标题： Cantor谱对于一类$C^2$似周期薛定谔算子 显示英文标题

标题： Cantor spectrum for a class of $C^2$ quasiperiodic Schrödinger operators

摘要： 我们证明了对于一类$C^2$似周期势，以及任何狄利克雷频率，相应的薛定谔算子的谱都是康托尔集。我们的方法是纯动力系统的方法，它依赖于对渐近稳定和不稳定方向的详细分析。我们还将它应用于一般的$\mathrm{SL}(2,\mathbb R)$系统，得到在某些单参数族中，一致双曲系统形成一个开稠密集。 显示英文摘要

摘要： We show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the spectrum of the corresponding Schr\"odinger operators is Cantor. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to general $\mathrm{SL}(2,\mathbb R)$ cocycles, and obtain that uniform hyperbolic systems form a open and dense set in some one-parameter family.