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Title:
Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles
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Abstract:
The purpose of this talk is to describe a recent result on the continuity of the Lyapunov exponents for analytic quasi-periodic cocycles. The new feature of this work is extending the availability of such results to cocycles that are identically singular (i.e. non-invertible anywhere), in the several variables torus translation setting. This feature is exactly what allows us, through a simple limiting argument, to obtain criteria for the positivity and simplicity of the Lyapunov exponents of such cocycles. Specializing to the family of cocycles corresponding to a block Jacobi operator, we derive consequences on the continuity, positivity and simplicity of its Lyapunov exponents, and on the continuity of its integrated density of states.
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