Global validity conjecture for the quantum Ising quasicrystal spectrum

Let $J_0,J_1>0$, and let $\{\sigma_k\}$ be the sequence of spectra whose Hausdorff limit is studied in the paper. Let $B_\infty(J_0,J_1)$ denote the limiting spectrum when this limit exists. Global validity conjecture. The conclusion of the main theorem holds for all $J_0,J_1>0$: the sequence $\{\sigma_k\}$ converges in the Hausdorff metric to a nonempty compact Cantor set $B_\infty(J_0,J_1)$ with continuous, nonconstant local Hausdorff dimension strictly between zero and one, zero Lebesgue measure, and Hausdorff dimension continuous in $(J_0,J_1)$. The theorem establishes these properties only when $J_1/J_0$ is sufficiently close to $1$ and $J_0\ne J_1$; the conjecture asserts that the parameter restriction is unnecessary.