THE FIBERED ROTATION NUMBER FOR ERGODIC SYMPLECTIC COCYCLES AND ITS APPLICATIONS: I. GAP LABELLING THEOREM

The paper proves m(1 − N(E)) = rot_f(E) mod µ(C(Θ,ℤ)) for generalized (matrix-valued) Schrödinger operators by defining the fibered rotation number via the S1×SU(m) covering of U(m), tracking the eigenphases of W_Λ along the Hermitian–symplectic cocycle, proving phase monotonicity in E, identifying Dirichlet eigenvalues through ker U_E(N+1), and then passing to the IDS limit (equation (2.5)); see the theorem statement and setup, the lift and rotation-number framework, the transfer matrix blocks and finite-volume identification, and the proof of Theorem 1.6 in Section 4 . The candidate solution reaches the same conclusion but uses a different route: a direct finite-volume identity Arg det W_L = 2π[mL − N_L(E,θ)] (mod 2πℤ) derived via W_L = (α̂+iγ̂)(α̂−iγ̂)^{-1}, an argument-principle count of zeros/poles tied to det α̂, and then Birkhoff/IDS limits. This differs in technique (argument principle vs. phase monotonicity), but aligns with the paper’s key identifications (Â_E/A_E conjugacy, W_Λ = (X+iY)(X−iY)^{-1}, Dirichlet eigenvalues via ker U_E(N+1), and IDS as the limit of normalized Dirichlet counts) . Minor notes: the paper casually states “Clearly, N is continuous in E,” which is generally not automatic for an arbitrary ergodic family (the IDS is monotone and right-continuous; continuity may need conditions), but this is not used critically in the main proof .