Invariant Measures of the Topological Flow and Measures at Infinity on Hyperbolic Groups

The uploaded paper states and proves Theorem 1.1: for any non‑elementary hyperbolic group Γ and strongly hyperbolic metric producing a Hölder cocycle κ, there is a transitive SFT (Σ0,σ) and a positive Hölder roof r0 whose suspension maps onto Fκ, flow‑equivariantly and bounded‑to‑one. The result is explicitly formulated and proved (see Theorem 1.1 and Theorem 4.9, with the construction via Cannon’s automatic structure, the suspension Π, the maximal component selection in Proposition 4.8, and the surjectivity argument using equilibrium states and absolute continuity) . By contrast, the candidate solution hinges on two unsupported steps. First, it asserts a time‑change conjugacy between flow spaces built from different metrics by claiming the Busemann-derived cocycles are cohomologous on ∂2Γ, but the proposed transfer function H(ξ,η) uses βo(e,⋅), which vanishes, so H≡0 and the claimed identity does not hold as written; more importantly, no general cohomology result between Busemann cocycles for different metrics is established in the paper. Second, it claims one can simply restrict to a maximal irreducible component of the CP02 coding and still cover the entire flow space; the paper emphasizes this is nontrivial and precisely the gap it closes by ergodic/pressure arguments to obtain a transitive cover of Fκ (see the discussion around known codings potentially being non‑transitive and the enhanced coding) . Therefore, the paper’s argument is complete and correct, while the model’s proof outline contains critical errors and missing justification.