ERGODIC OPTIMIZATION FOR CONTINUOUS FUNCTIONS ON NON-MARKOV SHIFTS

The uploaded paper (Shinoda–Takahasi–Yamamoto, 2024) proves Theorem A as stated: (a) R_H is a dense G_delta via open dense sets O_n of ergodic measures and the transfer theorem [32, Thm 1.1], and (b) for any f not in R_H, a small perturbation yields uncountably many fully supported ergodic maximizing measures with entropy > H. See their Section 3.2 for the full argument, including the use of Corollary 2.6 for density in M^e and [32, Thm 1.1] for openness/denseness transfer (); the path of high-entropy fully supported ergodic measures comes from Proposition 2.4 and Corollaries 2.5–2.6 (, ), and the convex-analytic ‘tangent functional’ (Theorem 3.1) plus the barycenter support lemma (Jacobs’ theorem; Lemma 3.3) give uncountably many maximizers (, ). The candidate solution correctly executes part (a), but its part (b) contains a crucial gap: it invokes a Urysohn-type construction to produce an observable ψ whose integrals vanish along a continuum {μ_t} and are negative off a neighborhood, thereby forcing all μ_t to maximize a perturbed potential. This step is unjustified; Urysohn’s lemma yields functions on the measure space, not observables on Σ whose integrals realize an arbitrary continuous function on M(Σ,σ). The paper’s proof avoids this by using the tangent measure and the barycenter support property to extract uncountably many maximizers without requiring all μ_t to maximize ().