VARIATIONAL PRINCIPLES ON SUBSETS OF NON-AUTONOMOUS DYNAMICAL SYSTEMS: TOPOLOGICAL PRESSURE AND TOPOLOGICAL ENTROPY

The uploaded paper proves exactly the variational principle the candidate addresses and does so via the same three pillars: (i) equivalence of Pesin–Pitskel and weighted pressures using a Vitali-type selection (Proposition 4.2), (ii) measure-to-pressure upper/lower bounds (Theorem A), and (iii) a Frostman-style construction yielding a measure with ball-decay estimates to obtain the reverse inequality and hence Theorem C. Theorem C states PB_f_{1,∞}(Z,ψ) = PW_f_{1,∞}(Z,ψ) = sup{P_μ,f_{1,∞}(X,ψ): μ(Z)=1} for nonempty compact Z, exactly matching the candidate’s claim. The paper’s statements and proofs explicitly cover these steps, including the Vitali lemma (with 5× expansion) and the needed ball estimates, e.g., μ(B_n(x,ε)) ≤ c^{-1} e^{−αn+S_{1,n}ψ(x,ε)} leading to P_μ ≥ α. See the main statements and proofs in the PDF: Theorem C and its proof outline, Proposition 4.2, and Theorem A and its proof (, , , , , ). The only minor adjustment is that the paper assumes Z is nonempty and compact, while the candidate wrote “for all compact Z”; otherwise, the arguments coincide.