Ergodic Optimization Restricted on Certain Subsets of Invariant Measures

The paper’s Theorem A establishes, under transitivity, shadowing, and upper semicontinuity of the entropy map, a residual set of functions whose constrained maximizer is unique, ergodic, full-support, and satisfies h_μ(T)+∫φ dμ=c; it also excludes physical-like measures in the C^1 transitive Anosov case. The statement and the proof skeleton are explicit in the PDF, with the key Baire-category engine (Theorem 3.2) and residuality of Δ_c∩Me and of full-support measures inside Λ_c (Lemmas 3.1 and 4.1), which together enforce (2)–(4) and the Anosov refinement (Lemma 4.2) . The candidate solution follows a similar Baire approach but commits a critical convexity/boundary error: it identifies ∂K_c with {μ: h_μ+∫φ=c} and argues that any maximizer with h_μ+∫φ>c lies in the relative interior of K_c, hence is excluded by a tie-breaking condition. This is false in general because points with h_μ+∫φ>c can still lie on the boundary of M(X,T), so the “interior-maximum” exclusion does not preclude such maxima; thus property (4) is not secured by the candidate’s argument. The paper’s proof avoids this gap by directly forcing maximizers into Δ_c via residual subsets inside Λ_c .