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Some remarks on the dynamics of the almost Mathieu equation at critical coupling. (English) Zbl 1443.37026

Summary: We show that the quasi-periodic Schrödinger cocycle with a continuous potential is of parabolic type, with a unique invariant section, at all gap edges where the Lyapunov exponent vanishes. This applies, in particular, to the almost Mathieu equation with critical coupling. It also provides examples of real-analytic cocycles having a unique invariant section which is not smooth.

MSC:

37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
39A70 Difference operators
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