The Julia-set inclusion for the magnetic Laplacian spectrum on the infinite Sierpinski gasket

Let $G_\infty$ be the infinite Sierpinski gasket lattice, let $\mathcal{U}$ be the spectral-decimation map, and let $\mathcal{J}(\mathcal{U})$ denote its Julia set. For parameters $(\alpha,\beta)\in\mathbb{T}^2$, let $S^\infty_1(\alpha,\beta)$ be the first spectral component in the decomposition of $\sigma(\mathcal{L}^{(\alpha,\beta)}_\infty)$. Julia-set inclusion. One has

$$S^\infty_1(\alpha,\beta) \supset \mathcal{J}(\mathcal{U}) \cap ((\alpha,\beta)\times\mathbb{R}).$$

This inclusion identifies the Julia-set slice as part of the magnetic spectrum and underlies the proposed Hofstadter–Sierpinski butterfly. The supplied context states the surrounding spectral decomposition as a theorem, but gives no resolution status for this inclusion.