Smooth rigidity for 3-dimensional volume preserving Anosov flows and weighted marked length spectrum rigidity

The paper’s main theorem (Theorem 1.1) states exactly the dichotomy claimed by the model: for C^r (r>2) 3D volume-preserving Anosov flows that are C^0 time-conjugate, either the conjugacy is C^{r*} or both flows are constant-roof suspensions over Anosov automorphisms of T^2 on a mapping torus. This is stated verbatim in the introduction and §1.3 of the paper and framed as an improvement over classical de la Llave–Moriyón/Pollicott smooth rigidity which required periodic eigenvalue matching; the new result removes that assumption except in the suspension case . The proof strategy in the paper uses a local tetrachotomy (Theorem 3.1) together with an Alternate/Positive-Proportion Livšic theorem and the Foulon–Hasselblatt dichotomy to force either the ‘A-cocycle’ (unstable Jacobian discrepancy) to be a coboundary (yielding periodic exponent matching and then C^{r*} via de la Llave–Marco–Moriyón–Pollicott), or else to reduce to the contact/constant-roof case, where contact implies C^{r*} (Prop. 3.2) and constant roof implies both are constant-roof suspensions . The model’s solution outlines the same chain (periodic data criterion + Livšic–Journé + the paper’s new rigidity), differing mainly in presentation: it phrases the new contribution as ‘automatic periodic-data matching unless the suspension obstruction,’ which is essentially the net effect of the paper’s argument. Minor imprecision aside, the model’s solution tracks the paper’s logic and conclusions closely.