From Thermodynamic and Spectral Phase Transitions to Multifractal Analysis

The paper proves a unique thermodynamic/spectral phase transition at t0 for the skew-product class Dr and gives a C1, concave multifractal entropy spectrum; the candidate solution recovers the same results using a sharper Lasota–Yorke/Hennion route and the Legendre-transform description. The only substantive divergence is the model’s additional claim of a sharp essential spectral radius formula r_ess(e^{-P(t)}L_t) = exp(P(t+1)−P(t)), which is not established in the paper (which instead uses a Campbell–Latushkin-type bound) and would need extra justification for “sharpness.” Aside from that overclaim, both arguments align on all main statements.