Entropy spectra of constrained historic behaviour: symbolic dynamics and a hotel revenue regime model

Let (ΣA,σ) be a topologically mixing subshift of finite type and let φ,ψ∈C(ΣA). For a value α and a compact interval J, the conditional historic set is defined by E(φ,ψ;α,J)={x∈ΣA:limn→∞Anφ(x)=α,Acc⁡{Anψ(x):n≥1}=J}. For Mφ(α)≠∅ and a nonempty compact interval J⊂Iα(φ,ψ), we prove htopB(E)=infβ∈JH(α,β),htopP(E)=supβ∈JH(α,β),htopUC(E)=htop(σ), where E=E(φ,ψ;α,J) and H(α,β)=infq,r∈R{P(qφ+rψ)−qα−rβ}. A four-state hotel occupancy–price model with prescribed occupancy scores and representative prices derived from Shanghai hotel statistics (2020–2026) illustrates the coexistence of a convergent average occupancy score and a nonconvergent average revenue score.