LAUSR

In this paper, we consider the spectrum of discrete quasi-periodic Schrödinger operators on $\ell ^{2}(\mathbb{Z})$ with the potentials $v\in C^{k}(\mathbb{T})$ . For sufficiently large $k$ , we show that the Lebesgue measure of the spectrum at irrational frequencies is the limit of the Lebesgue measure of the spectrum of its periodic approximants. This gives a partial answer to the problem proposed in Jitomirskaya and Mavi [Continuity of the measure of the spectrum for quasiperiodic schrödinger operator with rough potentials. Comm. Math. Phys. 325 (2014), 585–601]. Our results are based on a generalization of the rigidity theorem in Avila and Krikorian [Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. of Math. (2) 164 (2006), 911–940]; more precisely, we prove that in the $C^{k}$ case, for almost every frequency $\unicode[STIX]{x1D6FC}\in \mathbb{R}\backslash \mathbb{Q}$ and for almost every $E$ , the corresponding quasi-periodic Schrödinger cocycles are either reducible or non-uniformly hyperbolic.