# Ergodic Schrödinger Operators

## Abstract

We now suppose that (a_{n},b_{n}), n ε ZZ, is a stationary random process This means that the real random variables (a_{n} (ω), b_{n}(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 a_{n+1}=a_{n}ºθ, b_{n+1}, ºθ. In general we don’t write the variable ω and when a property depends only upon the common law of the sequence (a_{n}, b_{n}) 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} = ψ_{n-1}.

## Keywords

Lyapunov Exponent Borel Subset Invariant Probability Measure Pure Point Independent Case## Preview

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