ERGODIC OPTIMIZATION FOR BETA-TRANSFORMATIONS

The uploaded paper proves exactly the four statements the model outlines: (A) for beta-numbers β, Lock_α(β) is open and dense (Theorem 1.1), (B) for non-emergent β, Lock_α(β) is open and dense (Theorem 1.2), (C) for emergent β, C0,α(I) equals Crit_α(β) ∪ cl(Lock_α(β)) (Theorem 1.4), and (D) the set of β for which typical periodic optimization holds is residual and full measure (Corollary 1.3). The paper’s strategy aligns with the model’s: replace Tβ by the upper beta-transformation Uβ so maximizing measures exist and “harmonize” with the beta-shift, prove a Mañé-type lemma with revealed versions, and invoke Contreras’ TPO theorem on open distance-expanding restrictions to Cantor subsystems. See the paper’s statements and proof sketches: the Uβ compactness/closure harmonization and equivalence with the shift model, Theorems 1.1, 1.2, 1.4, and Corollary 1.3; the role of the set Rα(β) built via simple β and open expanding restrictions; and the emergent set being meagre and null via the specified β-shifts linkage. The model’s outline is faithful to these arguments, with only small phrasing imprecisions (e.g., the open expanding restriction is on Tβ|Hγβ for simple γ, not Uβ).