Critical energy conjecture for analytic quasi-periodic Jacobi operators

Let $\alpha$ be irrational, let $H_\theta$ be a quasi-periodic Jacobi operator with analytic sampling functions, and let $\mu$ denote the relevant spectral measure. For a non-singular operator, call energies with positive Lyapunov exponent supercritical and the remaining critical energies those in the critical regime; for a singular operator, let $\mathcal{Z}$ be the set defined in the global theory.

Critical energy conjecture.

(i) If the Jacobi operator is non-singular, the spectrum restricted to the set of critical energies is purely singular continuous for $\mu$-almost every $\theta$.

(ii) If the Jacobi operator is singular, the spectrum restricted to $\mathcal{Z}$ is purely singular continuous for $\mu$-almost every $\theta$.

Together with the global theory, this conjecture would give a full characterization of the spectral properties of both singular and non-singular analytic quasi-periodic Jacobi operators. The source presents it as an unresolved conjecture; the preceding discussion already establishes that, in the singular case, the absolutely continuous spectrum is empty.