measure zero. This generalizes earlier results for Schrödinger operators.

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Correspondence to Siegfried Beckus.

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Beckus, S., Pogorzelski, F. Spectrum of Lebesgue Measure Zero for Jacobi Matrices of Quasicrystals. Math Phys Anal Geom 16, 289–308 (2013). https://doi.org/10.1007/s11040-013-9131-4

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Keywords

  • Jacobi operator
  • Cantor spectrum
  • Lyapunov exponent
  • Dynamical system
  • Aperiodic subshift

Mathematics Subject Classification (2010)

  • 81Q10
  • 47B80
  • 37A30
  • 52C23