Anosov Vector Fields and Fried Sections

The paper proves existence of a canonical nonzero section τ_ν(iZ) of ν = det H(Y,F), its C^1 variation in families, and Gauss–Manin flatness when ∇^F is flat on M (Theorem 1.1), using truncated resonant spaces DZ,<a, the identity [i_Z, α∧]=1, and Fried/Ruelle zeta to glue across truncations and avoid rank-jump pathologies in degree-0 resonances . In contrast, the model asserts C^1 variation of the zero-resonant projector Π_0(s) and bundles E_0(s) across families, which the paper explicitly warns need not hold (the 0-resonant dimensions may fail to form a vector bundle) . The model’s single-fiber use of α with [i_Z, α∧]=1 to split the i_Z-complex on resonant states aligns with the paper’s algebraic step (cf. (6.103), (6.107)) , but its family claims bypass the essential zeta-function gluing and Gauss–Manin comparison developed in Sections 7–8 , yielding an incorrect conclusion about smooth/parallel dependence.