Genericity of exponential dichotomy for two-dimensional differential systems
Consider the two-dimensional, quasi-periodic, traceless linear system \( {\dot x} = A( \psi + \gamma t ) x \), where \( x \in \mathbb{R}^2 \), \(\gamma \in\mathbb{R}^k \) is the frequency vector, \( A : {\mathbf T}^k \to \text{sl}(2, \mathbb{R})\) is continuous, \({\mathbf T}^k :=\mathbb{R}^k /2 \pi \mathbb{Z}^k \) is the \(k\)-dimensional torus and \( \text{sl} ( 2, \mathbb{R})\) are the real matrices with zero trace. The authors show the genericity of such systems exhibiting exponential dichotomy (or hyperbolic splitting) in the \(C^0\)-topology. The dichotomy property is of great theoretical interest and it is closely related with spectral properties of certain linear differential operators, like quasi-periodic Schrödinger operators, see, e.g., \textit{R. Johnson} and \textit{J. Moser} [Commun. Math. Phys. 84, 403--438 (1982); erratum ibid. 90, 317--318 (1983; Zbl 0497.35026)] and \textit{J. Moser} [Commun. Math. Helv. 56, 198-224 (1981; Zbl 0477.34018)]
scientific article; zbMATH DE number 1993709
scientific article