Embedding of Analytic Quasi-Periodic Cocycles into Analytic Quasi-Periodic Linear Systems and its Applications

In this paper, we prove that any analytic quasi-periodic cocycle close to constant is the Poincar\'{e} map of an analytic quasi-periodic linear system close to constant. With this local embedding theorem, we get fruitful new results. We show that the almost reducibility of an analytic quasi-periodic linear system is equivalent to the almost reducibility of its corresponding Poincar\'e cocycle. By the local embedding theorem and the equivalence, we transfer the recent local almost reducibility results of quasi-periodic linear systems \cite{HoY} to quasi-periodic cocycles, and the global reducibility results of quasi-periodic cocycles \cite{A,AFK} to quasi-periodic linear systems. Finally, we give a positive answer to a question of \cite{AFK} and use it to prove Anderson localization results for long-range quasi-periodic operator with Liouvillean frequency, which gives a new proof of \cite{AJ05,AJ08,BJ02}. The method developed in our paper can also be used to prove some nonlinear local embedding results.Comment: 28 pages, no figur

Examples of Discontinuity of Lyapunov Exponent in Smooth Quasi-Periodic Cocycles

We study the regularity of the Lyapunov exponent for quasi-periodic cocycles $(T_\omega, A)$ where $T_\omega$ is an irrational rotation $x\to x+ 2\pi\omega$ on \SS^1 and A\in {\cal C}^l(\SS^1, SL(2,\mathbb{R})), $0\le l\le \infty$. For any fixed $l=0, 1, 2, \cdots, \infty$ and any fixed $\omega$ of bounded-type, we construct D_{l}\in {\cal C}^l(\SS^1, SL(2,\mathbb{R})) such that the Lyapunov exponent is not continuous at $D_{l}$ in ${\cal C}^l$-topology. We also construct such examples in a smaller Schr\"odinger class.Comment: 43pages, 2 figure

Dry Ten Martini Problem in the non-critical case

We solve the Dry Ten Martini Problem in the non-critical case, i.e., all possible spectral gaps are open for almost Mathieu operators with $\lambda\ne \pm 1$