Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts

The paper proves that Γ-strong continuous orbit equivalence is equivalent to eventual conjugacy for irreducible, non-permutation 0–1 matrices (Theorem 6.6), via an explicit construction: starting from Γ-scoe, it produces τ2 = h ◦ τ1^{-1} ◦ h^{-1} and postcomposes h by τ2 to obtain a new orbit equivalence whose forward cocycle is identically 1, which then yields eventual conjugacy (Proposition 6.3, Lemma 6.4, Proposition 6.5, Theorem 6.6) . By contrast, the candidate solution precomposes by τ∈Γ_A and asserts a specific transformation law for the forward cocycle that forces it to be 1; this step is incorrect and internally inconsistent (they alternately write c1' = c1∘τ + … and c1' = c1 + … without justification). The correct adjustment, as done in the paper, is to postcompose by τ2 on the B-side, not to precompose on the A-side .