Abstract
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.
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Recommended by Dr Irina Nenciu
Footnotes
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Dedicated to the memory of Russel A Johnson.
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Of course there are plenty of examples of real-analytic cocycles with two 'highly' discontinuous invariant sections (Oseledets' directions); one attracting the forward iterations and the other one attracting the backward iterations. See, e.g., [20] and references therein.