Thermodynamic Formalism on the Skorokhod space: the continuous time Ruelle operator, entropy, pressure, entropy production and expansiveness

The paper’s Theorem 1 asserts that for any strictly positive continuous kernel P on [0,1]^2 with ∫ P(x,y) dx = 1 and any Hölder potential V, there exist strictly positive left/right eigenfunctions ℓ,r that are themselves Hölder. However, the proof’s compactness step on H^α wrongly uses only uniform continuity of P and V; it does not show that (L+V+zI) maps H^α into H^α (no Hölder control of the image unless P is Hölder). Indeed, with P(x,y)=p(x) continuous but non-Hölder, one gets ℓ(x) ∝ p(x)/(1+λ−V(x)), which need not be Hölder, contradicting Theorem 1. The model’s solution delivers the correct existence/positivity of continuous eigenfunctions via a positive compact operator T_θ on C([0,1]) and shows Hölder regularity requires Hölder regularity of P. Hence the paper’s Hölder claim is false as stated; the model is correct and provides a minimal fix.