Communications in Mathematical Physics

, Volume 164, Issue 2, pp 421–432 | Cite as

Zero measure spectrum for the almost Mathieu operator

Article

Abstract

We study the almost Mathieu operator: (H α, λ, θ u)(n)=u(n+1)+u(n-1)+λ cos (2παn+θ)u(n), onl2(Z), and show that for all λ,θ, and (Lebesgue) a.e. α, the Lebesgue measure of its spectrum is precisely |4–2|λ‖. In particular, for |λ|=2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational α's (and |λ|=2) we show that the Hausdorff dimension of the spectrum is smaller than or equal to 1/2.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Lebesgue Measure 

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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Y. Last
    • 1
  1. 1.Department of PhysicsTechnion-Israel Institute of TechnologyHaifaIsrael

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