Topological pressures of a factor map for nonautonomous iterated function systems

The paper proves the NAIFS factor-inequality for topological pressure by a direct, fully topological argument using the fiber topological sup-entropy H(Φ;·) and equicontinuity-derived d*_n metrics. The main theorem states P(Φ, φ∘π) ≤ P(Ψ, φ) + sup_y H(Φ; π^{-1}(y)) and is proved in detail, including the definition of H(Φ;Y) and the final bounding of separated sums, culminating in the inequality (letting auxiliary parameters go to zero) . By contrast, the model’s solution hinges on (i) a variational principle for NAIFS pressure in exactly the form P = sup(h_μ + ∫φ dμ) over a suitable class of Φ-invariant measures on X and (ii) a nonautonomous Rokhlin-type entropy addition formula for factor maps. Neither ingredient appears in, nor is needed by, the paper; more importantly, the model does not justify these claims for the NAIFS framework defined here (pressure via word-averaged separated/spanning sums on X), where the appropriate “invariance” and relative entropy notions are delicate. Without rigorous, cited statements specialized to this NAIFS pressure, the model’s proof is not validated.