One-dimensional quantum Hamiltonian operator $H =T+V$ on
$\ell^2(\mathbb{Z})$ is assumed composed of a discrete
(negative) Laplacean $T=-\triangle$ and of a real-valued
quasiperiodic local potential $V(n)=f(\omega\,n+\theta)$ with
1-periodic $f(x)$ and integers $n=\ldots, -1,0,1,\ldots$. An
amazingly extensive literature dating back to Aubry and Andre
in 1980 is reviewed showing how exciting various subtleties of
the underlying spectral problem are. What is assumed is the
positivity of the Lyapunov exponent, with the purpose of
``roughening'' (i.e., of successfully relaxing the regularity)
of $f(x)$ from its most common analyticity down to the
$\gamma-$H\"{o}lder continuity with $\gamma>1/2$. What is
studied are the measures of spectra (as sets) and what is
proved is that at irrational $\omega$s, Diofantine or not,
these measures are limits of measures of spectra of periodic
approximants with rational $\omega$s. In the absence of
continuity of the Lyapunov exponents (weakened, in a way, to a
uniform upper-semicontinuity) the proof is based on the
ergodicity ideas in connection with the recurrent
transfer-matrix generation of the wave-function components (one
speaks about - iterated - cocycles) yielding the key property
of uniformity (in phases and neighborhoods) of the convergence
from above.

MR3148097  Jitomirskaya, Svetlana; Mavi, Rajinder
Continuity of the measure of the spectrum for quasiperiodic
Schrödinger operators with rough potentials.
Comm. Math. Phys. 325 (2014), no. 2, 585--601. 47A10 (47F05)