Weak Mixing Property for Linear Involutions

The paper proves that, for any dynamically irreducible generalized permutation π and Lebesgue-a.e. admissible length λ with g(S)>1, the associated linear involution T(π,λ) is weakly mixing, by adapting the Avila–Forni scheme to linear involutions via a Veech-type criterion, the Rauzy–Veech/Zorich renormalization on generalized permutations, and the minus (anti-invariant) Kontsevich–Zorich subbundle on the orienting double cover. This is stated as Theorem 2 and carried out using Zariski density/simplicity and uniform cocycle machinery (Lemma 2, Theorem 4, Lemma 4) to eliminate eigenvalues for almost every parameter, closing with a two-case argument based on H(π) and its orthogonal complement . The candidate solution asserts the same theorem and follows the same architecture: BL renormalization, identification with the KZ cocycle on the anti-invariant part, non-uniform hyperbolicity/simplicity, and Avila–Forni’s eigenvalue-elimination. Minor wording differences aside (e.g., citing EKZ for positivity versus the paper’s use of Treviño and BDG+21 for Zariski density), the approaches are essentially the same.