The Dirac operator for the Ruelle-Koopman pair on Lp-spaces: an interplay between Connes distance and symbolic dynamics

The paper proves W_{d∞}(µ,ν) ≤ d_p(µ,ν) ≤ λ√2·W_{d∞}(µ,ν), with λ = max{p,p′}, via an explicit computation of ∥[D_p,π(M_f)]∥ = ∥M_{f∘σ−f}K∥_{max{p,p′}} and a telescoping bound along the preimage-tree metric d∞ (Theorem 32). This hinges on the identification of the commutator norm with a tail-averaged quantity, effectively ∥(L|f∘σ−f|^{λ})^{1/λ}∥_∞, not with the sup norm ∥f∘σ−f∥_∞ for general p (see the commutator block formula (8)–(9) and Remark 13). Only for p=1 or p=∞ does one recover equality d_p = W_{d∞} (Proposition 15 and Theorem 32). The model’s key step incorrectly asserts ∥[K,M_f]∥ = ∥f∘σ−f∥_∞ for all p, collapses ∥[D_p,π(M_f)]∥ to the d∞-Lipschitz seminorm, and concludes d_p = W_{d∞} for every p, contradicting the paper’s bounds and the stated equality being restricted to p∈{1,∞} (and the paper even notes numerical evidence that the inequalities are strict away from these endpoints).