Beyond Bowen’s Specification Property

Both the paper and the candidate solution correctly aim to apply the Climenhaga–Thompson scheme: build a good collection G from center Birkhoff averages, prove specification on G and the Bowen property for Hölder φ, and show a strict pressure gap for the obstruction collections. The paper states Theorem 10.11 and points to the decomposition (9.3) and hyperbolicity estimate (9.4), then invokes the general uniqueness theorem for maps (Theorem 10.9) to conclude uniqueness when P+ ≠ P− , , . The model’s proof follows the same blueprint. However, both leave a key hypothesis of Theorem 10.9 unverified in the general Hölder case: the pressure of obstructions to expansivity P⊥exp(φ, ε) must be strictly less than P(φ). The notes assert that beyond §9 properties, only the Bowen property on G is additionally needed for Theorem 10.11, but they do not spell out how the h⊥exp control from §9 upgrades to a bound on P⊥exp(φ, ε) for arbitrary Hölder φ; the model omits this condition entirely . The rest of the ingredients (decomposition G via (9.3), hyperbolicity (9.4), and Bowen-on-G via Remark 10.10/Prop. 10.6) are consistent and well aligned with the paper , , .