Ergodic Schrödinger Operators
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.
KeywordsLyapunov Exponent Borel Subset Invariant Probability Measure Pure Point Independent Case
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