Localization as a consequence of quasiperiodic bulk-bulk correspondence

We report on a direct connection between quasi-periodic topology and the Almost Mathieu (Andre-Aubry) metal insulator transition (MIT). By constructing transfer matrix equations from limit of rational approximate projected Green's functions, we relate results $\text{SL}(2,\mathbb{R})$ co-cycle theory (transfer eigenvalue scaling) to consequences band theory. This reduction links eigenfunction localization MIT chiral edge modes Hofstadter Hamiltonian, implying localized phase roots in topological "bulk-bulk" correspondence, bulk-boundary correspondence 1D AAH system (boundary) its 2D parent Hamiltonian (bulk). differentiates Anderson disordered systems. Our are widely applicable systems beyond this paradigmatic model.