THE COHOMOLOGICAL EQUATION AND CYCLIC COCYCLES FOR RENORMALIZABLE MINIMAL CANTOR SYSTEMS

The paper proves Theorem 1.1 (finite-dimensional obstructions D_i in H^α with α>2; D_i(f)=0 iff f is a coboundary with u ∈ H^{α−2−ε} and a tame estimate; d_μ = dim Ȟ^1(X̂_B;ℝ)) by building a solenoid Ω_x, computing its leafwise cohomology, and projecting the “bumpified” class to a finite-dimensional space (Proposition 5.1), then deriving traces (Theorem 1.2) via a dense subalgebra W^∞_α stable under holomorphic functional calculus. These statements and their set-up are explicitly in the PDF (Condition 1, the order-bundle shift σ, renormalization, Ω_x ↔ X̂_B, Theorem 1.1/1.2, and the construction of D_i) . The candidate solution outlines an alternative, Bufetov/Forni-style renormalization proof using incidence-matrix cocycles and Oseledets, then constructs invariant distributions from unstable directions and solves the cohomological equation by a tower-parametrix with a 2+ε loss. That approach targets the same result and is broadly consistent with the literature, but contains misstatements that need repair: (i) it identifies Ȟ^1(X̂_B;ℝ) with continuous coinvariants C(X,ℝ)/(g−g∘φ), whereas the paper uses Čech cohomology (via locally constant functions/leafwise cohomology) and proves finiteness by passing to Ω_x; (ii) it asserts dim E^u equals dim Ȟ^1 without justification. With these caveats fixed, the model’s proof strategy can be made consistent with the paper’s result. Hence: both correct, but proofs differ and the model sketch needs amendments.