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Ergodic Schrödinger Operators

  • Philippe Bougerol
  • Jean Lacroix
Part of the Progress in Probability and Statistics book series (PRPR, volume 8)

Abstract

We now suppose that (an,bn), n ε ZZ, is a stationary random process This means that the real random variables (an (ω), bn(ω)) are defined on some complete probability space (Ω,a,ℙ) and that there exists an invertible measurable transformation θ of Ω, leaving ℙ invariant and such that an+1=anºθ, bn+1, ºθ. In general we don’t write the variable ω and when a property depends only upon the common law of the sequence (an, bn) we omit the index n and speak of the variables (a) and (b). We say that the family H(ω) of associated operators on H is ergodic if a θ invariant measurable subset of Ω is of zero or one ℙ measure. It’s easily seen that H° θ = U-1 H U where U is the shift measure. It’s easily seen that H o θ = U-1 H U where U is the shift operator on ZZ, (Uψ)n = ψn-1.

Keywords

Lyapunov Exponent Borel Subset Invariant Probability Measure Pure Point Independent Case 
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

© Birkhäuser Boston, Inc. 1985

Authors and Affiliations

  • Philippe Bougerol
    • 1
  • Jean Lacroix
    • 2
  1. 1.UER de MathématiquesUniversité Paris 7ParisFrance
  2. 2.Département de MathématiquesUniversité de Paris XIIIVilletaneuseFrance

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