Aubry's completeness conjecture for twist maps

Let $F$ be a twist map, let $c_1<c_2$, and let $\Lambda_{[c_1,c_2]}$ be the set of orbits corresponding to minimal configurations with $c\in[c_1,c_2]$. Let $\alpha$ denote the relevant cohomology function, so that $D\alpha:[c_1,c_2]\to\mathbb{R}$ is the graph of rotation numbers as a function of cohomology classes. A function is purely singularly continuous if it is continuous, nonconstant, and has derivative vanishing almost everywhere. Aubry's completeness conjecture. If $\Lambda_{[c_1,c_2]}$ is uniformly hyperbolic, then the graph of $D\alpha:[c_1,c_2]\to\mathbb{R}$ is purely singularly continuous. This is Aubry's complete devil's staircase conjecture, expressing that the rotation-number graph has no intervals of positive derivative; the paper states that the conjecture is proved under the stated uniform hyperbolicity hypothesis.