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

2015/01/22 by Wang, Yiqian, You, Jiangong
#37D25 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1501.05380

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 \cal C^ l (1 ≤ l≤ ∞) smooth quasi-periodic cocycles. As a consequence, the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is not \cal C^ l open.

Related

• Sharp Holder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles
• Density of positive Lyapunov exponents for symplectic cocycles
• Density of positive Lyapunov exponents for SL(2,R) cocycles
• Hölder continuity of Lyapunov exponent for a family of smooth Schrödinger cocycles
• Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles