The Weak and Strong Rigidities of Entropy Spectra

The paper proves that if two Gibbs systems on (possibly different) one-sided topological Markov shifts are measure-theoretically isomorphic, then their entropy spectra coincide. It does so by: (i) expressing the spectrum via the Legendre transform of β(q)=P(qf)−qP(f) for Hölder Gibbs measures (Theorem 2.8), and (ii) showing that an isomorphism forces equality of these β-functions using Jacobians built from the Ruelle–Perron–Frobenius data and a preimage-sum formula for pressure; this yields E(μ_f)=E(μ_g) (Theorem 3.2). The key steps—Definition of the spectrum (Def. 1.1), β-representation (Thm. 2.8), the isomorphism set-up (Def. 3.1), the Jacobian identity (Lemma 3.3), and the preimage-sum pressure formula (Lemma 3.4)—are all present and correctly executed in the paper . By contrast, the candidate solution’s Step 5 asserts that the L^q partition-sum growth rate τ_μ(q) is an isomorphism invariant for Bernoulli systems and is independent of the choice of finite generator. This is false in general: even among Bernoulli shifts with equal Kolmogorov–Sinai entropy (hence isomorphic), the τ-functions computed from their standard coordinate generators are τ(q)=log∑_i p_i^q, which differ for different symbol distributions. Thus generator-independence and isomorphism-invariance of τ_μ(q) do not hold as claimed. The remainder of the candidate solution rests on this step, so it fails.