Quasi-Periodic Schrödinger Cocycles with Positive Lyapunov Exponent are not Open in the Smooth Topology

Abstract

One knows that the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is open and dense in the analytic topology. In this paper, we construct cocycles with positive Lyapunov exponent which can be approximated by ones with zero Lyapunov exponent in the space of ${\mathcal{C}}^l$ ($1\le l\le \infty$) smooth quasi-periodic cocycles. As a consequence, the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is not ${\mathcal{C}}^l$ open.

arXiv comment45 pages, 3 figures

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Cited by in corpus (2)

• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Cantor spectrum for a class of $C^2$ quasiperiodic Schrödinger operators