A Dynamical Thouless Formula

The paper proves a dynamical Thouless formula for affine GL(2,R) cocycles via a real-variable, root-counting and winding-length argument, culminating in an equidistribution of polynomial zeros and a subharmonic-limit identification of L1(At) with the log-potential of the rotation-number measure (Theorem 1.1, proof around eq. (1.4)) . The candidate model presents a different potential-theoretic route based on subharmonicity, a claimed harmonicity off the real axis, and Avila–Krikorian’s “monotonic cocycles” framework to produce a complex rotation number and identify ΔL1 with 2π dρ. The conclusions match the paper’s main result. However, the model’s Step 2 invokes monotonic-cocycle complexification and holomorphic conjugacy in a general ergodic-base setting that is not established in the paper; the paper deliberately avoids this machinery and proves the result by purely geometric and potential-theoretic means (e.g., Proposition 2.23 and Lemma 2.14) . Still, no contradiction arises: the model’s outline would yield the same formula in regimes where the cited complex-analytic theory is valid. Thus, the paper is correct, and the model offers a different proof sketch that is plausible but relies on additional nontrivial external inputs not verified within the paper’s generality.