On rigidity properties of time-changes of unipotent flows

The uploaded paper (Artigiani–Flaminio–Ravotti, 2022) proves a dichotomy for measurable conjugacies between good time-changes of unipotent flows under a renormalizing Cartan: either A-translates of the graph joining converge to the product, or the conjugacy is algebraic up to a centralizer and a measurable U-translation. The candidate solution follows the same renormalize-and-shrink scheme: (i) compactness of A-translates of the graph joining, (ii) U×U-invariance of any weak-* limit using small-time control from quantitative mixing (derived from spectral gap), (iii) Ratner-type classification to obtain a homogeneous limit supported on the graph of an isomorphism after passing to finite index, and (iv) an upgrade from the limit joining back to the original conjugacy, yielding the affine form with centralizer factor. This mirrors the paper’s proof of Theorem A, including the key Lemma that limit points are U×U-invariant and the subsequent use of Ratner’s joinings/measure classification to produce an algebraic map ζ, followed by the final decomposition ψ = ζ·c(·)·u2^{t(·)}. One minor quibble: the candidate’s description suggests the centralizer could appear already in the homogeneous subgroup L for the limit joining; in the paper, the limit joining lives on the graph of an isomorphism, and the centralizer enters only when reconstructing the original conjugacy from the limit. Aside from this nuance, the logic and hypotheses align with the paper’s, and the arguments are essentially the same scheme.