Nonautonomous Dynamical Systems II: Variational Principles

Zhuo Chen, Jun Jie Miao

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the Bowen and packing variational principles for nonautonomous systems via Billingsley-type theorems, a weighted-measure/Frostman construction, and an ε→0 passage under equicontinuity. See Theorem 2.6 and Theorem 2.11 for the main equalities, the Billingsley inputs (Theorem 2.4 and Theorem 2.9), and the Frostman-type Lemma 6.2 together with the weighted pressure framework (Proposition 3.6) and ε-stability (Proposition 3.3; Proposition 3.9) . The candidate solution follows the same architecture (fixed-scale control on Bowen balls, Billingsley comparisons, Frostman/pressure-distribution construction, limit in ε), but it overstates a fixed-ε variational equality (e.g., PB(T,f,K,ε)=supμ∫P−μ(⋯,ε)), which the paper does not claim or prove. Aside from this overreach, the candidate’s steps correctly reproduce the paper’s conclusions.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

This work establishes clean variational principles for Bowen and packing pressures in a general nonautonomous setting using Billingsley-type theorems, a robust weighted-measure/Frostman framework, and careful ε→0 limits under equicontinuity. The results are timely and extend the autonomous theory convincingly. The presentation is clear; a few refinements regarding fixed-scale versus limiting statements would further aid readers.