DISTRIBUTIONS OF PERIODIC POINTS FOR THE DYCK SHIFT AND THE HETEROCHAOS BAKER MAPS

The paper proves Theorems 1.1–1.2 by coding the heterochaos baker map to the Dyck shift π(Λ)=Σ_D (not an SFT) and then transporting periodic-point statistics through Krieger’s conjugacies with two full shifts Σ_α, Σ_β. It uses Kifer’s large-deviation machinery to show that periodic points with H_n>0 (resp. H_n<0) equidistribute to ν_α (resp. ν_β), and finally pulls this back to μ_α, μ_β for the baker map; it also shows #Per_{α,n}=#Per_{β,n} via an exact symmetry/counting identity (2.5) and (2.6) (Theorem 1.1 and its Dyck-shift counterpart Theorem 1.2) . By contrast, the model’s solution incorrectly asserts a one-step mixing SFT coding with specific local adjacency rules, whereas the correct coding is the Dyck shift (not SFT) defined by balanced-bracket constraints . The model then invokes equidistribution and constrained variational principles for mixing SFTs; these do not apply as stated to the Dyck shift, and the claimed “Markov partition” and SFT adjacency are unsupported and contradicted by the paper’s π(Λ)=Σ_D identification. The model also mischaracterizes the excluded H_n=0 class as “subexponential,” whereas the paper gives its exact exponential growth with smaller rate via Hamachi–Inoue (2.6)–(2.9) . In short, the paper’s result and proof are sound, while the model rests on a false SFT premise and misapplied tools.